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I am reading the Scholarpedia article on Lyapunov exponents:

Given a dynamical system $$ \dot{\vec{x}}=\vec{F}(\vec{x}) $$ and a fixed point $\vec{x}_0$ such that $\vec{F}(\vec{x}_0)=\vec{0}$, the Lyapunov exponents are defined as the real parts of the relevant Jacobian eigenvalues. They measure the exponential contraction/expansion rate of infinitesimal perturbations. So far, so good.

The problem comes while reading the Properties section. The author says:

A strictly positive maximal Lyapunov exponent is often considered as a definition of deterministic chaos. This makes sense only when the corresponding unstable manifold folds back remaining confined within a bounded domain (an unstable fixed point is NOT chaotic).

This is what I’m not understanding. I was convinced, in fact, that, in a non-integrable dynamical system, chaos always arises in correspondence of unstable fixed points. What am I missing?

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2 Answers 2

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The definition you are quoting¹ only applies to the direct vicinity of a fixed point (boldface mine):

In this simple case, the LEs $λ_i$ are the real parts of the eigenvalues.

In general, Lyapunov exponents are properties of the dynamics, not of a certain point². Roughly speaking, they are a temporal average of the projection of the Jacobian to a specific direction along the trajectory. Analogously, chaos is a property of a dynamics or set of trajectories (a chaotic attractor, saddle, transient, or invariant set), not of a fixed point.

If you look at a stable fixed point, a trajectory within its basin of attraction will be very close to the fixed point for this average and thus you obtained the quoted definition¹. For an unstable fixed point, almost any trajectory will eventually move away from it and its type of dynamics (fixed point, periodic, chaos, …) depends on the structure of the phase-space flow in regions distant from the unstable fixed point. So, the nature of a fixed point does not tell you anything about a system being chaotic or not.

Your second quote alludes to the following: Chaos does not only require a positive Lyapunov exponent, but also a bounded dynamics. For example, $\dot{x} = x$ also has a positive Lyapunov exponent, but is not chaotic – it is unbounded and not folding back (also see this question on Math SE).


¹ “Given a dynamical system $\dot{\vec{x}}=\vec{F}(\vec{x})$ and a fixed point $\vec{x}_0$ such that $\vec{F}(\vec{x}_0)=\vec{0}$, the Lyapunov exponents are defined as the real parts of the relevant Jacobian eigenvalues.”
² “A notion of local/instantaneous/… Lyapunov exponents also exists, but it’s probably not what you’re asking about and doesn’t play into the definition of chaos. ³ “A strictly positive maximal Lyapunov exponent is often considered as a definition of deterministic chaos. This makes sense only when the corresponding unstable manifold folds back remaining confined within a bounded domain (an unstable fixed point is NOT chaotic)”

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  • $\begingroup$ Thanks a lot for your very detailed answer. Just few clarifications, please correct me if I'm wrong. Suppose you have a fixed point $\vec{x}_0$ which is associated to a Jacobian having at least one eigenvalue endowed with positive real part. Therefore, this fixed point is unstable. BUT chaos does NOT necessarily emerge. In fact the trajectory, escaping from the unstable fixed point may fall, for example, in the basin of an attractive fixed point. How does your answer change if we introduce the hypothesis that the dynamical system is Hamiltonian? $\endgroup$
    – AndreaPaco
    May 7, 2018 at 15:21
  • $\begingroup$ How does your answer change if we introduce the hypothesis that the dynamical system is Hamiltonian?Hamiltonian systems neither have stable nor unstable fixed points. $\endgroup$
    – Wrzlprmft
    May 7, 2018 at 16:13
  • $\begingroup$ Are you saying that Hamiltonian systems have just neutral-equilibrium fixed point? This sound strange to me. Can you please provide an explanation? $\endgroup$
    – AndreaPaco
    May 7, 2018 at 19:27
  • $\begingroup$ @AndreaPaco: Sorry, replace unstable fixed point with repellor in my previous comment. Anyway, Hamiltonian systems cannot have attractors (or repellors), as this would mean contracting (or expanding) phase-space volumes, which in turn would violate Liouville’s theorem. Also see the link in the previous comment. Hamiltonian systems can have saddle points. $\endgroup$
    – Wrzlprmft
    May 7, 2018 at 19:41
  • $\begingroup$ I know that Hamiltonian systems feature a zero-divergence flow. What I am interested in, is what happens near unstable fixed points, i.e. points associated to a Jacobian having at least one eigenvalue with positive real part. Are there additional conditions for the emergence of chaos? Thanks for your help. $\endgroup$
    – AndreaPaco
    May 7, 2018 at 19:47
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Classical chaos requires, in addition to sensitive dependence on initial conditions (positive maximum LE), that trajectories mix. For instance consider a single hard-sphere scatterer. When the impact parameter is +/- zero, then small perturbations will send the particle off in different directions. But to make the system chaotic you must confine the trajectories so that they can mix and fill out phase space (or real space) as in the Bunimovich stadium. I think that is the point of the property you quote.

Here is a picture from Hans-Jürgen Stöckman that illustrates the point. This notion goes over to quantum or wave chaos, where instead of trajectories, we look at the asymptotic properties of the nodal domains of Dirichlet eigenfunctions. In that case, "scars" show up as the ghostly remains of unstable periodic trajectories. Hope that is what you had in mind.

Billiards

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