Tagged Questions

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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 $\left\{a_n\right\}$
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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. ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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 ...
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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 ...
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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 ...
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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?
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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 ...
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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 ...
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 ...
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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 ...
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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 ...