0
votes
1answer
18 views

Is a cascaded chaotic system is still chaotic?

I am curious whether a new system which cascades two individual chaotic systems is always chaotic. My feeling says if two chaotic systems $S_1$ and $S_2$ satisfying the constraints that $${\rm ...
0
votes
1answer
46 views

Differential Equation: Periodicity of a circle with zero radius in polar coordinates

I am given the following diff. equation in polar coordinates: $$\dfrac{dr}{dt} = r(1 + a~\cos \theta - r^2) \\ \dfrac{d \theta}{dt} = 1$$ where $a$ is a positive number and is less than $1$. I am ...
0
votes
1answer
30 views

Periodic points of topologically conjugated functions in dynamical systems?

I'm working on a homework problem which seems obvious, but I am having a hard time proving/completing. The problem can be stated as follows: Let $f,g:$ $\mathbb R$ $\rightarrow$ $\mathbb R$ be ...
1
vote
0answers
51 views

Lyapunov-exponent and rate of convergence of continued fractions

I'm writing an essay on the Lyapunov-exponents of the Gauß map \begin{align} T:[0,1]&\to[0,1],\\ x&\mapsto \begin{cases}\frac 1 x \mathrm{mod}\,1 & \text{for $x>0$,}\\ 0 & \text{for ...
1
vote
2answers
50 views

Prove sensitivity to initial conditions numerically?

How can I prove sensitivity to initial conditions numerically? I mean directly from the computed data and neglecting the dynamical system that originated the data. The data comes from hybrid ...
2
votes
1answer
45 views

Are deterministic RNGs chaotic systems?

Deterministic random number generators (RNG) are designed to provide faithful approximations of a uniform distribution. Given that a deterministic RNG always gives the same sequence for a given ...
2
votes
0answers
36 views

Explicit form of strange attractors

Are there any examples of continuous-time dynamical systems possessing strange attractors for which there exist explicit formulas describing these attractors? Many thanks in advance and apologies if ...
2
votes
2answers
67 views

where did the term $\omega$-limit set originate from?

What it says on the tin. I've always used the phrase 'in the limit of all things' but hearing '$\omega$-limit' in a chaos theory class has me wanting to use the term. That said, I'd feel really ...
2
votes
0answers
109 views

“is topologically mixing” vs. “is topologically transitive” in the defition of chaos

Wikipedia gives essentially "is topologically mixing and has dense periodic periodic orbits" as the definition of chaos, and this paper shows that its (the paper's) definition of chaos is equivalent ...
1
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0answers
45 views

Discrete vs Continuous Dynamics

Where does the problem arise concretely in considering a discrete process as a sampling of a continuous process? Can we use continuous methods of solving say differential equations and find the ...
1
vote
1answer
75 views

Coupling in the circle map

I'm currently investigating Arnold tongues (areas in parameter space with rational rotation numbers $\rho$, ie. $\rho(\Omega, K) \in \mathbb{Q}$) arising when iterating the circle map $$ ...
4
votes
3answers
243 views

Discuss the convergence of $ \left \{ a_n \right\} $ where $ a_{n+1}=\frac{a_0}{2}+\frac{a_n^2}{2},n\geq 1 $

Let $$ a_{n+1} = \dfrac{a_0}{2} + \dfrac{a_n^2}{2} $$ where $ a_1 = \dfrac{a_0}{2} $ and $ n\geq 1 $ Discuss the convergence of $ \left\{a_n\right\} $
2
votes
1answer
56 views

How are definitions of chaos related?

Chaotic systems can be defined in many ways. One definition is that the system has a positive Lyapunov exponent, that is, two trajectories starting near each other will diverge exponentially quickly. ...
0
votes
1answer
144 views

How does chaos arise in Hamiltonian systems?

I have a question about how chaos arises in Hamiltonian systems. I've been taking a upper year undergrad class on dynamics and it has focused on chaos a lot. So far however we have only seen the more ...
0
votes
1answer
78 views

Possible to make a flow that forms horseshoes on a 2-dimensional manifold?

It it possible to have a flow $\phi(t,x)$ on a 2-dimensional manifold where for some $t > 0$, the map $g(x) := \phi(t,x)$ creates a horseshoe? By $\phi(t,x)$ I mean the solution to the ODE ...
1
vote
3answers
74 views

Generating Bifurcation Animations

https://en.wikipedia.org/wiki/File:Hopf-bif.gif Does anyone know how this animation was produced? I could make it by stitching together snapshots (what I'm doing) but this seems primitive, especially ...
3
votes
2answers
105 views

Chaos (and logistic functions); what is it and is it truly chaotic?

