# Questions tagged [dynamical-systems]

In dynamical systems, the motion of a particle in some geometric space, governed by some time dependent rules, is studied. The process can be discrete (where the particle jumps from point to point) or continuous (where the particle follows a trajectory). Dynamical systems is used in mathematical models of diverse fields such as classical mechanics, economics, traffic modelling, population dynamics, and biological feedback.

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### Notation for orbits in Ergodic Theory

Consider a map $T:X\to X$ from a set to itself. In ergodic theory, we say that the orbit of a point $x\in X$ is the set \begin{equation} \{x,Tx,T^2x,T^3x,\cdots\}.\end{equation} Is there any standard ...
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### Locally asymptotic stable equilibrium of a non-linear difference equations system

Theorem 4.8 in the book "discrete dynamical system" gives a sufficient condition for an equilibrium to be locally stable in a nonlinear difference system. It basically says the following: a ...
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### Jacobian of a trajectory given by a matrix exponential

I need to get the jacobian of the function $x(t) = e^{At} x_0$. so, I was thought about applying the vectorization and Kronecker product: $d \, vec \, x = (x_0^T \otimes I) \, d \, vec (e^At)$. But ...
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### Schwarzian derívate of a polynomial with different real roots

I’m working on the exercise 7.5.2 of the book Brin and Stuck introduction to dynamical systems. It says: Show that any polynomial with distinct real roots has negative schwarzian derívative. The ...
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### Extension of pseudo-orbit for continuous action

Let $G$ be a finitely generating set and $H\leq G$ be a subgroup with finite index, this means that there is finite set $\{g_i\}_{i=1}^n\subseteq G$ such that $G= \bigcup_{i=1}^n g_iH$. Let $A$ be a ...
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### Soft question - Index theory in nonlinear dynamics vs Complex analysis

The video https://www.youtube.com/watch?v=wZvFKcQ_3Rc&t=8s mentioned something called the Index Theory. I can't find it on wikipedia. Where could I find more about the theory? Here index is just ...
1 vote
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### Unipotent closure in classical groups

Let $G=\mathrm{SL}_n(\mathbb{R}),\mathrm{Sp}_{2n}(\mathbb{R}),\mathrm{Spin}_n(\mathbb{R})$ be a semi-simple simply connected classical group, $\Gamma\subset G$ a discrete and cocompact subgroup. Then ...
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### Question on the proof of $\mathrm{scl}(a) = |\mathrm{rot}(a)|/2$ in $\mathrm{Homeo}^+(\Bbb R)^{\Bbb Z}$

I'm currently reading a part of Calegari's scl and encounter a proof I can't understand well. Theorem 2.43. Let $\mathrm{Homeo}^+(\Bbb R)^{\Bbb Z}$ denote the full preimage of $\mathrm{Homeo}^+(S^1)$ ...
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### Is it possible to derive an explicit expression for the inverse branches of $f(x)=x + x^{1+p}\mod 1, p > 0, 0\leq x\leq 1$?

My dynamics course has talked a bit about the Pomeau-Mannville map, defined as $f(x)=x + x^{1+p}\mod 1,p > 0, 0\leq x\leq 1$. I was wondering whether there is any known trick or a special function ...
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### nonlinear odes: stabilizing terms in a subcritical pitchfork bifurcation

I am reading through Strogatz's book on nonlinear odes and dynamical systems. One thing that is a little confusing is his description of stabilizing higher order terms to control the dynamics of a ...
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### Differences between Poincare map and Poincare section

I am self-studying dynamical systems, and wanted to double-check my understanding of these concepts. In Strogatz's "Nonlinear Dynamics," the author plots a periodic solution to the forced ...
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### Reference for introductory topology for dynamical systems

I'm going to take an introductory class on dynamical systems at the level of Guckenheimer & Holmes, or Arnold's ODEs book, but I don't have any topology background. I know analysis at the level of ...
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### Do other resources exist on Rate-Independent Systems?

Are Rate Independent System new in the world of mathematics? I can't tell because I certainly am new to mathematics. I couldn't find any more resources than this book: Rate-Independent Systems: Theory ...
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### Inconsistency Between Approaches Solving System of Differential Equations

This requires a bit of setup, so bear with me. I'm working through a physics problem. The physics itself is irrelevant, but the setup gives you a system of differential equations. The details in how I ...
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### Modulational instability: Structures beyond linear stability analysis

Modulational instabilities occur in waves which occur in nonlinear system. A linear stability analysis is done to examine the dynamics in the short-time regime to observe the structures which ...
### $\ln(\exp(x) + \exp(y) + \exp(z)) = x + y + z$ , iterates of $x - \ln(\exp(x)-1)$ and generalizations
Let $x,y > 0$ Consider the equation $$\ln(\exp(x) + \exp(y)) = x + y$$ Now it is clear (by symmetry) that if we express $y$ as function of $x$ or $x$ as a function of $y$ that is the same function. ...
### Behaviour and limits of $f(n+1) = \frac{f^5(n)}{2} - f(n-1)$
Let $f(0) = 0,f(1) = \frac{1}{2}$ and $$f(n+1) = \frac{f^5(n)}{2} - f(n-1)$$ where $*^5$ is a power. Then it seems $$\sup f(n) = \lim \sup f(n) = \frac{1}{2}$$ and  \inf f(n) = \lim \inf f(n) = \...