# 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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### Assumption of continuously differentiable function in the Lyapunov Stability Criterion

According to the proof of Lyapnuov's theorem given in  the assumption of continuity of partial derivatives is necessary to prove asymptotic stability while for simple stability it is not. I wonder ...
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### how the angle between vectors changes in a topological equivalence

I'm studying dynamical systems and i have the next problem: I have two dynamical systems in continuous time, let's say $\dot{x} =X(x)$ and $\dot{y} =Y(y)$ where there is a topological equivalence (or ...
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### Question regarding contraction theory

I am following the paper (A Study of Synchronization and Group Cooperation Using Partial Contraction Theory) to understand how contraction theory used to analyze coupled oscillators. On page 211, the ...
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### Complexification Averaging Method

I want to study about Complexification Averaging Method. What books do you suggest to start with? I want to completely understand this method and why should we use it. Thank you.
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### Definition of essentially T-invariant function

A function is $T$-invariant if $f(T(x))=f(x)$ for all $x\in X$. In text book: Introduction to Dynamical system by Brin, it defines essentially $T$-invariant: if $f(T(x))=f(x)$ almost every for $x\in X$...
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### "Spanning" of solutions of ordinary differential equations

Suppose we have a switched ODE $$\dot{x} = A_{\sigma(t)}x,$$ where $A_{\sigma(t)}$ is a constant matrix given $\sigma(t)\in\mathcal{M}=\{1,2,\cdots,m\}$. If we fix the initial condition and can ...
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### Necessity of the hypotheses of Lyapunov asymptotic stability theorem

In my ordinary differential equations course we saw Liapunov's theorem for asymptotic stability. I have a doubt about the necessity of the "negative definite" assumption. The statement we ...
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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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### Tent map is topologically transitive

Let $T:[0,1] \to [0,1]$ be the function $Tx= 2x$, if $x \in [0, \frac{1}{2}]$ and $Tx = 2-2x$, if $x \in ( \frac{1}{2} , 1]$. We say that a map is topologically transitive if, for any pair $U, V$ ...
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### Best betting strategy for an unfair random walk with a skewed payoff

Say you start with bankroll $B$ and i.i.d. random variables $U_i$ with distribution $p=P(U_i=r)>.5$ and $q=1-p=P(U_i=-1)$. Your earnings from bet $i$ is $W_iU_i$, where $W_i$ is your wager at step ...
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### Ruling out limit cycles in 2 dimensions

Let $f:[0,1]^2 \rightarrow \mathbb{R}^2,$ where $f_1(x,y) = g(y)-x$ and $f_2(x,y) = g(x)-y.$ Here $g(\cdot)$ is a strictly decreasing polynomial function such that $g(0)=1$ and $g(1)=0.$ I am ...
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### Can one obtain a Poincaré Section without having the explicit solution of the system of differential equations?

I'm studying Poincaré Sections. Every example I see, it calculates the solution $x(t)$ and then it uses it to calculate the Poincaré map. I don't see the point of the Poincaré map then. What ...
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Intuitively, I think I understand this, but more formally if we have a smooth vector field $$F(x,y) = \begin{bmatrix} f(x,y) \\ g(x,y) \end{bmatrix}$$ such that for a smooth plane curve $C: x \... 1 vote 1 answer 41 views +100 ### a question on visualizing a state of DAE and a question from a continous time nonlinear dynamics Could someone explain me what is the fundamental difference between the dynamical system of the kind$\dot x = f(x)$and$E \dot x= f(x)$where$E$is a singular matrix with real entries. For the ... -1 votes 0 answers 29 views ### Find real-valued closed formulas for the trajectory x(t+1)=Ax(t), where Find real-valued closed formulas for the trajectory$x(t+1)=Ax(t)$, where $$A=\pmatrix{−4 & 3 \\ -3 & −4}\quad \text{and}\quad x(0)=\pmatrix{1 \\ 0}$$ (image link) 3 votes 1 answer 47 views ### Help Proving that Set Invariant for a Dynamical System. I'm practicing for an upcoming test, and this one has been giving me some problems. Suppose we have $$\begin{cases} \dot x = x^2 + 2y - 4 \\ \dot y = -2xy \end{cases}.$$ Let$R = \{(x,y) : |x| \leq ...
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Given a dynamic system $(X, \Omega, \mu)$ ($\Omega$ the $\sigma$-algebra and $\mu$ a measure), we assume there is a group $G$ acting on the system in the sense that, for each group element $g$, $g$ ...