Lesson 16: Dynamical Systems and Applications of Mechanics

Introduction

Lesson 15 ended with a vector field on phase space. Hamilton’s equations define on , the flow of that field carries each point of phase space to a later point, and Liouville’s theorem says the flow rearranges those points without changing the volume they occupy. A dynamical system is the study of that flow for its own sake, where the points go, which sets trap them, which points stay fixed, which curves close, and how the picture changes when a parameter is varied. The Hamiltonian case remains the model, but the language now covers any smooth vector field, Hamiltonian or not.

The first two subsections keep the contact with mechanics and with thermodynamics. Optics as a Mechanics Problem rewrites Fermat’s principle as a Hamiltonian system on an optical phase space; rays are the projections of Hamiltonian orbits, and the symplectic structure already met in Lesson 15 reappears as the conservation of étendue. Liouville’s Theorem and the Second Law of Thermodynamics then asks how a volume-preserving flow on phase space can be compatible with the increase of entropy. The resolution is that the fine-grained density is constant while the coarse-grained density, the only one an observer can measure, increases.

The remaining subsections develop the geometry of flows. A flow is the family of maps generated by a vector field. An autonomous system is one whose vector field does not depend explicitly on time. Vector Fields on Phase Space simply names the arena with the same of Lesson 15, or any manifold that has been chosen as the space of states. Fixed Points are the zeros of the field. Stability asks whether nearby points stay nearby. Linearization replaces the field near a fixed point by its derivative, and the eigenvalues of that derivative decide the local picture. Lyapunov Functions and Lyapunov Stability give a criterion that does not require the eigenvalues: a function that decreases along the flow forces asymptotic stability. Periodic Orbits are closed integral curves. Limit Cycles are periodic orbits that attract a neighborhood. The Poincaré–Bendixson theorem limits what a flow on the plane can do: if a trajectory stays in a compact region that contains no fixed points, its ω-limit set is a periodic orbit. Bifurcations are the changes in that portrait that occur when a parameter crosses a critical value. Conservative versus Dissipative Systems returns to the distinction already drawn between Hamiltonian flow (volume-preserving, no attractors) and gradient-like or frictional flow (volume-contracting, attractors possible). Sensitivity to Initial Conditions, Chaos, and Lyapunov Exponents describe the regime in which nearby trajectories separate exponentially, so that long-term prediction from imperfect data becomes impossible even though the equations are deterministic. Connections to Physics closes the lesson by pointing to the places these ideas reappear: stability of equilibria in mechanics, the approach to equilibrium in statistical mechanics, pattern formation, and the onset of turbulence.

The through-line is that Lesson 15 supplied the vector field and the volume it preserves. Lesson 16 asks what the integral curves of a vector field can do, and what remains true when the field is no longer Hamiltonian.

Optics as a Mechanics Problem

Fermat’s principle asserts that a light ray joining two points of an isotropic medium makes stationary the optical length

(16.1)

where n is the refractive index and ds is Euclidean arc length. The principle is already a variational statement of the same type as Hamilton’s principle. One may therefore introduce an optical configuration space, an optical Lagrangian, and, after a Legendre transform, an optical Hamiltonian whose integral curves project to the rays.

Figure 16.1: Two fixed points in a medium, a family of trial curves, and the optical length ∫ds n marked on each. The stationary curve is highlighted. Use it to open the subsection.

Take the configuration space to be ordinary Euclidean space, or a domain in it, with Cartesian coordinates . Parametrize a trial curve by an arbitrary parameter σ, and write the optical Lagrangian

(16.2)

The associated action is precisely the optical length. The Lagrangian is homogeneous of degree one in the velocities and is invariant under reparametrization, exactly as the relativistic free-particle Lagrangian of Lesson 15 was invariant under reparametrization of proper time. The fiber derivative

(16.3)

therefore satisfies the eikonal constraint

(16.4)

The image of the Legendre map is not all of ; it is the constraint surface on which the local wave number equals the refractive index. On that surface the optical Hamiltonian may be taken as

(16.5)

or any function that vanishes precisely on the surface. Hamilton’s equations generated by H, restricted to H = 0, reproduce the ray equation

(16.6)

In a homogeneous medium ∇n=0, the momentum p is constant, and the rays are straight lines. In a gradient-index medium the rays curve toward higher n.

Figure 16.2: A bundle of parallel rays in constant n, with the associated constant momentum vectors p drawn beside them. Phase-space volume is a rectangle in the (q,p) plane that simply translates.

Because the underlying variational principle is reparametrization-invariant, the value of the parameter along a ray is arbitrary. One may choose arc length, optical length, or any other convenient coordinate. After that choice the reduced equations on the constraint surface are an ordinary Hamiltonian system. The symplectic form inherited from remains non-degenerate on the surface once the constraint is taken into account, and the Hamiltonian flow preserves it. Consequently the optical phase-space volume—the product of a bundle’s cross-sectional area and its solid angle, weighted by —is constant along the bundle. That conserved quantity is the étendue—the optical phase space volume of a bundle of rays. Its conservation is Liouville’s theorem written in optical variables, and it is the reason no passive optical instrument can increase the brightness of an extended source.

Figure 16.3: A sharp boundary, an incident ray, reflected and refracted rays, and the tangential component of p continuous across the interface. The normal component jumps so that |p|=n on each side.

The construction is local. At a sharp interface n jumps, the variational principle must be supplemented by a refraction condition, and the momentum component tangent to the interface is continuous while the normal component adjusts so that |p| = n on each side. Snell’s law is that matching condition. In the Hamiltonian picture it is the statement that the optical phase space is still well-defined, but the Hamiltonian vector field acquires a discontinuity across the interface; orbits cross by a canonical transformation that preserves the tangential momentum.

The same formalism accommodates anisotropic media once the refractive index is allowed to depend on direction. The constraint surface |p| = n is replaced by a more general eikonal surface H(q,p) = 0, and the rays remain the projections of the Hamiltonian orbits lying on that surface. In every case the optical problem has been rewritten as a constrained Hamiltonian system on the cotangent bundle of the ambient space. The rays are the spatial projections of those orbits, the symplectic form supplies the conserved étendue, and the passage from Fermat’s principle to ray optics is identical in structure to the passage from the relativistic action to the free-particle Hamiltonian in Lesson 15.

Figure 16.4: In a two-dimensional (q,p) sketch, the circle (or curve) |p|=n(q) at a fixed q, with the Hamiltonian vector field tangent to it. This shows why the rays live on the eikonal surface.

The optical example therefore serves two purposes. It shows that the Hamiltonian apparatus is not confined to massive particles, and it supplies a concrete illustration of a reparametrization-invariant system whose constraint surface carries a preserved symplectic volume. Both features reappear when the configuration space itself becomes a space of fields.

Liouville’s Theorem and the Second Law of Thermodynamics

Liouville’s theorem states that the fine-grained density of representative points in phase space is constant along the Hamiltonian flow. The second law of thermodynamics states that the entropy of an isolated system does not decrease. The two statements appear to conflict only if one fails to distinguish the density that the flow preserves from the density that an observer can actually measure.

Let ρ(q,p,t) be the fine-grained density, where the number of ensemble points per unit phase-space volume at the point (q,p) at time t. Because , the continuity equation reduces to

(16.7)

The numerical value of ρ attached to each moving point is invariant. If the initial density is uniform on a region U, it remains uniform on the image , and the volume of that image equals the volume of U.

An observer, however, cannot resolve the fine-grained density once the flow has stretched and folded the support of ρ on scales smaller than the resolution of any practical measurement. One therefore introduces a coarse-grained density , obtained by averaging ρ over cells of a fixed partition of phase space (or by convolution with a smoothing kernel of fixed width). The entropy computed from the coarse-grained density,

(16.8)

is non-decreasing. Equality holds only when the fine-grained density is already constant on the scale of the cells. As soon as the Hamiltonian flow produces filaments thinner than the cell size, the average spreads into a larger effective volume, increases, and S rises.

Figure 16.5: Stretching and folding of a phase-space blob.

The underlying points are not created or destroyed, and the symplectic volume they occupy is exactly conserved. What increases is the volume of the region that must be counted as “occupied” once one refuses to resolve structure below a fixed scale. The second law is a statement about that refusal. It is not a violation of Liouville’s theorem; it is a consequence of applying Liouville’s theorem to a density that has been irreversibly averaged.

