Questions tagged [chaos-theory]

For questions in chaos theory.

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Example of Topologically Transitive Maps on $d$-dimensional Euclidean Space

What are some standard examples of topologically transitive maps on $\mathbb{R}^d$? I know of many on compacta but not on the entire space itself...
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Are chaotic systems generally non-differentiable w.r.t. initial conditions?

There is some important background to go over before this question will make any kind of sense, so before calling me out on my poor understanding of dynamical systems, chaos, and differentiabilty, ...
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Finding stable and unstable manifolds

Consider the system $x ^ { \prime } = 4 x + 2y ^ { 3 } \\ y ^ { \prime } = - 3 x$ Question: Find the stable and unstable manifolds around the fixed point $(0,0)$ and and sketch the phase portrait ...
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Showing that system has a periodic orbit and has a limit cycle.

$x ^ { \prime } = y + \frac { x } { \sqrt { x ^ { 2 } + y ^ { 2 } } } \left( 16 - (x ^ { 2 } + y ^ { 2 }) \right)$ $y ^ { \prime } = - x + \frac { y } { \sqrt { x ^ { 2 } + y ^ { 2 } } } \left( 16 - (...
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Plotting Phase Portrait for Nonlinear Damped Pendulum for larger damping

I'm asked to sketch the phase diagram near the equilibrium points of the nonlinear damped equation: $x ^ { \prime \prime } + k x ^ { \prime } + \sin x = 0$. I've found that for any integer $n$, $( n \...
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Does contuinity of time derivate implies continuity of the dynamical system?

Let $f$: $ \mathbb{R}^{n} \times \mathbb{R} \rightarrow \mathbb{R}^{n}$ $ (x,t)\rightarrow f(x,t)$ a dynamical system with: $\frac {df}{dt}$ define on every point, never null, and: $x \rightarrow \...
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54 views

Sarkovskii's Theorem

I asked my students a question in connecting with Sarkovskii's theorem in midterm exam: "Can a continuous function on $R$ have a priodic point of period 48 and not one of period 24?" This is similar ...
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Why does chaos theory talk too much about initial conditions? [closed]

Why does chaos theory talk too much about initial conditions? When the idea of attractors is the more important idea. It seems chaos theory revolves around the idea of an attractor not initial ...
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what is the Quanitity to measure the boundedness of non-linear recurrence relation

Consider the nonlinear first-order recurrence ${x_{n}=f(x_{n-1})}$ How can I find out if the recurrence stays bounded for an initial value of ${x_0}$. looking at the logistic map as an example, ...
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Determining the “chaoticness” of categorical time series data

For dynamical systems we can determine whether a system is chaotic by finding how much it diverges over time from two close starting points. We can also apply this concept to raw data instead of a set ...
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4th-order Runge-Kutta method for double pendulum numerical solution

I am trying to numerically solve the equations of motion of the double pendulum system using the 4th order Runge-Kutta method by a C++ code. system diagram Energy equations required equations for the ...
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How does the introduction of a wall in this chaotic single degree of freedom system double the primary steady-state response?

Consider the following system: System Image 1 This system has the following free body diagram: FBD 1 The equation of motion is given by $ \ddot{x}+2\omega_{n}\zeta\dot{x}+\omega_{n}^{2}x = \frac{F}{...
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Prove that non-attracting periodic orbits of $x \mapsto x^2 + c$ are in the Julia set [closed]

Let $c$ be complex number and let \begin{align} f_c : \mathbb{C} &\to \mathbb{C} \\ x &\mapsto x^2 + c \end{align} be the quadratic map. It is to be shown that if $O$ is a periodic orbit that ...
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Lorenz System - Stability of Fixed Points

I want to understand the stability of the fixed points of the Lorenz system of equations, given by: $$\dot x = \sigma\left(y-x\right)$$ $$\dot y = rx - y xz$$ $$\dot z = xy - bz$$ where $b, \sigma, ...
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Showing an equality using a dynamical system

I'm asked to show that $\sum_{n=0}^{+\infty} \frac{(x+y)^n}{n!} =\sum_{n=0}^{+\infty} \frac{x^n}{n!}\sum_{n=0}^{+\infty} \frac{y^n}{n!}$ using a dynamical system. However, I couldn't figure out how to ...
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Please help me prove this about Lyapunov Exponents!

