Questions tagged [chaos-theory]

For questions in chaos theory.

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Questions about Lyapunov Exponents of quasi-periodic orbits or in the conservative system

I have three questions about the two-dimensional discrete dynamical system. Is it true that the orbit is called quasi-periodic if its Lyapunov exponents include zero? If the system is conservative, ...
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12 views

What part of the Lyapunov spectrum is conserved in a time-delay embedding under Takens' theorem?

My understanding is that time-delay embeddings are often used to estimate the maximum Lyapunov exponent of a chaotic system for which we may not have full state measurements. It seems that you cannot ...
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13 views

Fixed, Stable Points in a Linear Map

I am reading a proof that goes through the basics of a linear map and how we prove that a fixed point is stable. My question arises from what I assume would be an attractor. If an input maps to itself ...
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how to prove that given a, b constants the function $f(x)=ax+b$ is not chaotic

I have thought about finding a stable orbit but the writing confuses me. Be $\epsilon>0$ I take $y\in$ b ($x, \delta$) and $\displaystyle{\delta=\frac{\epsilon}{|a^n|}}$ then for |x-y|<$\delta$ ...
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1answer
22 views

Laplace transform of solution of an integral equation for non-recursive logistic map computations

Consider the logistic map, $x_n = r \: x_{n-1} \left( 1 - x_{n-1} \right)$ If we generalize this to a complex function $f : \mathbb{C} \mapsto \mathbb{C}$, we get, $f \left( z \right) = r \: f \left( ...
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Periodic functions and Lyapunov exponents

Suppose $f(x)$ is periodic. Or even quasiperiodic. Does it follow that $f(x)$ has a zero top Lyapunov exponent? I'm thinking that if we say $f(x)$ is periodic, then $f(x)=f(x+2\pi)$ as an example. ...
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If you know why something happened, and why something didn't happen…

I'm not a mathematician, so go easy. I dropped out of high school. But I'm a programmer / logician / writer of algorithms. And I'm currently engaged in one of my most complex projects to date, which ...
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Generalizations of Feigenbaum universality for multidimensional maps and ones with multiple order parameters

Feigenbaum showed that for discrete 1D systems with a (smooth) unimodal evolution function, the route to chaos is universal, and depends only on the order of the map's maximum. Are there analogous ...
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1answer
42 views

Questions on doubling map D(x)

For the doubling map D on [0, 1): $$D(x)=\begin{cases} 2x & 0\leq x< \frac{1}{2} \\ 2x-1 & \frac{1}{2}\leq x <1 \end{cases}$$ List all points whose orbits end up landing on ...
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41 views

Equations for Mandelbrot bifurcation diagram

The Mandelbrot set is the set of all complex numbers $c$ that cause the function $z_{n+1} = z_n^2 + c$ to remain bounded within a circle of radius 2 when iterated from $z_0 = 0$. Looking at only the ...
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27 views

Mandelbrot set and logistic map connection

I'm currently writing an undergraduate thesis on chaos theory with a particular focus on the connection between the Mandelbrot set and the logistic map. I have found scattered posts on this site, ...
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14 views

Prove there exists no homotropy of a manifold in which a chaotic trajectory is periodic?

If you map the plane to the Riemann sphere or torus, you can technically prescribe infinity as periodic orbit. However, if your trajectory is chaotic, not even that case will occur, but what I ...
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22 views

What is the difference between these two systems?

I have these two models: Both show Temperature vs Time for 15 different input conditions (i.e. inlet fuel velocity). My question is: what is the difference between these two systems? Is one of them ...
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1answer
37 views

How to measure magnitude of chaotic behavior?

Not necessarily Lyapunov time. In a higher dimensional model, suppose you have many rooms in a house with various heating sources. This is then modeled by a large matrix of coefficients of heat ...
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Is a turbulent burner chaotic system?

This is a plot of temperature evolution with time of a specified flame zone of a highly turbulent combustion burner: Temperature vs Time Blue (Lo) = temperature evolution at a "low" speed ...
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1answer
87 views

Do chaotic systems exist that cannot be predicted even at the limit of inifinite precision initial conditions and infinite resources?

I have a layman person's understanding of the theory of chaos, that seems to indicate that using finite-precision initial conditions and finite computing resources, chaotic systems cannot be predicted ...
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31 views

Systems that Display Chaotic Behavior

I take a course in 'nonlinear dynamics and chaos'. For our final project, we have to choose a dynamical system in that is nonlinear and specifically one that displays chaotic behavior. I know that ...
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Continuous (Smooth) Fractal Zoom

This is my first ever post, I made an account to ask this question. Could have put it on a code forum but thought this challenge would be better suited to a mathematician with a programming foundation....
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1answer
52 views

are all the terms of this chaotic map 0s?