Figure 16.6: Fine-grained versus coarse-grained density.

The same mechanism operates in the approach to equilibrium. An initial density concentrated on a small region of the energy surface is stretched by the flow into ever finer filaments that eventually cover the surface densely. The coarse-grained density approaches the uniform density on that surface, and the coarse-grained entropy approaches its maximum. The fine-grained density remains as nonuniform as it began; only its oscillations have become unobservable. Recurrences, in which the filaments reassemble and the coarse-grained entropy temporarily drops, are not forbidden by the dynamics, but their waiting times are exponentially long in the number of degrees of freedom.

Figure 16.7: Entropy of the coarse-grained density.

Thus the conservation of phase-space points and phase-space volume is compatible with the second law once the distinction between fine-grained and coarse-grained descriptions is maintained. The Hamiltonian flow supplies the stretching and folding; the observer’s finite resolution supplies the irreversibility. The optical étendue of the preceding subsection is the same conservation law read in a different set of variables; the thermodynamic entropy is the same law read after an average.

Figure 16.8: An energy oval (or circle) initially containing a small patch of points; later panels show the points spread evenly around the oval after repeated stretching and folding.

Flows

A vector field X on a manifold M assigns to each point x∈M a tangent vector . An integral curve of X is a curve γ(t) in M whose velocity at every instant equals the value of the field at the point occupied by the curve

(16.9)

The arrows of the field are everywhere tangent to the integral curves.

Figure 16.9: Vector field and integral curves.

The flow of X, when it exists, is the family of maps that sends each initial point to the point reached by its integral curve after elapsed time t

(16.10)

Differentiating with respect to t recovers the defining equation

(16.11)

A region of initial conditions is carried by the flow to a (generally deformed) region of later states.

Figure 16.10: Flow map acting on a region.

When the vector field is complete, every integral curve exists for all real t, and the maps form a one-parameter group of diffeomorphisms of M

(16.12)

Completeness is not automatic. A smooth vector field on a noncompact manifold may drive some integral curves to infinity in finite time, exactly as a non-proper Hamiltonian may do on . In that case the flow is defined only on a t-dependent open set, and the group property holds only where both sides are defined.

Figure 16.11: Complete versus incomplete flow.

In the setting of Lesson 15 the manifold is phase space, , and the vector field is the Hamiltonian vector field . The integral curves are the solutions of Hamilton’s equations, and the flow is the time-evolution map already discussed under the heading of global solutions. The same definition applies verbatim to any smooth vector field on any manifold that has been chosen as the space of states. Autonomy of the field—its lack of explicit time dependence—is what makes a group rather than a more general two-parameter family. A time-dependent field still generates a flow, but the evolution from time to time depends on the starting instant, and the simple composition law is lost.

The orbit of a point x is the image of its integral curve. Fixed points are the orbits that consist of a single point; periodic orbits are those diffeomorphic to a circle. The flow partitions M into orbits. Two points lie on the same orbit if and only if one is the image of the other under for some t.

Because is a diffeomorphism (on its domain), it carries orbits to orbits and preserves the qualitative arrangement of the phase portrait where fixed points remain fixed points, periodic orbits remain periodic orbits, and the topological type of the decomposition into orbits is unchanged.

A passive change of coordinates on M rewrites the components of X by the chain rule but leaves the integral curves and the maps unaltered as geometric objects. An active diffeomorphism of M pushes the vector field forward and conjugates the flow. The geometric content of the flow—the orbits and the time parametrization along them—is therefore independent of the labels used to describe M.

Thus a flow is the integrated form of a vector field. Hamilton’s equations supply the vector field on phase space; the flow is the resulting motion of every point. The remainder of the lesson examines the possible orbit structures of such a flow, first for a general vector field and then for the special case in which the field is Hamiltonian.

Autonomous Systems

A dynamical system is autonomous when its vector field does not depend explicitly on time. The evolution equation takes the form

(16.13)

where the same vector is attached to a given point of the state space at every instant. The integral curves are therefore invariant under a shift of the time origin where if γ(t) is a solution, so is . Equivalently, the flow satisfies the group law

(16.14)

whenever both sides are defined. Time translation is a symmetry of the equations, and the entire future (and past) of every orbit is determined by its position alone.

Figure 16.13: An autonomous vector field is frozen in time; a non-autonomous field changes its arrows as the external clock advances.

A non-autonomous system lacks this symmetry. The vector attached to a point depends on the absolute time, so the evolution from to cannot be composed from the evolution over an equal interval that begins at a different instant. Periodic driving, explicit decay rates, or any external schedule that refers to a laboratory clock produces a non-autonomous equation.

Any non-autonomous system on a manifold M can be rewritten as an autonomous system on the extended space M×R. One adjoins the time coordinate s as an extra state variable and sets

(16.15)

The new vector field on M×R has no explicit time dependence, and its flow projects to the original non-autonomous evolution. The price is the loss of any compactness that M may have possessed where the extended energy surfaces, if they exist, are no longer compact, and the elementary recurrence theorems that rely on compactness become unavailable. For this reason the geometric results of the present lesson—stability, the Poincaré–Bendixson theorem, the existence of Lyapunov functions—are stated for autonomous systems. When a non-autonomous problem must be treated, it is converted to an autonomous one on the extended space only after the consequences of that extension have been noted.

Figure 16.14: Adjoining the time coordinate converts a non-autonomous equation on M into an autonomous equation on M×R. The extra dimension removes compactness.

In the Hamiltonian setting of Lesson 15 the same distinction appears. If the Lagrangian (and therefore the Hamiltonian) has no explicit time dependence, the system is autonomous, energy is conserved, and the flow on each energy surface is a one-parameter group. An explicit time dependence in H destroys both the conservation of energy and the group property; the extended phase space must again be introduced if one wishes to restore autonomy. Most of the orbit structures examined in the remainder of the lesson—fixed points, periodic orbits, limit cycles, and their stability—are therefore discussed under the standing assumption that the vector field is autonomous.

Vector Fields on Phase Space

The arena of the dynamics remains the phase space introduced in Lesson 15. A point of is a complete instantaneous state (q,p) with a configuration together with its conjugate momentum. A vector field on this space assigns to each such state an infinitesimal displacement

(16.16)

In adapted coordinates the components are simply 2n functions of (q,p). The integral curves of X are the possible motions; the flow , when it exists, advances every state by elapsed time t.

Figure 16.15: A general vector field on phase space. At each point the arrow indicates the instantaneous velocity of the state; the integral curves are the trajectories that follow those arrows.

Not every vector field arises from a Hamiltonian. The Hamiltonian vector field of Lesson 15 is the special case in which the components are recovered from a single function by

(16.17)

Equivalently, is the unique vector field satisfying

(16.18)

where is the canonical symplectic form. The defining relation forces with respect to the symplectic volume, so the flow of preserves phase-space volume (Liouville’s theorem). A generic vector field has non-vanishing divergence; its flow expands or contracts volumes and cannot be the Hamiltonian flow of any function H.

Figure 16.16:  Left: the Hamiltonian vector field of a harmonic oscillator, everywhere tangent to the level curves of H and divergence-free. Right: a generic vector field whose arrows converge, so that areas shrink under the flow.

The distinction is coordinate-independent. A passive change of adapted coordinates on mixes the components of any vector field by the chain rule, but it preserves the symplectic form and therefore preserves the class of Hamiltonian vector fields. A vector field that fails to preserve Ω in one adapted chart fails to preserve it in every adapted chart. Thus the geometric object “Hamiltonian vector field” is well-defined on the cotangent bundle, while a general vector field on phase space need not arise from any Hamiltonian.

When the state space is not a cotangent bundle—for example, when one studies a reduced system on an energy surface, or an abstract dynamical system on an arbitrary manifold—the same language applies. One still has a vector field, its integral curves, and its flow. The special features that follow from the existence of a Hamiltonian (conservation of energy and the preservation of a symplectic form) are simply absent. The remainder of the lesson develops the orbit structure of a general vector field and then recovers the Hamiltonian case as the subclass in which the field is divergence-free with respect to a preserved symplectic volume.

Fixed Points

A fixed point of a vector field X is a point at which the field vanishes

(16.19)

The constant curve is then an integral curve, and the flow satisfies for every t in the interval of existence. In the language of the preceding subsection, a fixed point is an orbit that consists of a single point. Equilibria, steady states, and critical points are synonymous terms.

