# Questions tagged [chaos-theory]

For questions in chaos theory.

429 questions
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### How can I calculate Feigenbaum's constant?

So I am trying to calculate Feigenbaum's constant for the logistic map: $$x_{n+1} = 4 \lambda x_n (1-x_n)$$ I am writing this through python and the main pieces I have for my code that are relevant ...
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### Why does Chaos Improve Evolutionary Algorithms? [on hold]

I have presented an evolutionary algorithm using chaos theory (chaotic numbers) to solve an optimization problem. The results of the experiments show that this algorithm is much better than the same ...
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### How to generate chaos?

I would like to generate chaotic behavior that would have reastic 1/f noise properties. For example, a time-series of price fluctuations or pendulum oscillations that would have chaotic properties. ...
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### Bernoulli Map: $f'(x) = 2$ Almost Everywhere and “Local Separation” Increases as $2^n$

The Bernoulli map is $$x_{n + 1} = f(x_n)= \begin{cases} 2x_n, & 0 \leq x_n < 0.5\\ 2x_n - 1, & 0.5 \le x_n \le 1 \end{cases}$$ I am told that (1) $f'(x) = 2$ almost everywhere and so (...
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### The Unpredictability of Chaos: An Intrinsic Property, or a Matter of Computing Power and/or Mathematical Knowledge?

I was reading that Chaos (in the mathematical sense) is deterministic, but not predictable. Is this unpredictability an intrinsic property of Chaos, or is it a practical matter of computing power and/...
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### Applying a linear transformation to a system of differential equations

I am reading the book: Nonlinear Oscillations, Dynamical System and Bifurcations of Vector Fields (Guckenheimer and Holmes), chapter 2: An introduction to chaos. About Van der Pol's equation, it can ...
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### Equivalent condition of a diffeomorphism having a dense orbit

Say $M$ is a manifold and $f: M \to M$ is a diffeomorphism. Assume also that, if we are given any nonempty open subsets $U$ and $V$, then there is $n \in \mathbb{Z}$ such that $f^n(U)$ intersects $V$....
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### Does the property of uniquely ergodic imply that the map has a unique chaotic attractor for all $c$ in $D$

Definition: Let $(X,B)$ be a measurable space and let $T:X→X$ be a measurable transformation. If there is a unique $T$-invariant probability measure then we say that $T$ is uniquely ergodic. Consider ...
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### Can a family of functions $F_{\lambda}$ experience a saddle-node bifurcation and also experience a period -doubling bifurcation?

Suppose $F_{C}=(x-C)^{2}$ is a family of functions. I found that at $C=-\frac{1}{4}$ there is a saddle-node bifurcation. So for $C<-\frac{1}{4}$ there are $0$ fixed points, $C=-\frac{1}{4}$ there ...
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### About topological conjugacy

The Smale horseshoe map $f$ is desribed in this page: What's the point of a Horseshoe map? A striking feature of this system is the stability of its dynamics: given any diffeomorphism $g$ ...
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### Is there a meaningful link between the golden ratio and chaos theory?

I heard it casually mentioned by strangers but am unable to find any information about this. Is there a meaningful link between the golden ratio and chaos theory? The closest thing I could find is ...
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### Why Lorenz attractor can be embedded by a 3-step time delay map?

I'm investigating attractor reconstruction of Lorenz system. I saw a bunch of work claiming that the time delay map $[x(t), x(t -\tau), x(t - 2\tau)]$ is sufficient to reconstruct the attracotr, e.g. ...
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### Question about Chaos Theory and Relation to Game…

I am working on a project (for social sciences) and I found a game, Parable of Polygons (link) I observe chaotic behavior, but am unable to mathematically explain this. Are there any ideas on how the ...
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### If the solutions to Lorentz equation diverge exponentially, how can they be confined in a strange attractor? [duplicate]

In the book of Chaosbook, at the beginning of chapter 6, it is given that [...]if the attractor is strange, any two trajectories $x(t) = f^t(x_0)$ and $x(t)+\delta x(t) = f^t(x_0 + \delta x_0)$ ...
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### Does the existence of a superstable fixed point imply a $-\infty$ Lyapunov exponent?

This is the Lyapunov exponent as a function of $r$ for the logistic map ($x_{n+1}=f(x_n)=r(x_{n}-x_{n}^2)$) The big dips are centered around points where $f'(x)=0$ for some $x$ in the trajectory used ...
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### Application of Correlation Dimension to Fractals other than Sets of Points

In Chaos Theory, the Correlation Dimension is defined to calculate the dimension of fractals. At least in the context where I've learnt it, it is applied to fractals made up of sets of points. Is it ...
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### Fixed Points and their Stability and Finally Bifurcation diagram

Find and Classify all fixed points and sketch vector field. What's the bifurcation? $$𝑥̇ = 𝑟𝑥 − \tan(𝑥)$$
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### Dynamic system $f(x) = 2x$ mod $1$

I am reading the following paper: ergodic theory of chaos and strange attractors, by J.-P. Eckmann (can be easily downloaded) My question is from an example on p....
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### Time dependence in chaos theory [closed]

One of the best, more original examples of chaos theory comes from Ray Bradbury’s A Sound of Thunder. People in the chaos-theory community have always said that the hunters that time safari allowed to ...