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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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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In the definition for sensitivity to initial conditions what exactly does the distance between trajectories mean?
I've seen this definition for sensitive dependance in Modeling Life (Garfinkel et al, 2010):
$$d(M_{t} - N_{t}) = e^{\lambda * t} * d(M_0 - N_0)$$
or alternatively from Wikipedia:
$$ {|\delta \mathbf {...
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Properties of the bifurcation diagram for the logistic function
Once the bifurcation diagram has been plotted ($x_{n+1}=rx_n(1-x_n)$), there are 3 elements or properties that I don't know haw to explain, and I have not found any article where they are explored. ...
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Linearity of system of differential equations?
I am learning how to solve differential equations (ordinary and partial)and why they are so important for physics.One thing I have noticed so far is that we know so little on the nature of the ...
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Period $3$ orbit of the logistic map $x_{n+1}=r \cdot x_n(1-x_n)$
One can proof, that the logistic map has an stable orbit of period three for $r=1+2\sqrt{2}$. This can be done by looking at the third iterated of $f$ and investigate it for stable fixed points. For ...
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Chaotic behavior of the logistic map at $r=4$.
With Sarkovskii's theorem I want to conclude the chaotic behavior of the logistic map $f(x)=r \cdot x(1-x)$. I can't find a value of $x$ which leads to a periodic three orbit. Does anyone know a value ...
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Periodic orbits of the logistic map
I have a question about the period orbits of the logistic map $f(x)=r \cdot x(1-x), r \in [0,4], x \in [0,1]$.
The bifurcation-diagram own for $r<3.5699$ only periodic orbits of period $ p=2^k,k \...
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Sharkovskii`s theorem and a period three orbit of the logistic map at e.g. $r=3.83$
Let $f:[0,1]\to [0,1]$ be continuous with a period three orbit.
We know with Sharkovskii`s theorem that f owns period orbits of any natural number.
If we take a look to the bifurcation-diagram of the ...
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Finding Critical Points of the Model $x=\exp(-rx)$ [closed]
Given the model $x_{t+1} = e^{-rx_t}$, how do I find the fixed/critical points? I know that critical points are $x = f(x)$ but how would you go about that here? Either as originally put or $\frac{\ln(...
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What is correlation dimension, actually?
I'm taking a course on chaotic dynamical systems, and we're talking about attractors with non-integer correlation dimensions, but I can't seem to find a satisfactory definition for this concept. ...
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Length of a circular permutation cycle
Question. Consider a transformation $T:\mathbb{R}^{m\times n} \mapsto \mathbb{R}^{m\times n} \,$ which collects the elements by columns and arranges them on rows.
After how many consecutive ...
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conjugacy of complex squaring map?
Defined here, it seems to me that if you start with a complex pair (z, z*) ie z and its conjugate, and iterate each one the same number of times that the conjugacy of the pair remains?
Possibly ...
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Double Pendulum Simulation Accuracy?
I have what I think is a very simple doubt, but one that I've never seen explicitly addressed.
It's a classic coding activity to simulate a double pendulum. You can code this simulation up more or ...
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real/integer quadratic mapping's real/integer k-periodic points
The quadratic mapping is discrete dynamical system
$$x_n = x_{n-1}^2 + c$$
There is systematic reference about periodic points when $x_n$ and $c$ are both complex. $k$-periodic points are $x_{n+k} = ...
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Distinguishing between chaos and multiperiodic oscillations from the Fourier spectrum
Consider a system which exhibits multiperiodicity, say with oscillations of the form
$$x(t) = \sum_{n=0} c_n \cos(n \omega_0 t), \qquad \lim_{n \to \infty} c_n = 0$$
The Fourier transform $\tilde{x}(\...
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When is the Lorenz system the Lorenz attractor?
I am currently studying chaos theory but am unsure what the difference is between the Lorenz system and the Lorenz attractor. Is the Lorenz attractor just the solution of the Lorenz system with $\beta=...
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How many different chaotic attractors are there?
Let's assume we have a $d$-dimensional dynamical system
$\dot{\mathbf{x}}=F(\mathbf{x})$
with $F: \mathbb{R}^d \rightarrow \mathbb{R}^d$.
Further assume $F$ can be expanded in a Taylor series such ...
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How would I solve a chaotic function for x?
I need a chaotic function and a way to unsolve it for a project I have, but I don't know how I would go about undoing a chaotic function such as ...
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Takens’ embedding theorem for multistable systems
In an informal way, Takens’ theorem states, that we can reconstruct a (chaotic) attractor of an $N$-dimensional dynamical system with a delay embedding of just one trajectory $x_i(t)$:
$(x_i(t),x_i(t-\...