Figure 16.17:  Fixed points appear as points at which every arrow of the vector field has zero length. Nearby integral curves may approach the point, leave it, or circulate about it.

In adapted coordinates the condition is a system of 2 n scalar equations for the 2 n unknowns (q,p). For a generic vector field the solutions are isolated. A continuous family of fixed points can occur, but it is unstable under an arbitrarily small perturbation of the field where the zeros move and, typically, annihilate in pairs or disappear. The same count holds on an arbitrary manifold where the zeros of a section of the tangent bundle are isolated for a generic choice of section.

When the vector field is Hamiltonian the fixed points acquire an additional interpretation. They are precisely the critical points of the Hamiltonian,

(16.20)

because if and only if (16.20) is satisfied. On a compact energy surface the existence of at least one critical point is guaranteed by elementary topology, yet the Hamiltonian structure imposes relations among them. The Hessian of H at a non-degenerate critical point has eigenvalues that appear in ±λ pairs; a local minimum of H produces a center (to linear order), while a saddle produces hyperbolic directions. These constraints are absent for a general vector field, whose linearization at a fixed point may have any set of eigenvalues.

Figure 16.18:  For a Hamiltonian vector field the fixed points are the critical points of H. A non-degenerate minimum yields a center; a saddle yields a hyperbolic fixed point.

A fixed point may be created or destroyed when a parameter in the vector field is varied. The simplest such event is a saddle-node bifurcation, in which a pair of fixed points (one attracting, one repelling along a line) coalesce and annihilate as the parameter crosses a critical value. In the Hamiltonian case the corresponding event is constrained by the preservation of the symplectic form where fixed points appear or disappear in a manner compatible with the pairing of eigenvalues, and the value of H at the critical point typically changes with the parameter. These parameter-dependent changes are examined under the heading of bifurcations; the present subsection only identifies the fixed points themselves as the zeros of the vector field and, in the Hamiltonian setting, as the critical points of H.

The local behavior near an isolated fixed point is completely determined, to first order, by the derivative . That linearization, and the stability properties it controls, are the subjects of the next subsections.

Stability

A fixed point is Lyapunov stable if points that begin sufficiently near remain near for all future time. Precisely, for every neighborhood U of there exists a neighborhood V⊂U such that

(16.21)

(on the interval of existence of the flow). The orbit may wander inside U, but it cannot escape. The fixed point is unstable if this condition fails: some points arbitrarily close to eventually leave every prescribed neighborhood.

Figure 16.19:  Lyapunov stability. Every neighborhood U contains a smaller neighborhood V such that orbits starting in V remain in U for all future time.

Asymptotic stability is the stronger requirement that nearby orbits not only stay near but converge to it

(16.22)

Every asymptotically stable fixed point is Lyapunov stable, but the converse is false. A center—such as the origin of the harmonic oscillator—is Lyapunov stable yet not asymptotically stable as orbits remain on their ellipses and do not approach the origin.

Figure 16.20:  Left: a center is Lyapunov stable but not asymptotically stable. Right: a spiral sink is asymptotically stable; nearby orbits converge to the fixed point.

Hamiltonian systems cannot possess asymptotically stable fixed points. Liouville’s theorem asserts that the flow of preserves phase-space volume. Asymptotic stability, however, requires that a whole neighborhood of be mapped into smaller and smaller neighborhoods, which is possible only if volumes contract. The same obstruction applies to any volume-preserving flow. Consequently every fixed point of a Hamiltonian vector field is either Lyapunov stable (typically a center arising from a local minimum of H or unstable (typically a saddle). Asymptotic stability appears only when the vector field is dissipative.

The definitions extend immediately to more general invariant sets. A periodic orbit, or any compact invariant set, is Lyapunov stable when nearby orbits remain nearby, and asymptotically stable when they converge to the set. The same volume argument shows that a Hamiltonian flow on a compact energy surface cannot possess an asymptotically stable periodic orbit of lower dimension than the surface itself. Attracting limit cycles are therefore characteristic of dissipative systems and are excluded from the conservative dynamics of Lesson 15.

Stability is a qualitative property of the orbit structure. It is independent of the parametrization of time and of the choice of coordinates on the state space. The practical tests that decide stability—linearization at the fixed point, and the construction of a Lyapunov function—are developed in the subsections that follow.

Linearization

Near an isolated fixed point the vector field may be replaced, to first order, by its derivative. Let satisfy , and set . Taylor expansion yields

(16.23)

where is the linear map given in coordinates by the Jacobian matrix of the components of X. The linear system

(16.24)

is the linearization at . Its solutions are completely determined by the eigenvalues and eigenvectors of A.

Figure 16.21: In a sufficiently small neighborhood of a hyperbolic fixed point the nonlinear trajectories are topologically equivalent to those of the linearization.

The eigenvalues of A govern the local phase portrait of the linear system. An eigenvalue with negative real part produces exponential decay along its eigenspace; a positive real part produces exponential growth; a purely imaginary pair produces rotation (about a linear center). When every eigenvalue has nonzero real part the fixed point is called hyperbolic. The Hartman–Grobman theorem asserts that a hyperbolic fixed point of a smooth vector field is topologically equivalent to its linearization where there exists a neighborhood of and a homeomorphism of that neighborhood, sending orbits of the nonlinear flow to orbits of the linear flow, that preserves the sense of time. The homeomorphism need not be differentiable, so distances and eigenvalues are not preserved, but the qualitative arrangement—which orbits approach the point, which leave it, and which do neither—is the same.

Figure 16.22: The linear classification in the plane: sink (all eigenvalues with negative real part), source (positive real part), saddle (eigenvalues of opposite sign), and center (purely imaginary eigenvalues). Only the first three are hyperbolic.

When one or more eigenvalues lie on the imaginary axis the linearization is inconclusive. A nonlinear term may turn a linear center into a weak spiral, or may leave it as a center; the Hartman–Grobman theorem does not apply, and a finer test (normal-form calculation or a Lyapunov function) is required. In the Hamiltonian case the spectrum is constrained where eigenvalues appear in ±λ and quartets, and at least two eigenvalues vanish when the fixed point lies on a regular energy surface (the directions tangent to the flow and to the surface). A non-degenerate minimum of H produces a linear center; the nonlinear terms preserve the center because of the conservation of H.

The linearization therefore supplies a complete local classification precisely when the fixed point is hyperbolic. In all other cases it remains the first step of the analysis, to be supplemented by the global or higher-order methods of the following subsections. The same construction applies verbatim to a periodic orbit once a Poincaré return map has been introduced: the derivative of the return map plays the role of A, and its eigenvalues (Floquet multipliers) decide linear stability.

Lyapunov Functions and Lyapunov Stability

Linearization decides stability only when every eigenvalue has nonzero real part. When eigenvalues lie on the imaginary axis, or when one wishes to avoid computing them, a direct test is available. A Lyapunov function for a fixed point is a smooth real-valued function V defined on a neighborhood of that satisfies three conditions:

- and V(x) > 0 for ,

the orbital derivative X(V) = dV(X) obeys X(V)≤0 throughout the neighborhood,

(for asymptotic stability) X(V)≤0 for .

If such a function exists, is Lyapunov stable. If the third condition also holds, is asymptotically stable.

Figure 16.23: Level sets of a Lyapunov function V. The vector field points inward or is tangent to the levels, so X(V)≤0. Orbits cannot cross outward and therefore remain near the fixed point.

The geometric content is immediate. The sublevel sets {V ≤ c} form a nested family of neighborhoods of . Because X(V)≤0, the flow cannot carry a point out of any sublevel set, where once an orbit enters {V ≤ c} it remains there for all future time. Choosing c small enough that the sublevel set lies inside a prescribed neighborhood U produces the neighborhood V required by the definition of Lyapunov stability. When X(V)≤0 off , the orbit must descend through successively lower levels and therefore converge to the unique minimum at .

In the Hamiltonian setting the Hamiltonian itself often serves as a Lyapunov function. At a non-degenerate local minimum of H one has , nearby, and . Thus satisfies the first two conditions, and the fixed point is Lyapunov stable. The vanishing of the orbital derivative is forced by conservation of energy; the same conservation prevents the strict decrease needed for asymptotic stability, in agreement with Liouville’s theorem. For a general (possibly dissipative) vector field one must construct V by hand. Common candidates are quadratic forms suggested by the linearization, or total energy modified by a small dissipative term.

