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 ...

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80 views

Local stability + global attractivity = global asymptotic stability?

I was wondering how could I prove such a property stated in [Angeli, 2004]. For instance, consider the system $\dot{x}=f(x)$, where $f:\mathbb{R}^n\to\mathbb{R}^n$ is Lipschitz continuous. Claim. ...
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1answer
65 views

Stability of $\dot{x}=-A(t)x$

I already have an ODE of $A(t)$, that is $\dot{A}=-G(A(t)-A^*)$, where $G$ and $A^*$ are constant positive definite matrices. Thus I can deduce that $A(t)$ exponentially converge to $A^*$. Now I take ...
2
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1answer
22 views

almost periodic visits of a compact set of a circle group rotation

Let $\mathbb{T}$ be the circle group and $R:\mathbb{T\rightarrow T}$ an irrational rotation (hence a minimal system). Suppose there exists a compact set $K\subset\mathbb{T}$ such that for every ...
3
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2answers
66 views

Properties of matrix exponential

I know that the solution to system $x' = Ax$ is $e^{At}$, and I'm aware of various methods to calculate the exponential numerically. However I wonder if there are some analytical results. Namely, I'm ...
2
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0answers
25 views

How to measure intensity of attractors

Given a dynamic system with attractors, is there any way to measure the intensity of attractors? I mean intensity by the faster one point far away from the attractor moves to it, the higher intensity ...
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0answers
53 views

$2^n$ in base 10 and Dynamical systems

So in my Dynamical System course there is two problems about $2^n$ that I don't know how to solve first showing that there exists n such that $2^n$ written in 10 decimal forms starts with $123456789$ ...
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39 views

Shift and ergodic measures

Let $X = \{0,1\}^{\mathbb{N}}$ endowed with the product topology and $\sigma : \left\{ \begin{array}{ccc} X & \to & X \\ (\epsilon_n) & \mapsto & (\epsilon_{n+1}) \end{array} \right.$. ...
2
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2answers
61 views

Eigen values and vectors

$$ X'(t)= \begin{bmatrix}x'(t)\\ y'(t) \end{bmatrix} = \begin{bmatrix}-5 & -2\\-1 & -4\end{bmatrix}\begin{bmatrix}x(t)\\ y(t) \end{bmatrix}$$ Sketch the directions field for the system, plot ...
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1answer
53 views

Differential equation equilibrium points

Find the equilibrium points of the system $$\dfrac{dx}{dt} = y \\ \dfrac{dy}{dt} = x-x^3-y.$$ sketch the phase portrait.
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16 views

topological entropy of function with 3 period

How to prove that the topological entropy of the function with 3 period is strictly positive? I know that the function with 3 period implies chaos, but I cannot prove the topological entropy of it is ...
2
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0answers
53 views

Homoclinic orbit Hamiltonian system

The question is for which $a \in \mathbb{R}$ the system: $x'' + x - x^3 + a = 0$, has a homoclinic orbit. I let $y = x'$ so the system becomes: $x' = y$ $y' = x^3 - x - a$ and determined the ...
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1answer
31 views

“mass creating” flux in a conservation equation

$\phi: \mathbb{R} \to \mathbb{R}$ is a given, smooth gradient field and I come from the equation $\frac{\partial}{\partial t} u(x,t) = \text{div}(\phi(x,t))$ for $x\in \mathbb{R}$, $t\geq 0$ and some ...
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0answers
33 views

Convergence rate of Lyapunov exponent

For a random dynamical system the Lyapunov exponent is defined as: $$\lambda(x) = \lim_{n\to\infty} \sup \frac{1}{n}\log||A_n \cdots A_1||,$$ where $A_i$ are i.i.d. random matrices. ...
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1answer
20 views

Attractor return range

I am trying to render an image with return values from an attractor calculation and need to map the output to the screen. a and b values are generated randomly between -3 and 3 c and d values are ...
2
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1answer
93 views

Lyapunov stabilty, elementary question

Let’s say I have a system 1/(T1s+1) or any other n-th order polynomial and a PI controller (KP and TI). I already know that the system is stable but for, let’s say, educational purposes (not ...
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1answer
72 views

Showing that a gradient flow cannot have a homoclinic solution

My problem is, given a $V: \mathbb{R}^2 \rightarrow \mathbb{R}$ that is atleast $C^2$, consider its gradient flow $$ \dot{x} = -\nabla V,\quad \text{ or }\quad \dot{x_i} = -\frac{\partial V}{\partial ...
2
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1answer
79 views

invariant measure under irrational rotation on $S^1$

Prove that if $T:S^1 \to S^1$ is an irrational rotation, then the only probability measure on $S^1$ that is $T-$invariant is the lebesgue measure or a multiple or it. We are considering the ...
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1answer
55 views

