New answers tagged chaos-theory
Clarifications and addition: It's true that not all nonlinear systems are chaotic, but that all chaotic systems are nonlinear (or infinite-dimensional linear). The sensitivity to initial conditions is an important point and the commenter raises a good question. Consider the Lorenz system in a non-chaotic parameter regime (or for that matter, any stable ...
As you have commented, there is a minor but very important omission. The sensitivity to initial conditions is infinitely big. Or better, say, two vectors from the state space, modeling the initial conditions, no matter how close they are would eventually diverge from each others corresponding trajectories.
Not all nonlinear systems are chaotic. However a chaotic system is necessarily nonlinear. There doesn't exists a definition for chaos but using the one given by Strogatz, ref 1: Chaos is aperiodic long-termed behavior in a deterministic system that exhibits sensitive dependence on initial conditions. Like explained in the text: aperiodic long-termed ...
I recommend Turbulence: The Legacy of A. N. Kolmogorov by Uriel Frisch. It explains how turbulence is a top-down behavior, large scale turbulence in turn causes turbulence at smaller and smaller scales. This continues until a small enough scale is reached where the energy from the turbulence serves to create more heat on the molecular level. Because of this ...
It's largely a matter of definition. Here are some quotes. From Springer Reference: Definition There is no universally recognized definition of chaotic flows. Flows with properties that are neither constant in time nor presenting any regular periodicity are normally referred as chaotic. Fluid turbulence is generally found to be chaotic. It is ...
A dynamical system is a system that evolves by a rule over time. As opposed to modeling a system with a PDE, the system is modeled by some type of iterated function, $f^t(x)$. Smooth implies that the function being iterated is differentiable, so we are not talking about an iterated system like cellular automata.
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