Figure 16.24: Left: for a Hamiltonian minimum, H itself is a Lyapunov function with ; orbits stay on levels. Right: a strict Lyapunov function forces orbits to cross levels inward and converge to the fixed point..

The test is sufficient but not necessary. There exist stable fixed points that admit no smooth Lyapunov function of the classical type, yet the criterion remains the most practical tool when the linearization is inconclusive or when one seeks an explicit basin of stability. Once a Lyapunov function has been exhibited, the sublevel sets supply concrete neighborhoods on which the stability estimates hold, without any further reference to eigenvalues. The same idea extends to periodic orbits and to more general invariant sets: a function that decreases along the flow outside the set forces asymptotic approach to the set. Those extensions, and the relation to linearization, occupy the subsections that follow.

Periodic Orbits

A periodic orbit is a closed integral curve of the vector field. Equivalently, it is an orbit of the flow that is diffeomorphic to a circle so that there exists a point x and a period T>0 such that

(16.25)

and for all 0<t<T. The smallest such T is the minimal period. Every point of the orbit shares the same period, and the image of the orbit is a smoothly embedded circle that is invariant under the flow.

Figure 16.25: A periodic orbit is a closed integral curve. The flow simply rotates points around the circle with period T.

In one degree of freedom every compact regular level set of a Hamiltonian is a periodic orbit. The level set H=E is a closed curve, the Hamiltonian vector field is tangent to it and nowhere zero, and the time required to traverse the curve once is the period

(16.26)

The period generally depends on the energy. In higher dimensions periodic orbits are no longer forced by the topology of the energy surface. Their existence must be proved by other means; by a variational argument that produces a closed extremal of the action, by a fixed-point theorem applied to a Poincaré return map, or by an explicit symmetry that reduces the problem to one degree of freedom.

Figure 16.26: In one degree of freedom every compact regular energy curve is a periodic orbit. The period typically changes from one level to the next.

Stability of a periodic orbit is decided by the behavior of nearby orbits. The orbit is Lyapunov stable if every neighborhood of the circle contains a smaller neighborhood that is carried into the first by the flow for all future time. It is asymptotically stable if nearby orbits converge to the circle as e->∞. Linearization about the orbit is performed by means of a Poincaré return map, where a local hypersurface transverses to the orbit is crossed repeatedly by nearby trajectories, and the derivative of the first-return map at the intersection point has eigenvalues (called Floquet multipliers) whose position relative to the unit circle determines linear stability. One multiplier is always equal to 1, corresponding to the direction tangent to the orbit itself; the remaining multipliers govern attraction or repulsion in the transverse directions.

Hamiltonian mechanics again imposes restrictions. Because the flow preserves phase-space volume, a periodic orbit cannot be asymptotically stable, as an open set of points cannot collapse onto a lower-dimensional circle. In an integrable system the periodic orbits typically appear in continuous families parametrized by the values of the integrals. In a non-integrable perturbation most of these orbits are destroyed; those that survive are isolated and are either elliptic (surrounded by invariant tori) or hyperbolic. The detailed survival and destruction of periodic orbits under perturbation is the subject of KAM theory and of the Poincaré–Birkhoff theorem; the present subsection only identifies the orbits themselves and the elementary constraints that volume preservation places on their stability.

A periodic orbit may be created or destroyed when a parameter is varied. The simplest local mechanism is a Hopf bifurcation, where a fixed point loses stability as a complex-conjugate pair of eigenvalues crosses the imaginary axis and a small periodic orbit is born. Global mechanisms involve the collision of periodic orbits with one another or with saddle points. Those bifurcations, and the special role of isolated attracting cycles in dissipative systems, are taken up in the subsections that follow.

Limit Cycles

A limit cycle is a periodic orbit that is the ω-limit set of every point in some neighborhood of the orbit. Nearby trajectories approach the cycle as t->∞ (an attracting, or stable, limit cycle) or as t->-∞ (a repelling, or unstable, limit cycle). In the former case the cycle is asymptotically stable as an invariant set, where there exists a neighborhood U of the circle such that

(16.27)

The approach may be monotonic in a transverse coordinate or may involve oscillations; the essential point is that an open set of initial conditions is funneled onto the same closed curve.

Figure 16.27: An attracting limit cycle. Orbits from both inside and outside converge to the closed curve as t->∞.

Limit cycles are impossible for a Hamiltonian flow. The preservation of phase-space volume precludes the collapse of an open set onto a lower-dimensional subset. Consequently every periodic orbit of a Hamiltonian system is either surrounded by a continuous family of other periodic orbits (the integrable case) or is isolated yet non-attracting (elliptic or hyperbolic). Attracting limit cycles appear only when the vector field is dissipative, so that volumes contract and the long-term dynamics can concentrate on a set of measure zero.

The standard example is the van der Pol oscillator

(16.28)

or its equivalent first-order system on the plane. For μ>0 the origin is an unstable focus, and a unique stable limit cycle surrounds it. Trajectories starting near the origin spiral outward; trajectories starting far away spiral inward; both approach the same closed curve.

Figure 16.28: Phase portrait of the van der Pol oscillator. The origin is unstable; a unique attracting limit cycle captures all other orbits.

Existence of a limit cycle in the plane can often be established without an explicit solution. If one exhibits a trapping annulus—an annular region that the flow enters and never leaves, and that contains no equilibria—then the long-term behavior inside the annulus is severely restricted. In the plane the only possible ω-limit sets compatible with the absence of fixed points are periodic orbits; thus the annulus must contain at least one limit cycle. (The precise statement is the Poincaré–Bendixson theorem, taken up in a later subsection.) In higher dimensions no such elementary restriction exists; the long-term behavior inside a trapping region may be far more complicated than a single closed curve.

The contrast with Hamiltonian mechanics remains sharp. Volume preservation forbids attracting limit cycles, while dissipation permits them. An isolated closed orbit that captures an open set of initial conditions is therefore a signature of a non-conservative vector field. The conditions under which such an orbit can appear, disappear, or change its stability when a parameter is varied are examined later; the present subsection only identifies the orbits themselves and the geometric reason they cannot occur in the Hamiltonian setting of Lesson 15.

KAM Theory

An integrable Hamiltonian system with n degrees of freedom possesses n independent integrals in involution. The common level sets of the integrals are, where they are compact and regular, invariant tori of dimension n. On each such torus the flow is linear in suitable angle coordinates and that motion is quasiperiodic, with frequencies determined by the values of the integrals. In one degree of freedom the tori are simply the periodic orbits already examined; in higher dimensions they fill open regions of the energy surface.

A small Hamiltonian perturbation generally destroys the complete integrability. Most of the invariant tori disappear. Kolmogorov–Arnold–Moser (KAM) theory asserts that a positive-measure set of them nevertheless survives, provided the frequencies of the unperturbed torus satisfy a Diophantine condition

(16.29)

with γ>0 and τ>n-1. The surviving tori are only slightly deformed; the motion on each remains quasiperiodic with frequencies close to the original ω. The union of the persisting tori has positive measure in phase space, yet its complement is open and dense. Between the tori lie regions filled with chaotic orbits and with chains of elliptic and hyperbolic periodic orbits.

Figure 16.29: In a Poincaré section the surviving KAM tori appear as closed curves. Between them lie chaotic layers and chains of periodic islands. The tori occupy positive measure; the gaps are open and dense.

The surviving tori are Lagrangian submanifolds. Their persistence is established by constructing a near-identity canonical transformation that straightens the perturbed vector field on a Cantor set of frequencies. At each stage of the construction one solves a linear equation for the correction; the Diophantine condition on the frequencies guarantees that the small denominators remain controlled, and a rapidly convergent iteration produces a limiting transformation defined on a positive-measure set. Because the construction is local in a space whose coordinates are the values of the action variables—an action space, it applies equally to an autonomous Hamiltonian near a non-degenerate integrable system and to a periodically forced one-degree-of-freedom system (the setting in which many numerical illustrations are drawn).

Volume preservation is essential. The persisting tori are barriers to transport in the two-degree-of-freedom energy surface, where an orbit that begins between two tori remains trapped between them forever. In higher dimensions the tori no longer separate phase space, and orbits may drift slowly across the gaps (this is called Arnold diffusion). The diffusion is exponentially slow in the perturbation parameter and is invisible in most practical computations; the dominant observable effect remains the survival of a large set of quasiperiodic motions.