Here, $\lambda(y_0)$ denotes the Lyapunov exponent of the logistic orbit starting at $y_0$. I tried starting like this: We have $F(x_k)=ax_k(1-x_k)$. Hence $|F'(x_k)|=|a-2a x_k|$, but I have no idea ...
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Poincaré sections and one dimensional maps

New to dynamical systems and chaos theory: many textbooks start the discussion on chaos with one-dimensional maps and its associated orbits. Some more math-heavy textbooks begin the discussion with ...
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Lyapunov Exponent of the Logistic Map

So I was working on this problem: Consider the logistic map $F(x)=ax(1-x)$ on the interval $[0,1]$. Show that for $3<a<1+\sqrt6$, $\lambda(y_0)=\frac12\ln|a^2-2a-4|$ for all $y_0\in(0,1)\...
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How to find the Interval $[a,b]$ for a contraction mapping for $f(x)=x-\cos x$ where $a,b$ exists in $[0,1]$?

So, using Newton's method, I know the fixed point for the contraction mapping is approximately $0.739$. However, I'm not sure how to go about finding the interval for the contraction mapping. Every ...
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1answer
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Somewhat easy chaos theory question/summation

Show that for $1 <\mu< 3$ and $\mu\neq2$, the Lyapunov exponent of the logistic map $F_\mu(x)=\mu x(1-x)$ is given by $\lambda(x)=\ln|2-\mu|$. My work: We have $F_\mu(x(k))=\mu x(k)(1-x(k))$. ...
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61 views

Proof problem: Show that f is a 3-cycle (redo)

I'm trying to figure out how to solve this problem where, Let $f:R\rightarrow$ $R$ have a cycle {${a_1, a_2, a_3, a_4, a_5}$} where $f(a_i) = a$i+1 , $i= 1,2,3,4$ and $f(a_5) = a_1$. If $...
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Distinguish the stable two-period points (basin of attraction) in a discrete dynamic system

The dynamic system simple as: $T :\begin{cases} n=f(n,m)\\ m=g(n,m) \end{cases}$ As we know, as initial value (n,m) varying, the system will have different cycle basin structure, just like the ...
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How is chaos related to stability?

I am having trouble understanding the concept of chaos in relation to stability: In chaotic systems, does the presence of chaos and a strange attractor indicate that there is no equilibrium? I read ...
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Find the orbit of a periodic point and show parity

I'm confused with problem 1 and 4 (this is chaos theory). I don't know how to begin and I'm having trouble following the example.
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26 views

Prove the map has a 10-cycle but no 6-cycles

Given the map $\tilde{f}:[1,13]\to[1,13]$, and $\tilde{f}(x)=\begin{cases} f(x)+8;& 1\le x\le 5\\ x-8;& 9\le x \le 13\\ \end{cases}$, which looks like the figure: Prove that the map has a 10-...
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Can synchronized chaotic systems be periodic or stochastic?

I am confused regarding the terms chaotic, periodic, and stochastic in the context of coupled chaotic systems: In general, if chaotic systems are coupled together and synchronized then does the ...
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Beginner level clarifications on chaos theory & its control

I am new to the world of chaos theory & control. I am trying to wrap my head over the actual implication of what synchronization would mean for physical systems such as a network of nodes where ...
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Is tossing of a coin deterministic experimemt?

This is a question that I practically encountered while I was playing a game: Is tossing a coin a deterministic experiment? It might seem silly to ask but I had some thought over it. By the ...
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Two interpretations of Chaos?

Broadly speaking, I cannot pin down what is meant by Chaos. I understand that (informally) if a dynamical system is highly sensitive to initial input data then this system is said to be chaotic. Eg ...
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Lorenz equations with negative parameters

Consider the Lorenz system $$\dot{x}(t) = \sigma(y-x) \, ,$$ $$\dot{y}(t) = x(\rho-z) - y \, ,$$ $$\dot{z}(t) = xy-\beta z \, .$$ Usually one considers the parameters $\sigma, \rho,$ and $\beta$ to be ...
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Lyapunov exponent relationship with chaotic behavior

Many papers used positive Lyapunov exponent as an indicator that a map has chaotic behavior and having sensitive dependence on initial conditions. See for example: Hua, Z., Zhou, Y., Pun, C. M., &...
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Is multifurcation possible?

