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Consider the general linear system $$\dot{x}=Ax$$ where $A\in\mathbb{R}^n\times\mathbb{R}^n$, and $x\in\mathbb{R}^n$. Many sources assert that such a system cannot be chotic for any $n$, for example

https://en.wikipedia.org/wiki/Chaos_theory

How to know whether an Ordinary Differential Equation is Chaotic?

I'm looking for a simple reasoning, either by a proof, or by geometrical considerations, that will show that such systems can never be chaotic.

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It is easy to show that such systems are subject to the superposition principle, i.e., a linear combination of two solutions is again a solution: If $\dot{x} = Ax$ and $\dot{y} = Ay$, and $z=αx+βy$ is our linear combination, then $$ \dot{z} = \frac{\mathrm{d}}{\mathrm{d}t} (αx+βy) = α\dot{x} + β\dot{y} = αAx + βAy = A(αx+βy) = Az .$$ Suppose the dynamics is sensitive to initial conditions, i.e., any small perturbation will blow up (until it reaches the size of the attractor). Now, we can decompose the perturbed solution into the unperturbed solution and the perturbation, which must be again a solution due to the superposition principle. In particular we can also scale down any solution arbitrarily to use it as a perturbation. As this perturbation must blow up to extent, so must the solution, which is thus unbounded.

More formally, sensitivity to initial conditions means that for some $δ>0$ (at most the diameter of the attractor) and any $ε>0$, any trajectories starting with a distance of $ε$ will become $δ$ apart after a sufficiently large time: $$∃δ>0 : ∀ε>0: ∀x,y,|x(0)-y(0)|≤ε : ∃t: |x(t)-y(t)|≥δ,$$ where $x$ and $y$ are solutions of your system. Now choose $y=x+\frac{ε}{|x(0)|} x$, which is a solution due to the superposition principle. Inserting this into the above yields: $$∃δ>0 : ∀ε>0: ∃t: ε \frac{|x(t)|}{|x(0)|}≥δ.$$ Thus $x$ is unbounded.

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First note that chaotic dynamics must be both, bounded and sensitive to initial conditions.

Linear systems are one of the few differential equation systems that we can solve analytically. The solution is a linear combination of exponential functions and sines¹. These are either:

  • sensitive to initial conditions but unbounded. An example for this is the solution to $\dot{x} = x$.
  • bounded but not sensitive to initial conditions, namely converging to a fixed point, limit cycle (periodic dynamics), or limit torus (quasi-periodic dynamics). An example for this is the solution to $\dot{x} = -x$.

¹ which in turn are just a linear combination of two exponential functions with complex arguments

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