I'm currently studying discrete dynamic models and I am now reading about the logistic function $x_{n+1} = ax_n(1-x_n)$. Below there is a picture what happens with different values of a: These are ...
3
votes
1answer
58 views

Examples of systems conforming the Lorentz Attractor

Might sound like a trivial question but would you please show me some examples of real systems conforming the Lorentz Attractor? It can be any kind of system, just a little list. It can be a system ...
4
votes
2answers
147 views

Chaos without period doubling

I have been studying the Duffing oscillator rather intensively lately, mainly based on the theory in of the book by Guckenheimer and Holmes. From all that I have gathered, it seems that most dynamical ...
2
votes
1answer
117 views

How important are the following undergrad courses when trying to pursue studies in chaos theory/dynamical systems?

I'm currently a physics major with a year left, and deciding whether to switch into mathematical physics, mathematics or applied mathematics. I'm definitely switching into one of them, as I can meet ...
1
vote
1answer
294 views

Lorenz equations and find a minimal trapping region.

Consider Lorenz's equations $x^{'}= \sigma (y-x)$ $y^{'}= (rx-y-xz)$ $z^{'}= (xy-bz)$ $\sigma, r, b>0$ are parameters of the system. The question is as follows Show that there is a certain ...
1
vote
2answers
122 views

Cat map like maps without period

Is there any area-preserving chaotic map other than Arnold cat map which can be applied on a rectangle as well as being reversible but not periodic?
2
votes
1answer
104 views

Behaviour of deformation retraction of an unbounded interval to a point around t=1.

Let us have an unbounded interval, notably the whole real line no matter how close to one you are. Now we would like to construct a deformation retraction to one point, say $0$. $f_t:\mathbb{R} ...
3
votes
1answer
100 views

The extent of chaos

In chaotic systems the typical situation is that at a low level trajectories of points are wild, but overall there is a nice statistical description of the system. For example, consider the ...
9
votes
2answers
2k views

How to go about studying chaos theory/dynamical systems/fluid dynamics in grad school with a physics background?

I'm turning to you as I find myself in need of help as to how to best go about studying something related to chaos theory/dynamical systems/fluid dynamics in postgraduate school. I'm currently in my ...
6
votes
2answers
187 views

Prove that Anosov Automorphisms are chaotic

Let me add some detail first. An Anosov automorphism on $R^2$ is a mapping from the unit square $S$ onto $S$ of the form $\begin{bmatrix}x \\y\end{bmatrix}\rightarrow\begin{bmatrix}a && b \\c ...
1
vote
1answer
423 views

Find the values of r at which bifurcation occus and classify those as saddle node, transcritical, or pitchfork bifurcation

$$f(x)= rx-\frac{x}{1+x}$$ Find the values of $r$ at which bifurcation occus and classify those as saddle node, transcritical, or pitchfork bifurcation. I found the fixed points as ...
1
vote
2answers
452 views

Use linear stability analysis to classify the fixed points of the following system.$ f(x)=ax-x^3$ for $a>0$, $a=0$ and $a<0$.

Use linear stability analysis to classify the fixed points of the following system. $f(x)=ax-x^3 $ where a can be positive, zero or negative. I have found that for $a>0$ we have $2$ fixed points ...
3
votes
3answers
583 views

Beneficial to touch theoretically deeper texts earlier in core areas e.g. analysis, algebra?

Desired future direction: Dynamical System(Chaos), PDE More beneficial to read theoretically deep, modern and masterpiece texts earlier, (e.g. levels like UTX/GTM/GSM/LNM/CSAM) ? Especially in core ...
2
votes
1answer
295 views

Classification of bifurcations

I'm looking at this following equation $\dot{x} = \frac{dx}{dt} = \frac{x^2}{x^2 + 1} - rx$, and trying to classify the bifurcations that appear when the parameter $r$ is varied. I'm pretty much ...
1
vote
1answer
177 views

Plot bifurcation diagram from time series chaotic data

I have equations for Chua's circuit and need to plot bifurcation diagram. From the things I have read so far, I need to use 1-dimensional map to get the bifurcation diagram, but I have trouble ...
1
vote
1answer
87 views

Shuffling cards and the horseshoe map

I wonder if there is a connection between the dynamics of repeated cut & shuffle operations on a deck of cards, and topological chaotic maps such as the horseshoe map? I ask this entirely naively. ...
2
votes
1answer
103 views

Examples when Resonance Overlap fails to predict the onset of Chaos

In a Hamiltonian system Chirikov's resonance overlap criterion approximately predicts the onset of chaotic behavior. Furthermore in a system where resonances overlap, the strengths of the resonances ...