KAM theory therefore supplies the precise sense in which an integrable system is stable under small Hamiltonian perturbations. A positive-measure set of initial conditions continues to execute quasiperiodic motion on slightly deformed tori; the complementary set produces the chaotic layers whose existence was already anticipated by Poincaré. The theory does not describe the structure inside those layers, nor does it apply to dissipative perturbations, which destroy all tori and replace them by attractors. It remains the fundamental result that reconciles the abundance of periodic and quasiperiodic motions in nearly integrable Hamiltonian systems with the topological instability of the integrable foliation itself.

The Poincaré Section

We have had several references to the Poincaré section, what is it?

A Poincaré section is a local hypersurface transverse to the flow that is used to convert continuous-time dynamics into a discrete map. Let γ be a periodic orbit of period T, and let Σ be a smooth hypersurface of codimension one that intersects γ transversely at a point . For every point x∈Σ sufficiently close to the positive orbit through x returns to Σ after a time τ(x) close to T. The first-return (or Poincaré) map

(16.30)

is then well-defined and smooth on a neighborhood U of . Fixed points of P correspond to periodic orbits of the flow; periodic points of P of period k correspond to periodic orbits of the flow whose period is approximately k T.

Figure 16.30: A local Poincaré section Σ transverse to a periodic orbit. The first-return map P sends each point of Σ to the next intersection of its orbit with Σ.

The construction removes the trivial direction tangent to the flow. The derivative is a linear map on the (2n-1)-dimensional tangent space to Σ; its eigenvalues are the Floquet multipliers of the periodic orbit. One multiplier of the full monodromy matrix is always equal to 1 (the flow direction) and is absent from . The remaining multipliers determine the linear stability of the orbit exactly as the eigenvalues of determine the linear stability of a fixed point.

In a Hamiltonian system the return map inherits a symplectic structure. If the section is chosen inside a regular energy surface, P is symplectic with respect to the restricted symplectic form. Consequently its multipliers appear in reciprocal pairs λ,1/λ. An attracting limit cycle is again impossible where a symplectic map cannot send an open set of the section into a smaller set. The same volume argument that forbids asymptotically stable periodic orbits for the flow reappears as the statement that a symplectic Poincaré map cannot possess an asymptotically stable fixed point.

Figure 16.31: On an energy surface the Poincaré map is symplectic. Fixed points of the map are periodic orbits of the flow; they are typically elliptic (surrounded by invariant curves) or hyperbolic.

The section need not be tied to a single periodic orbit. Any local hypersurface transverse to a non-vanishing vector field defines a return map on the open set of points that return to it. In a two-degree-of-freedom Hamiltonian system a surface of section inside a compact energy surface is two-dimensional, and the return map is an area-preserving map of a planar domain. Invariant curves of that map are the intersections of invariant tori with the section; chaotic orbits appear as clouds of points that fill regions between the curves. The great practical advantage of the construction is dimensional reduction where a continuous flow in four-dimensional phase space (or three-dimensional energy surface) is replaced by a discrete map in two dimensions, which can be visualized and computed directly.

The Poincaré section therefore converts questions about periodic orbits and invariant tori into questions about fixed points and invariant curves of a lower-dimensional map. All of the local analysis developed for fixed points—linearization, Lyapunov functions, stability—transfers verbatim to the fixed points of P, while the global orbit structure of the flow is reflected in the orbit structure of the map. The same device will be used without further comment whenever a periodic orbit or a family of invariant tori is under examination.

The Poincaré-Bendixson Theorem

In the plane the long-term behavior of a smooth flow is severely restricted. Let X be a vector field on an open set of , and let γ be a positive semi-orbit whose closure is compact and contains only finitely many fixed points. Then the ω-limit set of γ is one of three objects: a fixed point, a periodic orbit, or a graphic (a finite union of fixed points joined by heteroclinic orbits). In particular, if the compact invariant set contains no fixed points, it must be a single periodic orbit.

Figure 16.32: A trapping annulus free of equilibria must contain at least one periodic orbit. All orbits that remain in the annulus approach that orbit or a finite collection of such orbits.

The theorem is local to the plane. Its proof uses the Jordan curve theorem where a periodic orbit separates the plane into an interior and an exterior, and a trajectory that crosses a local transversal can do so in only one direction. These topological facts fail in higher dimensions, where the ω-limit set of a bounded orbit may be more complicated than a fixed point or a periodic orbit. Consequently the theorem supplies an existence proof for limit cycles that is available only for planar (or spherical) flows.

A practical corollary is the trapping-annulus criterion. Suppose one exhibits a compact annular region A that is positively invariant (the vector field points inward on both boundary components) and that contains no fixed points. Then every orbit that enters A remains in A forever, its ω-limit set is nonempty, compact, connected, and free of equilibria, and therefore consists of one or more periodic orbits. At least one of those orbits is a limit cycle that attracts an open set of initial conditions. The same conclusion holds for a trapping disk that contains a single unstable focus where the boundary of the disk is crossed inward, the focus repels, and a limit cycle must occupy the region between them.

Figure 16.33: Left: a trapping annulus. Right: a trapping disk about an unstable focus. In both cases the Poincaré–Bendixson theorem guarantees a periodic orbit.

Hamiltonian systems in the plane are compatible with the theorem but yield no attracting cycles. Every compact regular level curve of H is already a periodic orbit, and the ω-limit set of a point is simply the level curve on which it lies. The trapping regions of a Hamiltonian flow are the regions between two level curves; the “limit cycles” they contain are the intervening energy curves, none of which attracts an open set. The theorem therefore recovers the familiar collection of periodic orbits without producing the asymptotically stable cycles that appear only when the flow fails to preserve area.

The result remains the principal tool for proving the existence of periodic orbits in planar systems when an explicit solution is unavailable. Once a trapping region free of equilibria has been constructed—by a Lyapunov function, by an energy estimate, or by direct inspection of the vector field on the boundary—the existence of a limit cycle follows at once. In higher dimensions no substitute of comparable simplicity exists, and one must turn to other methods.

Bifurcations

A bifurcation occurs when a qualitative change in the orbit structure is produced by the variation of a parameter. Let be a smooth family of vector fields depending on a real parameter μ. A bifurcation value is a value at which the phase portrait is not structurally stable, where arbitrarily small changes in μ alter the number, the stability type, or the mutual arrangement of fixed points and periodic orbits. The set of bifurcation values is typically a discrete collection of isolated points or, in a multiparameter family, a collection of curves and surfaces in parameter space.

The simplest local bifurcations of fixed points are visible already in one dimension. In the saddle-node bifurcation a pair of fixed points—one stable, one unstable—coalesce and annihilate as μ passes through . In the transcritical bifurcation two fixed points exchange their stability by passing through each other. In the pitchfork bifurcation a single fixed point loses stability and two new fixed points are born (supercritical) or disappear (subcritical). In each case the linearization at the critical parameter possesses a simple zero eigenvalue, and the higher-order terms decide which of the three scenarios occurs.

Figure 16.34: Local bifurcations of fixed points. Solid curves denote stable branches, dashed curves unstable branches. Left: saddle-node. Middle: transcritical. Right: supercritical pitchfork.

In two or more dimensions the same bifurcations occur inside a center manifold—the invariant surface tangent to the eigenspace of the critical eigenvalues—while the transverse directions remain attracting or repelling. An additional planar bifurcation appears when a complex-conjugate pair of eigenvalues crosses the imaginary axis. In the Hopf bifurcation a fixed point changes stability and a small periodic orbit is born. The orbit is attracting when the bifurcation is supercritical and repelling when it is subcritical. The radius of the orbit grows like the square root of the distance from the bifurcation value, and the period tends to 2 π/|Im λ|.

Figure 16.35: Supercritical Hopf bifurcation. As the parameter crosses the critical value a stable focus becomes unstable and a small attracting limit cycle appears.

Hamiltonian systems admit only those bifurcations compatible with the preservation of a symplectic form. A zero eigenvalue must have even algebraic multiplicity, and the birth of a periodic orbit in a Hopf-like transition is constrained by the existence of a first integral. The typical Hamiltonian bifurcation of a periodic orbit is the saddle-center bifurcation, in which an elliptic and a hyperbolic orbit coalesce and disappear. Dissipative systems, by contrast, permit the full range of attracting and repelling cycles.