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Multistable systems and their chaotic attractors
Let's assume we have a multistable dynamical system
$\dot{x}=F(x)$
Furhter assume it has at least two (chaotic) attractors $A_1,A_2$ with basins of attraction $B(A_1),B(A_2)$.
How much information ...
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Is there a way to plot my Euler outputs in a graph with the code i currently have?
enter image description here
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Why is the chaotic Lorenz System bounded in an ellipsoid
In all proofs I've found, there is a Lyapunov function; my question is: shouldn't a Lyapunov function be 0 on an equilibrium point?
In this case, I am using a Lyapunov function to determine the ...
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Significance of topological transitivity in chaotic systems
I am currently undertaking a research project on chaos theory during my final year of undergraduate study. I came across the three conditions in order for a system to be chaotic. Both having dense ...
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Continuity of the Largest Lyapunov exponent?
If I have a $C^1$-flow $\phi_t$ on $\mathbb{R}^n$. And $A$ is a minimal set of the flow ($A$ is a compact invariant set containing no proper positively invariant subset). In addition, $x\in A$ is a ...
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Three questions about the Space Filling nature of "Incommensurate Non-chaotic Trajectories" and "Chaotic Trajectories" in Phase Space
Before I ask my question, I would prefer to briefly explain what I know about Trajectories in Phase Space and about the Nature of Space-Filling Curves, so that any gap in my understanding, if any, ...
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Computing the $k$th iteration of the tent map in one step
Suppose we have the tent map, $T_\mu(x) = \mu \min(x, 1-x)$ with parameter $\mu = 2$ (so it is topologically conjugate to the logistic map with parameter $4$).
For any point $x_0 \in [0, 1] \cap \...
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Computing the $k$th iteration of the logistic map in one step
Suppose we have the logistic map with parameter 4, that is take
$$
f(x) = 4x(1-x).
$$
For any point in $x_0 \in [0, 1] \cap \mathbb{Q}$, does there exists an equation I can use to compute $f^k(x_0)$ ...
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How can the Lebesgue-Measure be an ergodic measure on [0, 1]?
In my class on ergodic theory there was a theorem, that all ergodic $F$-invariant measures $\mu$ for a Borel-function $F: X \rightarrow X$ are extreme points of the set $M_F(X) =$ {all $F$-invariant ...
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Is there a theoretical characterisation/closed form of the Lorenz-63 attractor?
For my research I'm learning about the Lorenz-63 system, and I have a relatively simple question that I can't find a straightforward answer to: are there any theoretical results about the shape of the ...
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Which values can be reached by period doubling bifurcation of the logistic Map?
We know, that the first period doubling bifurcation oscilates between $\frac{1}{2r}(r+1 \pm \sqrt{(r-3)(r+1)}$. We speak of the logistic map.
We see, that there is a quadratic term. Does that mean, ...
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Finding the original logistic map, given a binary sequence
Logistic maps can be used to generate a pseudo random sequence of binary digits. Here is an article.
But is there a method to reverse this?
Q1 : Given a sequence $b_0 b_1 b_2 ... b_n$, is it possible ...
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Prerequisites for the law in Feigenbaum universality
Situation: $$ x_{n+1}=f_{\mu}(x_n) ~ , ~ x_0 \in I \subset \mathbb{R}$$
Here, $f_{\mu}$ is a family of self-mapping functions from $I$ to $I$. We further know that we get period-doubling (for example ...
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How to quickly generate an equilibrium point of a strange attractor numerically?
In many strange attractors (for example the Lorentz system, given appropriate parameter values), the point that is described by the system's equations of motion seems to approach a manifold with a ...
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What is the relation between Poincaré sections and chaotic behaviour?
I've been studying Poincaré Sections. Here are some Poincaré Sections plots from the double pendulum.
I've read that, intuitively, when plotting a chaotic orbit through a Poincaré Section, it will &...
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How to understand the largest Lyapunov exponent?
I've posted the question in the physics site too.
It is said that
..the largest Lyapunov exponent, which measures the average exponential rate of divergence or convergence of nearby network states.
...
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Plotting the bifurcation diagram for Ikeda map
I'm trying to plot the bifurcation diagram for Ikeda map. I wrote a code in Python to get the points of this diagram, but it seems that for $u > 1$ the points diverge and my code doesn't work ...
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Show the system has one equilibrium point
I was wondering how we would show that the system:
$$\frac{dx}{dt}=-x^3+2x-4y \\
\frac{dy}{dt}=-y^3+2y+4x$$
has only one equilibrium point.
I have seen cases where the system is, for example:
$$\...
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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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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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