Questions tagged [chaos-theory]

For questions in chaos theory.

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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 &...
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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))...
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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. ...
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1 answer
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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 ...
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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, ...
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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 ...
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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 ...
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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 ...
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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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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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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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Find periodic points of quadratic polynomial (doubling angle map)

I have a question related to doubling map defined as: $D(x)= 2 x(\mod 1), x \in [0,1) $ and quadratic polynomial $Q_0(z)=z^2$, $z \in \mathbb{C}$. The doubling map is chaotic, and it can be shown ...
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Can we apply Singer's theorem to find maximum number of attracting cycles for certain mapping

Let $h(x) = \mu \cdot \sin(x) $, $0\leq x \leq \pi$, $0< \mu <\pi$. Find maximum number of attracting cycles. I used Singer's theorem (from Elaydi, Discrete Chaos) which says: "Let $f$ be ...
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Approximating a function with sufficient accuarcy with chaotic attractors

Imagine, we have a complicated function $f : \mathbb{R}^n \rightarrow \mathbb{R} , y = f(t)$ Now, consider a number of maps, possibly chaotic : $g_i : y_{n} \rightarrow y_{n+1}$. Question : Can we say,...
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Computing a Poincaré Map and Section

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?
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Does this paper make valid statements regarding Non linear dynamics?

This paper claims non linear dynamics has something to do with the brain, does he claims it makes in "Tension Domain " make sense? Is there really anything called "Phase Tension ...
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Numerical Convergence of a dynamic chaotic system [duplicate]

I'm doing a pendulum problem using Verlet method, aiming to illustrate the chaotic behaviour. For that reason, I perform two simulations with $\Delta \theta$ of about 10^{-10} between them. The ...
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Neutral saddle cycle and its interpretation in codim 2

Suppose the following scenario. 1-parameter limit cycle family at paramater value $\mu_1$ undergo neutral saddle bifurcation, i.e. Floquet multipliers satisfying $\mu_i\mu_j = 1$ and it touches ...
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(Hints only please) Prove the function $L(x)$ is chaotic on $[0,1]$.

Prove that the function $$L(x)=\left\{\begin{array}[cl] \displaystyle 3x & \text{if } x\leq \displaystyle\frac{1}{3} \\ \displaystyle\frac{3}{2}-\frac{3}{2}x & \displaystyle\text{if } x>\...
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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 ...
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1 answer
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Relation between topological entropy and metric entropy

It is well known that, for a dynamical system $T$ on a metric space $(X,d)$, the variational principle connects the definition of metric entropy and topological entropy. In other words, if $$M(X,T) := ...
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No Chaos in $\mathbb{R}^2$

While reading some basic introductory texts in nonlinear dynamics, it was asserted that no chaotic behaviour for flows can occur in $\mathbb{R}^2$ because of the Poincare-Bendixson Theorem. ...
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2 votes
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Non-trivial integrals of motion of a double pendulum?

I was always assuming that a double pendulum does not have any non-trivial integrals of motion besides the Hamiltonian, but after asking this question I tried to simulate the motion of the pendulum, ...
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Double pendulum is non-ergodic?

I got really curious about ergodicity in double pendulum. The diagram below represents the time it takes for a double pendulum to "flip" when started from different initial positions. The ...
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Are there "half step", or other subdivisions, of the Mandelbrot iteration function?

The Mandelbrot Set is generated by iterating $f(z)=z^2+c$. Is there a "half-step" function $g(z)$ such that $g(g(z))=f(z)$? What is the method for finding such half-iteration functions? ...
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Henon map and its parameter analysis

I am trying to create an encryption algorithm for educational purpose. Where I use Henon map to encrypt a message. In the Wikipedia section I got, "The map depends on two parameters, $a$ and $b$, ...
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Solutions depending continuously on the initial condition do not imply a non-chaotic state?