If $g$ is invariant under an ergodic map then it's almost everywhere constant

Let $(X,B(X),\mu)$ be a probability space, where $X$ is compact metrizable, and $B(X)$ are the Borel sets. Let $f:X\to X$ be a measurable function such that: i) $\forall A\in B(X)$ ...
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0answers
35 views

Vector fields near non defined points

As far as my short knowledge on the analysis of vector fields goes, one is often interested on studying such objects near equilibrium points. So the theory I've read concerns v.f.'s $X$ in a ...
2
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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 ...
1
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1answer
88 views

A theorem about oscillation in Arnold's mathematical methods of classical mechanics

There is a theorem in page 100 of Arnold's Mathematical Methods of Classical Mechanics, which says that: If $\cfrac{dx}{dt} = f(x) = Ax + R_2(x)$, where $A = \cfrac{\partial f}{\partial x}|_{x = ...
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1answer
33 views

Mapping one integral curve onto another

Let $v$ be the vector field $\sum_{i=1}^n x_i\dfrac{\partial}{\partial x_i}$, and let $f:\mathbb{R}^n\rightarrow\mathbb{R}$ be the projection map $(x_1,\ldots,x_n)\rightarrow x_1$. Show that $v$ ...
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1answer
50 views

Laplace Linearization near equilibrium solutions

Find the approximate solution near the equilibrium solution. $$x'=5x-x^2 -xy, \qquad y'=xy-2y. $$ Equilibrium $(2,3)$, I believe.
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1answer
64 views

dynamical system with constraint equation

I am not sure how to solve a dynamical system with some constraint equation. For simplicity, let us consider the following system $x'=-xy\\ y'=\frac{x}{2}\\ x+y^2=1$ The system is 1 dimensional. If ...
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52 views

Flows and vector fields in differential geometry

given vector fields $f = f_{0} + L(x)$ and $g = g_{0} + G(x)$ of $\mathbb{R}^{n}$ where $L$ and $G$ linear, can someone help with finding the flow $\psi^{t}$ of $f$ and the bracket of $f,g$ Thanks ...
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1answer
45 views

is the system exponentially stable? uniformly stable?

Consider the state equation: $$ \frac{\partial}{\partial t}x(t)= A(t)x(t), \: x(\tau)=x_0 $$ $$ A(t) = \begin{pmatrix} -1 & k(t) \\ 0 & -1 & \\ \end{pmatrix}, ...
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52 views

Example of a continuous function with only one fixed point and no periodic points

Does anyone know how to go about finding an example of a function with no periodic point and only one fixed point. This f is continuous in an interval I, and I c f(I). As an addendum, can this fixed ...
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0answers
15 views

Dynamic Optimisation where control “gets lost”

I have some problems with this exercise. Would be glad about some help! $$\max V(u)=\int_0^2(2x(t)-3u(t))dt$$ $$\text{subject to }x(0)=5 \text{ and } 0\le u(t)\le2 \text{ and } \dot x(t)=x(t)+u(t) ...
2
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1answer
84 views

Function from Cantor Set to itself.

I am stuck in getting rational functions (except identity) defined from Cantor set to itself. Please help me to get out these functions.
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35 views

Can $x'(t)\leq \mu |x(t)|,\forall t\geq 0$ imply $x(t)<\mu |e^{-\mu t}|$?

Assume $\mu>0$ If it can't, is it possible to give a good exponential bond(better with negative constant in front of t.)?
2
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1answer
92 views

Prove $0$ is an exponentially stable equilibrium of the system $x'=f(x)+g(x)$ if $f(0)=g(0)=0$

Besides the conditions in the title, we have: $0$ is an exponential equilibrium of the system $y'=f(y)$ $|g(x)|\leq \mu|x|,\forall x \in \mathbb{R}^n$ $\mu$ is sufficiently small! What I have ...
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2answers
40 views

Does “$\exists \delta >0$ S.T $||x(0)-x_e||<\delta\Rightarrow \displaystyle \lim_{t\rightarrow \infty}||x(t)-x_e||=0$” imply stability?

Recall the definition of stable and Asymptotically stable: A fixed point $x_e$ of a vector field is called (Lyapunov) stable if $\forall \varepsilon>0,\exists \delta(\varepsilon)$ such that ...
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2answers
51 views

Linear Hamiltonian systems and complex quartets

In the text Hamiltonian Systems: Chaos and quantization by Alfredo M. Ozorio De Almeida section 1.2 there is a discussion of the possible types of eigenvalues for Linear Hamiltonian systems. One ...
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0answers
35 views

Find the Liaponov function for the given system

I have been stuck on finding a Liaponov function for the following system. Any suggestions? $x^\prime=3y-3x^2$ $y^\prime=3x-3y^2$
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1answer
196 views

stable, center and unstable manifolds

‎let ‎‎$‎f:M‎\rightarrow ‎M‎$ ‎be a‎ ‎‎‎‎diffeomrphism and‎‎‎ ‎$ ‎\Lambda‎ $ a‎‎ ‎hyperbolic ‎set. ‎We ‎can ‎give a‎ ‎characterization ‎of ‎(local) ‎stable ‎and ‎unstable ‎manifolds ‎by‎: ‎‎for ...
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0answers
39 views