I have recently read on bifurcation in chaos theory. Now, I have a simple question (which hopefully has a simple answer): is this possible to have a family of functions that can model multifurcation (...
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logistic map averaged values graphed

For the logistic map: https://en.wikipedia.org/wiki/Logistic_map For a given x, taking the average of n values on the logistic map, gives a converging value. ie for the x domain 3.56 <= x < 4, ...
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Logistic map confusion

Logistic map is a simple example of discrtete dynamical systems defined as $$x_{i+1}=\lambda*x_{i}*(1-x_{i}).$$ It is known that for $\lambda=4$ this map shows a chaotic behavior for $x\in (0,1)$. ...
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If $1-\frac{f(x)}x$ converges to $0$ on (finite) iteration, when does $f(x)$ also converge on iteration?

If $1-\frac{f(x)}x$ converges to $0$ on (finite) iteration, when does $f(x)$ converge on iteration? Let $f:\Bbb Q\to\Bbb Q$ I will write it more precisely to make plain that I'm considering ...
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Chaos and continuous dependency of ODE solution on initial condition

From the textbook of nonlinear dynamics1, a theorem about continuous dependency of ODE on initial data: Theorem: My question What's the relationship between chaos (the so-called sensitivity on ...
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Underlying structure of phase diagrams

I'm doing research for my master work and notice that there is no information about the curves (seem to be smooth) and they are very clear on the diagrams. Bifurcation Diagram Inside of bifurcation ...
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Are there any methods to evaluate if draws of a specific lottery are truly random?

Just to be on the safe side this question is NOT about predicting winning numbers in a lottery etc. I just wonder (since there are people that think that xyz lottery company does not reward jackpots ...
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Does the set of points where the logistic map is chaotic have positive measure?

Let $r$ be a real number in $(0,4)$, let $x_0$ be any real number in $(0,1)$, and define a sequence: $$ x_{n+1} = rx_n(1-x_n). $$ This is the logistic map. For some choices for the value of $r$, the ...
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Mathematics of symmetry in chaos [closed]

I'm a neuroscience PhD student. Recently I became so interested in the mathematical equations involving order and disorder and complex plane. I read a book named "Symmetry in chaos" by Martin ...
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Question about an example of topologocally mixing map

How to show that the doubling map $T:[0,1) \to [0,1)$ given by $T =2x $ (mod $1$) is topologically mixing? Topologically mixing means that for any pair of open sets $U,V$ there is a large $N$ such ...
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How does Smale's horseshoe map work?

I am reading on the Horseshoe map, but all I have found are qualitative explanations. These are great, but I think I would understand it better if I also had access to an example with an explicit ...
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Proving a system is not chaotic

Suppose I have a discrete dynamical system defined as a map $f: X \to X$ where $X = [0,1]^N$ i.e. the $N-$dimensional unit hypercube. The map $f$ is discontinuous (specifically it is piecewise affine,...
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120 views

Difference between logistic map and logistic equation

I'm doing a research project on Chaotic Encryption using the logistic map(?). I'm still in my early stages, and my professor made the following question: What's the difference between the logistic ...
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Physical representation of the Lakes of Wada

While on the wikipedia page for the Lakes of Wada, it stated that an object (picture 1) was able to provide a representation (picture 2) of the Lakes of Wada. How is this picture a representation of ...
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Can the Mandlebrot set be extended to quaternions?

What would the mandlebrot set look like with quaternions? Does there exist any any visualizations of what it would look like? Are there any visualizations in 3D (maybe with 1, i, and j being space ...
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What is the angle between the two wings of Lorenz's strange attractor? [duplicate]

When reading about chaos theory, I am often presented with a picture of a butterfly shape formed by two spirals - Lorenz' strange attractor. I presume that there is a specific function that generates ...
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1answer
99 views

How to make the function of the planes from the lorenz attractor

How can I make the equation for the planes of the wings of the lorenzattractor, I know that the critical point should be used. But I don't know how I should make this plane, the figure shows in ...
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1answer
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I understand how Orbits are defined but I don't understand one of the statements about orbits.

“The orbit of a point x in [0,1] for the tent map T is orbit(x) = {Tn(x): n = 0,1,2,3,...}” This makes sense to me as a definition, suppose our tent maps μ = 2, than for n = 1, orbit(1/2) = {1,0} ...
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48 views

This is an application of Hartman Grobman Theorem?

Let $p \in M$ be a hyperbolic periodic point of $f \in Diff^r(M)$. Show that given $n \in \mathbb{N}$ there exists a neighbourhood $V$ of $p$ such that any periodic point of $f$ in $V - \{ p \}$ has ...

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