I'm trying to implement the algorithm proposed on this paper: A Simple Method for Image Encryption Using Chaotic Logistic Map but i cannot figure out the equation (7), i'm pretty sure that $X_0=0$, ...
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Monte-Carlo probability of chaotic dynamics - logistic map

Consider the map of the interval $x_{n+1}=4\mu x_{n}(1-x_{n})$, where $\mu\in[0,1]$. Using a Monte-Carlo technique, calculate the probability of finding a chaotic dynamicsin the parameter interval $\...
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Is 'level one chaos', as Yuval Noah Harari describes, a real concept in chaos theory?

In Sapiens by Yuval Noah Harari, it is mentioned that Chaotic systems come in two shapes. Level one chaos is chaos that does not react to predictions about it. The weather, for example, is a level ...
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1answer
70 views

Asymptotic mean in an one-dimensional chaotic system

I have the following one-dimensional dynamical system $$x_{t+1}=\begin{cases} ax_t,\quad\mbox{if }x_t\leq 1,\\ ax_t^{c},\quad\mbox{if }x_t\geq 1,\\ \end{cases}$$ where $x_0>0$ is an arbitrary ...
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How does Lyapunov time measure divergence?

"By convention, it is defined as the time for the distance between nearby trajectories of the system to increase by a factor of e." Okay, but in a Lorentz attractor, the distance between two ...
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1answer
20 views

Does linearization interfere with analysis of chaotic behavior?

If one has a nonlinear 3+ system of differential equations, then it is linearized, is the linearization capable of accurately portraying the chaotic behavior in some manner?
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29 views

value of parameter a such that $\frac{1}{2}$ is a 3-periodic point

Let \begin{cases} T_a(x) = ax & \text{ if} & x \in [0; \frac{1}{2}],\\ T_a(x) = a(1 − x) & \text{ if} & x \in [\frac{1}{2}; 1]. \end{cases} Find a parameter value of $a$ for which ...
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75 views

Book on Dynamical Systems

I am a Mathematics undergraduate and want to study about Dynamical Systems. Following are the courses that I have gone through: Logic and Set Theory Real Analysis(Including study of Topology of ...
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1answer
55 views

Proof that equicontinous and suriective dynamical system is distal

Let us have dynamical system $(X,f)$ (what means that $X$ is compact metric space with metric $d$ and $f: X \to X$ is continuous function). Moreover $f$ is surjective and equicontinous so when we take ...
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Bakers map : show that the distance between two points is less than $2^{-(N-1)}$

Baker Map The baker map $f: \Omega \rightarrow \Omega$ is given as follows, \begin{equation} f(x_1, x_2) = \left( 2x_1 \mod{1}, \frac{1}{2}(x_2 + [2x_1]) \right) \equiv (x_1',x_2') \end{equation} ...
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1answer
17 views

properties of a truncated tent maps on unit interval

given the following truncated tent map: $T_h: [0,1] \rightarrow [0,1], x \mapsto min(h, 1-2|x -1/2|)$ for $0 \leq h \leq 1$ my script in dynamical systems states the following properties: $T_1$ has ...
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29 views

What kind of local bifurcation occurs here?

I have encountered a bifurcation diagram for my five-dimensional non-linear ode system? I am really confused to identify the kind of bifurcation that occurs in the knowns forms of local bifurcations ...
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1answer
47 views

Are complicated systems more likely to be chaotic?

Though the simple pendulum isn't chaotic, the double pendulum is. Is it true in general that more complicated differential equations are more likely to be chaotic than simple differential equations? ...
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Nonlinear dynamics and Curl

I'm taking an introductory course in nonlinear dynamics inspired by the lectures of Steven Strogatz on the subject. The main goal of the course is to provide tools that enable one to qualitatively ...
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1answer
37 views

Fractal Surface Area of Lung

I'm preparing for an upcoming exam and one of the questions is the following; "A typical human lung has a volume of approximately 5 litres, and a fractal dimension of approximately 2.97. One way ...
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About calculation of the Lyapunov spectrum of network of oscillators, for heterogeneous nodes

Maybe this is nonsense and wrong but if we calculate the spectrum of the Lyapunov exponents (LEs), we can assign each LE to each node, right? If we have heterogeneous oscillators, we can have some ...
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Bios and the creation of complexity

I found some information about Bios series on this PDF file: https://www.societyforchaostheory.org/resources/files/00005/Bios_tutorial.pdf There is an equation and some constant which generates Bios, ...
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Can I learn and potentially make any contributions to the field of Chaos theory if I don't pursue higher education?

