# 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.

6,260 questions
Filter by
Sorted by
Tagged with
0 votes
1 answer
24 views

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

According to the proof of Lyapnuov's theorem given in [1] the assumption of continuity of partial derivatives is necessary to prove asymptotic stability while for simple stability it is not. I wonder ...
0 votes
0 answers
32 views

### 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 ...
0 votes
1 answer
27 views

### 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 ...
• 397
0 votes
0 answers
8 views

### 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.
• 111
1 vote
0 answers
15 views

• 63
2 votes
1 answer
29 views

### 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 $x\in X$...
• 321
3 votes
1 answer
64 views

### "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 ...
• 129
1 vote
1 answer
36 views

### 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 ...
• 86
4 votes
0 answers
57 views

0 votes
1 answer
19 views

• 1,397
3 votes
1 answer
71 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. ...
• 1,397
2 votes
1 answer
78 views

### 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 $U, V$ ...
• 61
4 votes
1 answer
129 views

### 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 ...
2 votes
0 answers
23 views

### 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 ...
• 105
1 vote
0 answers
20 views

### 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 ...
• 113
0 votes
0 answers
18 views

1 vote
0 answers
12 views

### Questions about Ratio Set of a Dynamic System

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 $g$, $g$ ...
2 votes
1 answer
32 views