Questions tagged [chaos-theory]

For questions in chaos theory.

607 questions
Filter by
Sorted by
Tagged with
13 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 &...
• 113
7 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))...
69 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. ...
• 1,387
35 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
73 views

1 vote
53 views

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 ...
36 views

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 ...
• 11
1 vote
49 views

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-...
• 43
1 vote
58 views

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 ...
• 223
26 views

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 ...
• 223
12 views

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,...
• 145
33 views

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?
• 31
28 views

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 ...
27 views

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 ...
12 views

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 ...
• 141
1 vote
45 views

44 views

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. ...
48 views

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, ...
• 263
89 views

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 ...
• 263
38 views

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? ...
• 1,373
32 views

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 $b$, ...
22 views

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 $x(0)\in(-\epsilon, \epsilon)$,...
• 1,387
52 views

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 ...
• 3,171
44 views

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?
• 11