I am a Computer Science student right now, and I'm currently perusing my bachelors. Most of my classes are related to programming, algorithms, and logic. I have a desire to learn Chaos theory for ...
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How to distinguish chaos from inaccurate?

If you have a system of ODE that gives rise to a chaotic system, you can easily find that the solutions either explicit or implicit are drastically different from real-world results. Yet, this does ...
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49 views

is the number of digits in the decimal expansion of $2^x$ periodic?

I graphed the number of digits in the base $10$ expansion of the series $2^x$: At first, it looks like a repeating pattern in the plot but when I overlay and shift a sequence on top of that graph, it ...
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38 views

Book suggestion. [duplicate]

I am a phd scholar doing research in the area of mathematical modelling related to dynamical systems. I want to study manifold theory and does not know anything about it. So, can anyone suggest some ...
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47 views

How do I prove that the function $f(x)=tan(x)$ is Devaney chaotic on $\mathbb{R}$?

I need to prove that the function $f(x)=tan(x)$ is Devaney chaotic in $\mathbb{R}$. I know I need to prove that $f$ is transitive, has sensitive dependence on initial conditions and that the set of ...
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1answer
38 views

Lyapunov exponents in bounded systems [duplicate]

This is the definition used to numerically calculate the maximal Lyapunov exponent: $$ |\delta Z(t)| \approx e^{\lambda t} |\delta Z_0| $$ $$ \lambda \approx \frac{1}{t} \ln\left(\frac{|\delta Z(t)|}{|...
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1answer
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Find a $3$-cycle for a continuous function where $f(a) = b, f(b) = c, f(c)= d, f(d) = e, f(e) = a$.

I am working on dynamical systems (more specifically Sharkovskii) and I have to show there exists a $3$-cycle for a continuous function with $f(a) = b, f(b) = c, f(c)= d, f(d) = e, f(e) = a$ where $a&...
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Are there “well-behaved” functions which iterate infinitely without repeating while also being bounded?

In this video the presenter shows that for strictly increasing functions $f(x)=f^{-1}(x) \implies f(x)=x$ because when you iterate $f$ it either increases forever or decreases forever. We cannot solve ...
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1answer
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How to prove that a continuous function with a 5-cycle $\{a_1,a_2,a_3,a_4,a_5\}$ has two 3-cycles

I need to prove that a continuous function $f$ which has a 5-cycle $\{a_1,a_2,a_3,a_4,a_5\}$ with $a_1<a_2<a_3<a_4<a_5$ and $f(a_1)=a_2, f(a_2)=a_3, f(a_3)=a_4, f(a_4)=a_5, f(a_5)=a_1$ has ...
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Under what conditions can “generic property” delete in Takens' Embedding theorem

Formally, the Takens embedding theorem goes as follows: Let $M$ be a compact manifold of dimension $m$. For pairs $(\phi,y)$, where $\phi : M \rightarrow M$ is a smooth diffeomorphism (an invertible ...
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49 views

Bifurcations associated with a continuous dynamical system.

Let us consider the system \begin{align*} \frac{dx}{dt} &= -a \frac{xy}{1+x} - x + y\\ \frac{dy}{dt} &= -bx + cxy - d\frac{xy}{1+x} + ey \end{align*} $(0,0)$ is one of the fixed-point of the ...
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1answer
67 views

Chaos in duffing equation and universality

I was simulating a chaotic system (driven damped anharmonic oscillator) on my pc: $$ \begin{align} \dfrac {dx}{dt} &= v \\ \dfrac {dv}{dt} &= -x-x^3-0.3v+F \cos 2t \\ \end{align} $$ For ...
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19 views

What models employ the logistic map function, besides population growth?

The logistic map $x_{n+1} = r x_n(1-x_n)$ is used to predict controlled population growth such as in labs, and displays chaotic behavior, visible on a bifurcation diagram, for values beyond $3$ I ...
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65 views

Is the fixpoint $x=0$ stable, attracting, repelling for $f(x)= x\sin(1/x)$?

I'm working on exercise 1.5.8 from Goodson's Chaotic Dynamics. We have the function $f(x) = x\sin(\frac{1}{x})$ for $x\neq 0$ and $f(0)=0$. I have already proven that $x=0$ is not an isolated point. ...
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Detecting bifurcations in the logistic map

Having just finished James Gleick's book "Chaos", I thought that I would have a play with examining the behaviour of the Logistic Map myself. I plan to use an FFT to create a spectrogram of ...

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