Global bifurcations involve orbits of finite length rather than the linearization at a single point. A homoclinic bifurcation occurs when a branch of the stable manifold of a saddle coincides with a branch of its unstable manifold; a periodic orbit may be born or destroyed as the manifolds split. A heteroclinic bifurcation involves a connection between two distinct saddles. These global events are not determined by any finite jet of the vector field at a point; they require the examination of the entire orbit.

The catalogue of elementary bifurcations supplies the local mechanisms by which fixed points and periodic orbits can appear, disappear, or change stability. Once the critical eigenvalues and the leading nonlinear terms are known, the phase portrait on either side of the bifurcation value is determined. The same classification organizes the transitions observed in concrete models, from the onset of a limit cycle in a dissipative oscillator to the coalescence of periodic orbits in a nearly integrable Hamiltonian system. Higher-codimension bifurcations, in which several conditions are satisfied simultaneously, produce more elaborate diagrams but rest on the same geometric principles.

The detailed study of bifurcations is a major topic in what is called catastrophe theory.

Conservative vs. Dissipative Systems

A vector field is conservative when it is the Hamiltonian vector field of some function H, or more generally when its flow preserves a volume form. In the setting of Lesson 15 the canonical symplectic volume is preserved by every Hamiltonian flow, so every Hamiltonian system is conservative. Energy itself is an integral of the motion whenever H has no explicit time dependence. The phase portrait is organized by the level sets of the integrals: trajectories lie on the common level sets and cannot cross them.

Figure 16.36: Left: a conservative flow preserves the enclosed areas of the nested level curves. Right: a dissipative flow contracts areas; nearby orbits converge to an attractor.

A vector field is dissipative when its flow contracts volumes. In coordinates the divergence is negative on average,

(16.31)

so that the volume of any region decreases under the flow. The long-term dynamics is then concentrated on a set of measure zero—an attractor. Attracting fixed points, attracting limit cycles, and more complicated attracting sets become possible. Asymptotic stability, which is forbidden for volume-preserving flows, is the typical local behavior near a dissipative attractor.

The distinction is sharp for the objects already studied. A fixed point of a conservative system is never asymptotically stable; at best it is Lyapunov stable (a center arising from a local minimum of H). A periodic orbit of a conservative system cannot attract an open set of initial conditions. In a dissipative system both phenomena occur routinely for a spiral sink is an asymptotically stable fixed point, and a limit cycle is an asymptotically stable periodic orbit. The van der Pol oscillator examined earlier is dissipative; the harmonic oscillator of Lesson 15 is conservative.

Figure 16.37: Left: the harmonic oscillator (conservative). Right: a dissipative oscillator with an attracting limit cycle. Only the latter can capture an open set of initial conditions.

The same contrast appears in the linearization. The eigenvalues of a Hamiltonian linearization appear in ±λ pairs, so the sum of the real parts (the divergence) vanishes. A dissipative linearization may have all eigenvalues with negative real parts, producing an asymptotically stable node or focus. When a parameter is varied, a conservative system can lose stability only by the birth of a center or by a Hamiltonian bifurcation; a dissipative system can lose stability through a Hopf bifurcation that creates an attracting limit cycle.

Volume contraction also supplies a practical test. If a trapping region can be exhibited and the divergence is strictly negative inside it, the ω-limit set has measure zero and, in the plane, must be a fixed point or a limit cycle. No such conclusion is available for a volume-preserving flow: the trapping region between two energy curves is filled with periodic orbits, none of which attracts an open set.

Thus the conservative systems of classical mechanics preserve both energy and phase-space volume, while dissipative systems—those that model friction, resistance, or other irreversible processes—contract volume and permit attractors. The geometric mechanisms of stability, periodic orbits, and bifurcations operate in both settings, but the presence or absence of volume preservation decides which of those mechanisms can produce asymptotic stability. The remainder of the lesson examines the consequences of this distinction for sensitivity to initial conditions and for the long-term organization of the phase portrait.

Sensitivity to Initial Conditions

Two nearby initial conditions determine two nearby orbits. The distance between those orbits may remain bounded, may grow linearly with time, or may grow exponentially. When the separation grows exponentially,

(16.32)

with a positive Lyapunov exponent λ, the system is said to exhibit sensitive dependence on initial conditions. An arbitrarily small uncertainty in the present state is then amplified until it becomes macroscopic; long-term prediction from imperfect data becomes impossible even though the equations themselves are deterministic.

Figure 16.38: Sensitive dependence. Two orbits that begin a distance apart separate exponentially. After a time of order the separation is no longer small.

In a Hamiltonian system the symplectic structure constrains the possible rates of separation. Liouville’s theorem forbids uniform volume expansion, so exponential growth in some directions must be compensated by exponential contraction in others. The exponents λ that measure these rates appear in ±λ pairs. Sensitive dependence is therefore possible—indeed typical for non-integrable Hamiltonian flows—yet it coexists with the preservation of phase-space volume. The same pairing is absent in a dissipative system, where all nearby orbits may converge to an attractor while, on the attractor itself, sensitive dependence persists.

The practical consequence is a finite prediction horizon. If the initial state is known only to accuracy and a prediction of accuracy δ is required, the useful forecast time is at most

(16.33)

Beyond that horizon the evolved uncertainty exceeds the tolerance. Improving the initial measurement lengthens the horizon only logarithmically. The phenomenon is already visible in elementary examples, for example, the hyperbolic fixed point of a saddle has a positive exponent equal to the unstable eigenvalue; the chaotic layers that appear between KAM tori in a nearly integrable system possess a positive exponent on a set of positive measure.

Figure 16.39: Left: regular motion; nearby orbits on neighboring tori remain close. Right: sensitive dependence inside a chaotic layer; an initially tight bundle of orbits spreads across the layer.

Sensitivity does not require a strange attractor, nor does it require dissipation. It is a property of the linearized flow along a given orbit, so that if the variational equation possesses a positive Lyapunov exponent, the orbit is sensitive. The global organization of the phase portrait—whether the sensitive orbits are confined to thin layers between tori, or whether they fill a positive-volume chaotic sea—depends on the existence of integrals and on the strength of the perturbation. Those questions are taken up in the discussion of chaos that follows. The present subsection only isolates the local mechanism: exponential separation of nearby orbits, its compatibility with volume preservation, and the resulting logarithmic limit on predictability.

Chaos

A flow is chaotic when it possesses a compact invariant set so that the motion is sensitive to initial conditions, topologically transitive, and filled with a dense set of periodic orbits. Sensitivity has already been isolated, where nearby points separate exponentially. Transitivity means that some orbit (in fact a residual set of orbits) comes arbitrarily close to every point of the set; the orbit wanders throughout the region rather than remaining confined to a proper subset. The dense periodic orbits supply a skeleton of regular motions that is nevertheless invisible in any practical observation, because the periodic points are unstable.

Figure 16.40: A chaotic invariant set. One long orbit wanders throughout the region (transitivity). Unstable periodic orbits are dense but practically invisible.

The three conditions are independent. A hyperbolic saddle has sensitive dependence but its invariant set is a single point, not transitive in any interesting sense. An irrational flow on a torus is transitive and has no periodic orbits, yet nearby orbits separate only linearly. Only when all three hold simultaneously does one speak of chaos. In a Hamiltonian system the invariant set cannot be an attractor of positive codimension, because volumes are preserved; chaos appears instead as a chaotic sea of positive measure, or as a chaotic layer of positive measure between surviving KAM tori. In a dissipative system the same three conditions may be realized on a strange attractor of measure zero that nevertheless captures an open set of initial conditions.

The logistic map at parameter value 4,

(16.34)

provides an elementary discrete-time illustration. The interval [0,1] is invariant, the map is conjugate to the angle-doubling map, and every subinterval is eventually expanded across the whole interval. Periodic points of every period are dense, the orbit of almost every point is dense, and the separation of nearby points doubles at each iteration. The continuous-time analogue is a flow that possesses a transverse section on which the return map exhibits the same features.

Figure 16.41: Cobweb plot of the logistic map at the chaotic parameter value 4. Successive iterates fill the interval irregularly.

