# Chaos implies Nonlinearity?

Why, for finite dimensions, is nonlinearity a precondition for chaos?

This article (Linear Chaos? By Nathan S. Feldman) offers an example of an infinite dimensional chaotic map, which is linear. While it also suggests a hint (in section 2) about why finite dimensional linear maps cannot be chaotic, the reason is not immediately clear to me. Could one maybe relate nonlinearity to the fulfilment of the 3 criteria for chaos (mentioned in the paper)?

Moreover, could arguments made for a map/discrete dynamical system be naturally carried on to continuous dynamical systems/differential equations? I assume one would begin by looking at the flow function, but I have no idea how to proceed.

• The article explains exactly why this is true in Proposition 2.1. Commented May 6, 2018 at 19:10
• This seems to be more of a pure math question than a physics question. It'd probably make more sense at SE.Math.
– Nat
Commented May 6, 2018 at 19:13
• I wasn't able to see how exactly it does, hence the question. Additionally, the answer proposed is for a map. How does one extend it to a continuous dynamical system? Commented May 6, 2018 at 19:14
• @Primo-uomo Does that mean that you don't understand the proof, or that you don't understand how the proposition itself leads to the conclusion that finite-dimensional linear maps cannot be chaotic? Commented May 6, 2018 at 19:22
• @probably_someone I understand the proof in itself. I have difficulties with the latter. Additionally, I would like to understand how this translates to the case of continuous dynamical systems. Commented May 6, 2018 at 19:33

Yes, it does.

My take is that, without nonlinearity, folding is missing.

One of the main mechanisms behind classical chaos is the so-called stretch and fold. It can be visualized as a blob of initial conditions being stretched and then folded over itself by the mapping: stretching leads to a divergence of close trajectories (the hallmark of chaos), while folding keeps them bounded (and dense).

A linear system may be able to produce stretching, but this alone corresponds to a trivial behavior, divergence. But, in the spaces we're considering, a linear system cannot produce folding:

[none of the] orbits of a linear operator in finite dimensions [...] are dense in the space

Why can't linear functions produce folding?
- The reason is that folding means that an interval is mapped on itself twice, and that requires a non-monotonic function (and linear functions are monotonic), such as the logistic map's parabola.

Notice, though, that topology can provide the folding that a linear transformation alone can't, see, e.g., Arnold's cat map, which is a linear map on a torus.

if $f$ has a dense set of periodic points and is transitive, then $f$ must have sensitive dependence on initial conditions. Hence only the first two conditions of the definition of chaos need to be verified when showing that a particular function $f$ is chaotic.
As for differential equations, any linear ODE, $\dot{\mathbf{x}}=\mathbf{A}\mathbf{x}$, can be solved. More pictorially, the understanding of maps can be somewhat transfered to flows by means of Poincaré maps and, especially, Poincaré recurrence plots. See, e.g., Carroll's A review of return maps for Rössler and the complex Lorenz or Crutchfield's slides from the lecture Example Dynamical Systems.