In Glendinning's book Stability, Instability and Chaos (theorem 1.2), he said that if in the ode $\dot x = f(x,t)$, $f$ is smooth near $(0,0)$, and the initial condition $x(0)\in(-\epsilon, \epsilon)$,...
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How to show maps conjugate to tent maps are not structurally stable

I'm given the maps $F_4(x)=4x(1-x)$ on $[0,1]$, $G(x)=4x^3-3x$ on $[-1,1]$ and $H(x)=8x^4-8x^2+1$ on $[-1,1]$. I managed to show that all these maps exhibit chaotic behavior and are conjugate to the ...
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Why does the Mandelbrot Set only bifurcate on real values?

If you take a look at the figure below, the Mandelbrot Set seems to only bifurcate when it collapses into a real value (without an imaginary component). Why does this occur?
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Stable and unstable manifold for a 2D dynamical system

In "Differential Equations" by Stephen Wiggins (Example F.45) it is said that for the simple dynamical system $$\dot x=\lambda x$$ $$\dot y=\mu y$$ where $(x, y) \in R^2$ and $\lambda, \mu &...
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1 answer
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What is the probability density function on solutions to the Lorenz system?

Consider the Lorenz system with parameters $\sigma = 10$, $\rho = 28$, and $\beta = \frac{8}{3}$: $$\dot{x} = 10(x - y)$$ $$\dot{y} = x(28 - z) - y$$ $$\dot{z} = xy - \frac{8}{3}z$$ I'm interested in ...
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Is correlation dimension vs embedding dimension analysis and K2 entropy valid for a discrete variable?

I have a dataset generated from a finite symbol space. Is testing how the correlation dimension grows as a function of the embedding dimension (for a suitable lag) valid to check for the presence of ...
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Answers to Hirsch and Smale Differential Equation Textbook

I was wondering if anyone studied from Hirsch, Smale, Devaney differential equations, dynamical systems, and an introduction to chaos (2nd edition). Where did you get the answers to the excerise? I ...
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1 vote
1 answer
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Lyapunov exponent of a stable p-cycle.

I'm following an example from Nonlinear Dynamics and Chaos (Strogatz) that asks to show how if $f$ has a stable $p$-cycle then the Lyapunov exponent $\lambda<0$. I understand why this must be the ...
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Conditions for a transcritical bifurcation to occur in a 1D dynamical system

Suppose we have a 1D dynamical system $\dot{x}=f(x,\mu)$. What are the most general conditions on $f$ and its derivatives which guarantee that a transcritical bifurcation occurs? I have seen ...
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Is there an example of a continuous chaotic mapping?

I studied a bit of chaos in college, but from my learning, I've never known a mapping that does not consist of discrete orbits. For example, take the Mandelbrot set, where the points on the mapping ...
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What is the difference between Lyapunov Exponent vs time and Lyapunov Exponent vs parameter?

What is the difference between Lyapunov Exponent vs time and Lyapunov Exponent vs parameter? I noticed that some plot graph of Lyapunov Exponent but are not usually the same x-axis. Some plot graph ...
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Is there deterministic chaos outside of physical systems?

Wikipedia introduces deterministic chaos as [Small differences in initial conditions yielding widely diverging outcomes] can happen even though these systems are deterministic, meaning that their ...
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Existence of Non-constant Orbitally Stable Periodic Solution

Given the equation $x''+(a^2-1)(x^2-4)x'+2x=0$, I am asked to find the all values of $a$ such that the equation has a non-constant periodically stable solution. My attempt: First off, let's determine ...
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What are the monodromy and saltation matrices of an ODE system?

Assume we have the following ODE system: $$\mathbf{y}^{(n)} = \mathbf{F}\left(x,\mathbf{y},\mathbf{y}',\mathbf{y}'',\ldots, \mathbf{y}^{(n-1)} \right),$$ which is unfolded into: $$ \begin{pmatrix} y_1^...
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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 ...
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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. ...
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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 ...
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Proving existence of roots

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)) = ...
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Consider the set, recursively built, starting from the unit interval and removing the first $\frac{1}{3}$ at each step. Find the similarity dimension. [closed]

My thinking for this question is that it is just a slight variation of the standard Cantor set and will therefore have the same similarity dimension. My logic is that at each new step, the interval ...
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