# Questions tagged [chaos-theory]

For questions in chaos theory.

607 questions
Filter by
Sorted by
Tagged with
4 views

### What is the relation between Poincaré sections and chaotic behaviour?

I've been studying Poincaré Sections. Here are some Poincaré Sections plots from the double pendulum. I've read that, intuitively, when plotting a chaotic orbit through a Poincaré Section, it will &...
6 views

### How to prove Boundedness of two 3D coupled chaotic c=system resulting into 6D system? [closed]

I have a two 3D chaotic system and I couple them to make a 6D system. How could I prove boundedness of the coupled nonlinear differential equation? The system description is like this: xdot(t)=f1(x(t))...
68 views

### How to understand the largest Lyapunov exponent?

I've posted the question in the physics site too. It is said that ..the largest Lyapunov exponent, which measures the average exponential rate of divergence or convergence of nearby network states. ...
33 views

### Plotting the bifurcation diagram for Ikeda map

I'm trying to plot the bifurcation diagram for Ikeda map. I wrote a code in Python to get the points of this diagram, but it seems that for $u > 1$ the points diverge and my code doesn't work ...
1 vote
68 views

### Show the system has one equilibrium point

I was wondering how we would show that the system: dx/dt=-x^3+2x-4y dy/dt=-y^3+2y+4x has only one equilibrium point. I have seen cases where the system is, ...
49 views

### 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?
27 views

### 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 ...
104 views

### 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 ... 78 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 ...
26 views

### 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
38 views

### 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 ...
44 views

### 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
6 views

1 vote
17 views

### Lyapunov dimension

I have a nonlinear differential equation system composed of 4 equations. I calculated Lyapunov's dimension of each of the states to be a little bit over 3 (say 3.11, 3.1, 3.13, 3.14). How can I ...
1 vote
73 views

22 views

### Calculating the Lyapunov exponents for the seir epidemic model

I am trying to numerically calculate the Lyapunov exponents for the seir epidemic model given as:  s^{'} = b - bs - \beta si \\ e^{'} = \beta si - (\alpha + b)e \\ i^{'} = \alpha e - (\gamma + b) i ...
35 views

### Equation for curves of high density in bifurcation diagram

Is there an equation for the high density curves in the chaotic regions of the bifurcation diagram for the logistic map? I'm talking about the sinusoidal-looking dark curves in the following picture. ...
1 vote
64 views

### Measures of expanding maps of the circle and their coding

It is well known that the dynamics of linear examples $f(x)=mx(mod1)$ for natural $m\geq 2$ is semi conjugated to the full shift on the space of one-sided sequences of digits $\{0,…,m−1\}.$ Is it true ...
1 vote
I have the following arbitrary function which is the result of solving an iterative map for any period two fixed points (ie. for $g(x_n) = x_{n+1}$, I am trying to find $k$-values for which g(g(x)) = ...