Henri Poincaré writings

I have heard that Poincaré writings were very intuitive in its approach and not very formal in the arguments. I'm searching for something like this to complement my study of dynamical systems. I ...
2
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1answer
49 views

Drawing a bufircation diagram

$\dot x=x(\mu+x-2)(\mu+2x-x^2)$ The first thing I did was to check the fixed points in the $(\mu,x)$-plane: $x=0$ $x=2-\mu$ (saddle node at 0 when $\mu$) $x_{1,2}=1+-\sqrt{\mu+1}$ (no fixed point ...
1
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1answer
55 views

Hopf bifurcation - System Transformation

I have the following system: $$\dot u=\mu u-v+u^2-v^2-u(u+v)^2 \\ \dot v=u+\mu v-uv-v(u^2-uv+v^2)$$ My lecture notes about Dynamical Systems says that this system can be transformed into $$\dot ...
2
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2answers
71 views

Doubts related to a phase plane diagram.

I want to draw phase plane diagram of the following differential equation $$\frac{d^2y}{dx^2} - 2\frac{dy}{dx} + 10 y = 0.$$ Please check if my approach is correct. I have some doubts about it. ...
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1answer
48 views

Show that the system $\Sigma(SAS^{-1},SB,CS^{-1},D)$ is observable/controllable iff $\Sigma(A,B,C,D)$ is observable/controllable

I am given the two linear systems: \begin{eqnarray} \Sigma_1: \dot{x}&=&Ax+Bu\\ y&=&Cx+Du \end{eqnarray} and \begin{eqnarray} \Sigma_2: \dot{x}&=&\bar{A}x+\bar{B}u\\ ...
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0answers
30 views

$A\in \Bbb R^{n\times n}$ and $B \in \Bbb R ^{n\times m}$. Show that $\exists p\in\Bbb R ^m$ s.t. $(A,Bp)$ is controllable iff $(A,B)$ is controllable

Let $A\in \Bbb R^{n\times n}$ and $B \in \Bbb R ^{n\times m}$. Show that there exists a vector $p\in \Bbb R ^m$ such that $(A,Bp)$ is controllable iff $(A,B)$ is controllable. Here when I say $(A,B)$ ...
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1answer
38 views

Can you find an expression for $F_{12}(t,\tau)$ in terms of $F_{11}(t,\tau)$ and $F_{22}(t,\tau)$?

I have a problem with this...I can not figure out how to solve it..! can you help me? thank you!! Show that if $A(t)$ is partitioned as $$ A(t) = \begin{pmatrix} A_{11}(t) & A_{12}(t) \\ ...
2
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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
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1answer
96 views

How can Hotelling reduce the Euler-Lagrange equation in his calculus of variations mine problem?

In a 1931 paper Hotelling gives the discounted profit of a mining operation as: $$P=\int_{0}^{\infty} \dot{x} p(x,\dot{x},t) e^{-rt} \:\:dt$$ Note that this is, for the most part, a typical calculus ...
3
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1answer
55 views

what is so great about having an invariant measure?

I am a student who just started to learn basic concepts of ergodic theory. It seems like that given a dynamical system, people are very excited to find various invariant measures of the system. But ...
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1answer
53 views

Energy functions and Lyapunov

So I found that $(\pm 1,0)$ and $(0,0)$ are steady states and its trace of the linear system is always $-1$. This implies all three points are sinks (fixed points). Is the question for (a) ...
0
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1answer
48 views

Why is this Lyapunov function positive?

$x'(t) = y$ $y'(t) = -x + x^3 + y$ $(0,0)$ is a steady state. According to the solutions, $E(x,y) = V(x) + y^2/2 = x^2/2 - x^4/4 + y^2/2$ is a Lyapunov function. It then says $E(x,y) > 0$ ...
4
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0answers
148 views

Difference between dynamic system and complex system

I know this might not be an easy question, I've already read the wikipedia page, and there is an interesting view: Therefore, the main difference between chaotic systems and complex systems is ...
2
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1answer
51 views

How to prove this ODE system is stable at origin?

A dynamical system in polar coordinates is: $$\Theta'=1, r'= r^2\sin(1/r), r>0, r'=0\mbox{ if }r =0.$$ How to show this is stable at origin? Intuitively, I really can't believe it because I ...
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0answers
23 views

Construction of time flow with three fixed points

I'm trying to construct a time flow $f:\mathbb{R}\times [-1,1]\to [-1,1]$ (ideally smooth) such that $-1,0,1$ are fixed points (that is $f(t,x)=x$ for all $t\in\mathbb{R}$ and $x=-1,0,1$) of which ...