# Questions tagged [chaos-theory]

For questions in chaos theory.

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### Is this system chaotic?

Consider a point P on a rectangle ABCD and a line segment with slope s passing through P. The rectangle ABCD has sides of length a and b. Starting from P, a point moves along the line segment until it ...
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1 vote
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### On physical measure on post-critical set for unimodal map

Let $f: [0,1] \to [0,1]$ be an infinitely renormalizable unimodal map, let $c$ denote its critical point. It is well-known that the post-critical set $\omega(c)$ of $f$ (which is the omega limit set ...
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### the relation between chaos and fractal basin

Does fractal boundary of basin of attraction has something to do with chaos? I think fractal boundary must lead to chaos, and how about the other way round?
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### Performing linear stability analysis for nonlinear discrete system by approximating function for large values of the varying bifurcation parameter

Here's my system, \begin{gather*} N_{t+2}=N_t\exp{[r(1-\frac{N_t}{K})]}\frac{1-e^{-aP_t}}{aP_t} \\ P_{t+1}=N_t[1-\frac{1-e^{-aP_t}}{aP_t}] \end{gather*} In the research paper, it states that ...
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### Intersection of stable and unstable manifolds.

Let $$\dot{x}=F(x)$$ be an autonomous (i.e. it does not depend on $t$) system with $F: \mathbb{R}^n \to \mathbb{R}^m$ a regular as you want vector field. Suppose also that $0$ is an hyperbolic ... 117 views

### Convergence of iterative map

I have the following iterative mapping: $$x_{n+1} = (N-n)^{-1} \frac{x_n}{f(x_n)} \left(C - \sum_{i=1}^n f(x_i)\right)$$ defined for $n \leq N$ and where $C > 0$ is some constant. I am trying to ...
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### Constructing a continuous map for an interval with some points of period x, but none of period 3

I was given this question where I have to construct a continuous map f:I -> I (interval) with a point of period 4, but none of period 3. I know that thanks to Sharkovskii's theorem that if it had a ...
1 vote
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### Is Devaney chaos maintained by topological semi-conjugacy?

A dynamical system is called Devaney chaotic is it is (i) transitive, (ii) periodic points are dense, and (iii) the system depends sensitively on initial conditions. My question is if Devaney chaos is ...
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### Limit cycles, simply and non-simply connected regions

I have a few questions about the negative Bendixon criterion. In order to present my doubts, I organize this post as follows. First, I present the theorem and its interpretation. Second, I present a ... 1 vote
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### Cyclic Composition Operators on Hardy Space

Let $H^2$ denote the Hardy space on the complex disk $D\subseteq \mathbb{C}$. Recall that for a function $f:D\rightarrow D$ the associated composition operator $C_f$ is defined by  \begin{aligned} ...
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1 vote
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### Can quadrics be applied to the n-body problem?

Gravitational orbits within the 2-body problem can be visualized as conics on the surface of a double cone. Is it reasonable to imagine that 3-body systems can be visualized as quadrics on the surface ...
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### A function where an infinitesimal difference in intial conditions grows into a finitesimal difference in final conditions, within finite time?

The typical functions I see with finite Lyapunov times are merely exponential; they only generate (e times larger) infinitesimal differences in final conditions from infinitesimal differences in ...
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1 vote
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### Relationship between two objects with the same fractal dimension.

I've been doing some research into fractal dimension (namely Minkwoski-Bouligand dimension) for my university dissertation and have found two seemingly unrelated objects that have the same Minkowski-...
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I have a dynamical system where I know $x$ and $y$ as functions of time. How do I go about finding the Poincaré Map and Section in the phase space $y$ by $\dot{y}$? Like what are the steps I take?