Chaos does not contradict determinism. The vector field remains smooth, and the flow is uniquely determined by the initial condition. What fails is practical predictability beyond a finite horizon set by the largest positive Lyapunov exponent. In the Hamiltonian setting the failure is compatible with the existence of a full set of formal integrals (the Birkhoff normal form) that nevertheless diverge; the true integrals, when they exist, are only continuous or worse. The geometric picture is therefore one of intermingled regular and chaotic regions, where we experience islands of stability surrounding elliptic periodic orbits, surrounded by chaotic layers whose thickness grows with the strength of the perturbation, until, for sufficiently strong perturbations, the chaotic sea occupies most of the energy surface.

The definition isolates the minimal geometric ingredients. Subsequent subsections quantify the sensitive dependence by means of Lyapunov exponents and examine the concrete realizations of chaos that appear in physical models. The essential point is already visible: deterministic evolution on a compact set can amplify microscopic uncertainties until the macroscopic state is unpredictable, while still preserving every integral and every symplectic invariant that the equations require.

Lyapunov Exponents

The local rate of separation of nearby orbits is measured by the Lyapunov exponents. Let be an orbit and let v(t) be a solution of the variational equation along that orbit,

(16.35)

The Lyapunov exponent associated with the initial vector is

(16.36)

provided the limit exists. For a typical initial vector the limit equals the largest exponent . The full spectrum is obtained by considering the growth rates of volumes of parallelepipeds, where the sum of the first k exponents is the exponential growth rate of k-dimensional volume.

Figure 16.42: Along a typical orbit a generic tangent vector is stretched at the rate of the largest Lyapunov exponent. The slope of log|v(t)| versus t approaches .

A positive exponent implies sensitive dependence on initial conditions along the orbit. When the largest exponent is positive on a set of positive measure (or on an attractor), the system is chaotic in the sense of the preceding subsection. Conversely, if every exponent is non-positive and the orbit is bounded, nearby points do not separate exponentially. In an integrable Hamiltonian system every exponent vanishes and the motion is quasiperiodic on tori and separations grow at most linearly.

Hamiltonian structure again forces a symmetry. Because the flow preserves the symplectic form, the spectrum is symmetric, λ and -λ appear together. The sum of all exponents is zero, recovering Liouville’s theorem (the divergence of vanishes). In a dissipative system the sum of the exponents equals the average divergence and is therefore negative; contraction in the phase-space volume coexists with expansion along the attractor.

Figure 16.43: Left: the Lyapunov spectrum of a Hamiltonian system is symmetric about zero. Right: a dissipative spectrum has negative sum; volume contracts while sensitive dependence may persist on the attractor.

The exponents are constant on each orbit and, for a smooth flow, take only finitely many values on a set of full measure (this is called Oseledec’s theorem). They are invariant under smooth changes of coordinates. Consequently they are geometric invariants of the orbit. In practice they are computed by integrating the variational equation alongside the original system and periodically renormalizing the evolving vectors (or by means of the QR decomposition of the fundamental matrix). The largest exponent alone already decides the presence of sensitive dependence and sets the scale of the prediction horizon.

Thus the Lyapunov exponents convert the qualitative notion of sensitive dependence into a set of real numbers attached to each orbit. Their signs distinguish regular from chaotic motion; their symmetry or asymmetry distinguishes conservative from dissipative dynamics; and their magnitude fixes the practical limit of predictability. The final subsection examines how these quantities appear in concrete physical models.

Connections to Physics

The geometric structures developed in this lesson appear throughout classical mechanics and classical electrodynamics. In mechanics the phase space is the cotangent bundle of the configuration manifold, the vector field is Hamiltonian, and the flow preserves both energy and the symplectic volume. The possible long-term behaviors are therefore those compatible with volume preservation, with centers rather than spiral sinks, families of periodic orbits or KAM tori rather than attracting limit cycles, and chaotic seas of positive measure rather than strange attractors of measure zero.

The harmonic oscillator realizes the simplest regular case. Its phase portrait consists entirely of periodic orbits surrounding a single center; every Lyapunov exponent vanishes and the motion is predictable for all time. A weakly perturbed integrable system—the Hénon–Heiles model, or a planetary three-body problem with small eccentricities—retains a positive-measure set of KAM tori while developing thin chaotic layers. The Lyapunov exponents are zero on the tori and positive on a set of positive measure in the layers. Stronger perturbations destroy most tori and leave a chaotic sea that occupies most of the energy surface; the largest Lyapunov exponent is then positive almost everywhere.

Figure 16.44: Poincaré section of a weakly perturbed integrable system. Most of the section is filled by the intersections of surviving KAM tori; chaotic layers occupy the gaps.

Classical electrodynamics supplies a second, field-theoretic realization of the same geometry. The configuration space of the electromagnetic field is the space of vector potentials (modulo a gauge), and the phase space is the corresponding cotangent bundle. In the absence of charges the Maxwell field is an infinite-dimensional Hamiltonian system; the symplectic form pairs the electric field with the vector potential, and the Hamiltonian is the familiar field energy. Plane-wave solutions are the analogues of the harmonic oscillator, with every “orbit” is quasiperiodic and every Lyapunov exponent vanishes. When the field is coupled to charged particles, or when nonlinear constitutive relations are introduced, the same mechanisms that produce chaos in finite-dimensional mechanics become available. A charged particle in a prescribed electromagnetic wave, for example, can exhibit sensitive dependence; the phase-space volume of the particle is still preserved by the Lorentz force, yet the particle’s trajectory on the energy surface may be chaotic.

Figure 16.45: Motion of a charged particle in a monochromatic electromagnetic wave. Regular islands coexist with a chaotic sea; the underlying flow remains Hamiltonian.

In both domains the conservative character of the ideal theory forbids attractors. Dissipation enters only when additional irreversible processes—friction, Ohmic resistance, radiation reaction—are modeled by non-Hamiltonian terms. Those terms produce volume contraction, permit attracting limit cycles or fixed points, and can convert a chaotic sea into a strange attractor. The ideal equations of classical mechanics and source-free electrodynamics remain, however, strictly conservative since they preserve a symplectic structure, pair their Lyapunov exponents, and organize their long-term dynamics by the geometry of energy surfaces and invariant tori.

The lesson therefore closes where it began. The Hamiltonian formalism of Lesson 15 supplies the phase space, the symplectic form, and the volume-preserving flow. The present lesson has examined the orbit structures compatible with that flow—fixed points, periodic orbits, invariant tori, chaotic layers—and the quantitative measures (Lyapunov exponents) that distinguish regular from sensitive motion. The same structures reappear, essentially unchanged, in every conservative classical field theory.

How to Do This Stuff in Mathematica

The geometric structures of the lesson are most easily explored by direct computation. Mathematica supplies both the numerical integrators and the graphical primitives needed to visualize flows, maps, and the diagnostics that distinguish regular from chaotic motion. The subsections that follow give working code for the principal examples.

Iterative Function Systems

An iterated function system (IFS) is a finite collection of contracting maps. The attractor is the unique nonempty compact set invariant under the union of the maps. The chaos game samples the attractor by repeated random application of the maps.

Figure 16.46: A closed feedback loop for a one-step map .

How, then, do we construct an IFS image? Let us take the Koch curve as an example. In 1904 Helge von Koch presented this self-similar curve that is remarkable as it is constructible from elementary geometry.

Here we wish to produce the base shape:

We establish, from this, our basic shape:

Now we code for our next shape:

In[4]:=

nextShape[{{x1_,y1_},{x2_,y2_}}]:=
    Module[{rotation2D,x,y,basicShape, newShape},
rotation2D[x_,y_] = Module[{a,b},
        a = x/Sqrt[x^2+y^2];
        b = y/Sqrt[x^2+y^2];
        {{a,-b},{b,a}}];
        {x,y}={x2-x1,y2-y1};
        r = Sqrt[x^2+y^2];
        basicShape = {{0,0},{1/3,0},{1/2,Sqrt[3]/6},{2/3,0},{1,0}};
        newShape =  Map[{x1,y1} + r rotation2D[x,y].# &, basicShape]//N;
newShape]

Let's try it out.

In[5]:=

nextShape[{{0,0},{1,0}}]

This looks okay, now we have to use this to break up our line segments.

