I am reading James Meiss’ text Differential Dynamical Systems (SIAM). I am specifically interested in how he tells the story of chaos.
In the Preface, he mentions the following: That Chapter 5 focuses on invariant manifolds:
- stable and unstable sets
- heteroclinic orbits
- stable manifolds
- local stable manifold theorem
- global stable manifolds
- center manifolds
That the “stable and unstable manifolds, proved to exist for a hyperbolic saddle, give rise to one prominent mechanism for chaos — heteroclinic intersection”.
That Chapter 7 is background for understanding chaos (“Lyapunov exponents, transitivity, fractals, etc”):
- Lyapunov exponents / definition / properties
- strange attractors / Hausdorff dimension / strange, nonchaotic attractors
And that in Chapter 8 he’ll discuss Melnikov’s method (onset of chaos): sections 8.12 and 8.13.
He notes that he doesn’t discuss discrete dynamics (maps).
After the preface, doing a word search for “chaos” or “chaotic”:
Chaos next comes up in the examples in section 1.4: Meiss introduces an example called the “ABC flow” from Arnold 1965. He mentions this is a “prototype chaotic system” and introduces the idea that “nearby trajectories will often diverge exponentially quickly in time”. Then he defines the Lyapunov exponent.
Section 1.7 is about Quadratic ODEs: the simplest chaotic systems, after the Lorenz model is introduced in section 1.6. The introduction of the Lorenz model includes an image of the setup, a mention of convective rolls, the idea of the Galerkin truncation, etc. So he introduces this system by deriving the ODEs for it.
In section 1.7 he says “informally, chaos corresponds to aperiodic motion that exhibits ‘sensitive dependence on initial conditions'”. He’ll provide a formal definition in chapter 7. He mentions that 3-dimensional systems “are the lowest dimensional autonomous ODEs that can exhibit chaos”. There is a chart of Sprott’s quadratic chaotic differential equations (the simplest quadratic systems with chaos).
In section 4.1 Definitions, “orbits can be quasiperiodic, aperiodic, or chaotic”. When he introduces orbits, he introduces the idea of a periodic orbit as well as other options.
Meiss returns to chaos in section 5.2 Heteroclinic orbits. (See Diacu and Holmes 1996 for the story of Poincare, his mistake, and its correction). He defines a heteroclinic orbit as an orbit that is backward asymptotic to one invariant set and forward asymptotic to a different one. The homoclinic orbit (doubly asymptotic) is then a special case that is forward and backward asymptotic to the same invariant set.
In a 2d system, if a branch of W^u intersects a branch of W^s then the branches coincide. Orbits that separate phase space are called separatrices: “they separate phase space into regions that cannot communicate”. In section 8.13, we’ll see that higher-dimensional systems are different from 2d ones, and that this doesn’t have to happen! Meiss also defines “saddle connection” and mentions that Hamiltonian systems in the plane often have separatrices.
Chaos comes up again in section 5.5 Global stable manifolds. The global set comes from flowing the local set backward in time. For finite time, it will be smooth. To think about its structure in general, Meiss introduces the idea of an “embedding”. He also defines an “immersion” and notes that “an immersion is locally a smooth surface”. Note that immersions can cross themselves. The topologist’s sine curve is an example that “has infinitely many oscillations and accumulates upon the interval [-1,1] on the y-axis”. “We will see later that the global stable manifold can have this accumulation problem: indeed, this is one of the indications of chaos”.
Next, in section 6.6, when the Poincaré-Bendixson theorem is introduced, it is introduced as a statement that “There is no chaos in two dimensions”. So Meiss is building intuition for the idea of chaos from the very first day in the course, and is distinguishing between it and what happens in 2d, even as he introduces 2d.
Chapter 7 focuses on chaotic dynamics, so is where the story will be more fully built out. The chapter begins with quotes from Poincaré and Lorenz and an informal definition. To formalize it will require defining “unpredictable” and “sensitive dependence”. This chapter occurs before the chapter on bifurcations.
Two very simple linear examples with sensitive dependence are given to build some intuition for the stretching between nearby trajectories that happens for some initial conditions in a system with sensitive dependence, and to show that sensitive dependence alone is not “chaotic”.
Aperiodicity or “wanders everywhere” on an invariant set is the next idea introduced, leading to a definition of “transitive”. Then “a flow is chaotic on a compact invariant set X if the flow is transitive and exhibits sensitive dependence on X”. This gives us an idea of mixing and unpredictability.
For a lot of systems, their trajectories look chaotic “when solved numerically”. Chaos was verified in the Lorenz system for r = 28 in Tucker 2002.
Note: I also should read the text for references to the Lorenz system, because that system is used as an example.