In[6]:=

Partition[{{0,0},{1/3,0},{1/2,Sqrt[3]/6},{2/3,0},{1,0}}, 2,1]

We can now write a function to do this:

In[7]:=

nextShape[lis_List] := Map[nextShape, lis]

In[8]:=

nP = Flatten[nextShape[Partition[{{0,0},{1/3,0},{1/2,Sqrt[3]/6},{2/3,0},{1,0}}, 2,1]],1]

Here is what it looks like:

In[9]:=

Show[Graphics[Line[nP]],AspectRatio -> Automatic]

This will not work as we go deeper, instead we need:

In[10]:=

lineSequence[lis_List] := Module[{seq ={}},
                            For[m=1, m< Length[lis], m++,
                                If[lis[[m]]!=lis[[m+1]], seq = Append[seq, {lis[[m]]}]]];
                                seq = Flatten[Append[seq, {lis[[Length[lis]]]}],1];
                            seq]

In[11]:=

lineSequence[nP]

Here is what this looks like:

In[12]:=

Show[Graphics[Line[lineSequence[nP]]], AspectRatio -> Automatic]

Now that we have removed the repeated sequences, we can write a nice function:

Let's see what this looks like:

In[17]:=

Show[
    Graphics[
        Line[
            kochCurve[{{0,0},{1,0}},4]]], AspectRatio -> Automatic]

As of 2011 there is a built-in function KochCurve.

Let's try a different Koch function:

In[19]:=

kochCurve[pointsList_List, n_Integer] :=
Nest[
    lineSequence[
        Flatten[nextShape[Partition[#,2,1]],1]]&,pointsList,n]

Here is what this one looks like:

In[20]:=

Show[
    Graphics[
        Line[
            kochCurve[{{0,0},{1,0}},5]]], AspectRatio -> Automatic]

We can also perform these transformations on a triangle,

In[21]:=

Show[
    Graphics[
        Line[
            kochCurve[
                {{0,0},{1/2,Sqrt[3]/2},{1,0},{0,0}},6]]],
                AspectRatio -> Automatic]

And on a square.

In[22]:=

Show[
    Graphics[
        Line[
            kochCurve[
                {{0,0},{1,0}, {1,1}, {0,1}, {0,0}},5]]],
                AspectRatio -> Automatic]

The Julia Set

Invented by 25 year-old Gaston Julia and published in 1918, the paper that derived what we call the Julia Set dealt with the behavior of iterates of rational functions of a complex variable. Specifically, sets of points whose nth iterate remains bounded on the complex plane as n goes to infinity.

The Julia set of a quadratic map is the boundary of the set of points that remain bounded under iteration. For most c the set is a fractal; for c=0 it is the unit circle.

In[23]:=

JuliaTest = Compile[{x, y, {n, _Integer}, {c, _Complex}},
            Module[{z, num = 0}, z = x + y I;
            While[Abs[z] < 2.0 && num < n, z = z^2 + c; num++];
            num]];

Here is what this set looks like. The darker the color, the faster the iterates of a point in that region escape to infinity.

In[24]:=

DensityPlot[JuliaTest[x, y, 50, -0.75 + 0.5 I], {x, -2.0, 2.0},
{y, -2.0, 2.0}, PlotPoints → 100, Mesh → False,ColorFunction->Hue]

Here we zoom in on the coordinates centered around{0.1637, -0.7544}.

In[25]:=

DensityPlot[JuliaTest[x, y, 50, -0.75 + 0.5 I], {x, 0.1,0.2},
{y, -0.8,-0.7}, PlotPoints → 100, Mesh → False,ColorFunction->Hue]

We can change the exact mapping, too. Here we will use .

In[26]:=

JuliaTest = Compile[{x, y, {n, _Integer}, {c, _Complex}},
            Module[{z, num = 0}, z = x + y I;
            While[Abs[z] < 2.0 && num < n, z = z^3 + c; num++];
            num]];

In[27]:=

DensityPlot[JuliaTest[x, y, 50, -0.75 + 0.5 I], {x, -2.0, 2.0},
{y, -2.0, 2.0}, PlotPoints → 100, Mesh → False,ColorFunction->Hue]

The Mandelbrot Set

Similar to the Julia set is the famous Mandelbrot set, even though this set was already studied by the French mathematician Pierre Fatou around 50 years before Mandelbrot studied it. The Mandelbrot set is a set of points on the complex plane such that a sequence does not tend to infinity. That sequence is where and .

In[28]:=

MandelbrotTest = Compile[{x, y, {n, _Integer}},
            Module[{z, num = 0,c = x + y I}, z = x + y I;
            While[Abs[z] < 2.0 && num < n, z = z^2 + c; num++];
            num]];

Here is what it looks like, where the darkest red are those elements of the set, the other colors represent the number of iterations it takes to reach infinity.

In[2]:=

DensityPlot[MandelbrotTest[x, y, 50], {x, -2.0, 0.75},
{y, -1.25, 1.25}, PlotPoints → 100, Mesh → False,ColorFunction->Hue]

Discrete Dynamical Systems and the Logistic Map

Here we will study the bifurcation diagram of the logistic map, this has the form, . Each iterate is given by,

Now we build a list of points for the iterations, we will consider only the last 100 iterates despite considering 1,000 iterates.

Here is what it looks like.

In[3]:=

ListPlot[plist,PlotStyle->PointSize[0.0001],Frame->True,FrameLabel->{r,n}]

Note how the trajectory of the solution branches at t=3, then again just after t=3.4 and so on. Thus we can display the dynamical behavior of a discrete system.

Continuous Dynamical Systems and the Lorenz Attractor

In 1963 Edward Lorenz, an MIT meteorologist developed a simple model of a convecting fluid. This model is constructed around three famous equations:

where the dots over the variables indicate time derivatives. Here x is the rate of convective overturning, y is the horizontal temperature variation, z is the vertical temperature variation, σ is the Prandtl number, r the Reynolds number, and b is a scaling factor. This system of three first-order differential equations is the source of the famous butterfly effect.

Here is how we can solve this system, where σ= 10, b=8/3, and r = 28.

In[32]:=

sln1 = NDSolve[ {x'[t] == 10 (y[t] - x[t]),
           y'[t] == 28 x[t] - y[t] - x[t] z[t],
           z'[t] == x[t] y[t] - (8/3) z[t],
           x[0] == z[0] == 0, y[0] == 1},
         {x, y, z}, {t, 0, 100},MaxSteps -> Infinity]

Here is the strange attractor of the system.

In[33]:=

ParametricPlot3D[Evaluate[{x[t],y[t],
z[t]} /. sln1],{t,0,100},PlotPoints->10000]

Chaos Diagnostics

One measure of chaos is the dimension of a figure. Here is a table of common dimensionalities of figures.

What about fractals? What is their dimension? By definition, they are non-integer dimensions. What is the dimension of the Koch curve, for example? Here is what we defined as the basic shape,

It has a length of one unit. We can call this L. This is constructed by removing the middle third segment of a line and replacing it with the top of an equilateral triangle. This gives us 4 segments each of length L/3.  Thus we define a new variable ε=1/3. The line is partitioned into N separate segments, in this case N=4. Were we to make this a plane figure with the given shape on a side we would write,

What about for arbitrary dimension D? We would simply write,

We can solve this for D.

Thus the dimension of a figure in general is,

This arrives at the correct dimension as ε gets vanishingly small in the assumption that ln(1/ε)≫ln L.

Returning to our figure, we have, at any step, m, 4 segments each of 1/3 the segment length, so we can define the dimension as

One common measure of chaos is the Lyapunov exponent, measuring the divergence between a reference trajectory x and an adjacent trajectory offset by some arbitrarily small displacement δ x. The distance of this displacement is the norm,

From this we can arrive at,

Here is a program to calculate the Lyapunov exponent for the logistic map.

One test of chaos occurs whenever this number is greater than 0.

Here is what the Lyapunov exponents look like for the logistic map.

Here is the bifurcation diagram and the plot of the Lyapunov exponent one above the other.

Further Reading

Joseph L. McCauley, (1997), Classical Mechanics, Cambridge University Press, reprinted with correction in 1998. This is a really good introduction to advanced mechanics.

Jorge V. José and Eugene J. Saletan, (1998), Classical Dynamics: A Contemporary Approach, Cambridge University Press. A fairly complete study of geometrical mechanics.

Richard H. Enns, George C. McGuire, (2001),Nonlinear Physics with Mathematica for Scientists and Engineers, Birkhauser Boston.

Nino Boccara, (2007), Essentials of Mathematica With Applications to Mathematics and Physics, Springer.

Gerd Baumann, (2005), Mathematica for Theoretical Physics, Classical Mechanics and Nonlinear Dynamics, Second Edition, Springer.

Wolfram Research, (2026), Mathematica version 15.0.