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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Systems of second order differential equation

i'm following a course in Hamiltonian systems and regarding the part of linear systems I found this exercise from a book and need to solve it. My ideas are just after the test of the exercise. ...
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72 views

Arnold's proof of Liouville's Theorem on integrable systems

My question happens to be almost identical to the one left unanswered/closed here, which gives a bit of background information - it may not be necessary. I hope the reason it was closed on ...
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1answer
60 views

Hamiltonian system; breakdown for different level set values

I have a system of differential equations defined by the hamiltonian of the scalar function $H=y^2+e^{-xy}-c$, for some $c>0$. I am asked to describe what happens for $c=1$. I can tell there is a ...
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51 views

Poincaré Index of a periodic orbit

I am trying to formalize the following proof on Perko's Differential Equations and Dynamical Systems, which says that a periodic orbit has index +1. My only problem is trying to prove that the map ...
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45 views

Stability of a time-varying system that converges to a time-invariant system

Consider a system $\dot x = A(t)x$, where $A(t)$ is a matrix satisfying $A(t)\rightarrow A$ exponentially fast. For each fixed $t$, $A(t)$ is Hurwitz (all the eigenvalues in the open left half plane). ...
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19 views

How to transform a coupled differential equation into a system with diagonal linear part

Consider the system given by $$iu_t +u_{xx}+2|u|^2u = -v+iu$$ $$iv_t +u_{xx}+2|v|^2v = -u-iu$$ I am trying to transform the system into a system with diagonal linear part. I can solve a problem like ...
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39 views

Oscillatory solutions if derivatives are are independent of each other?

Definition: An oscillatory solution is one where $(x(t), y(t))$ is a trajectory and $x(t)$ and $y(t)$ are not constant. Further, for any $n \in \mathbb{N}$ we have $x(t+nt) = x(t)$ and $y(t+nt) = ...
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28 views

underdamped oscillation with quadratic decay

I know that for a 2nd order linear differential equation system, there are 3 possible scenarios: over-damped, critically damped and underdamped. For the underdamped case the solutions are of the form: ...
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37 views

Understanding explicit computation of a stable manifold in $\mathbb{R}^2$

After introducing and proving the stable manifold theorem in $\mathbb{R}^2$, my instructor gave an example of a system for which it was possible to explicitly compute the stable manifold. I am trying ...
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281 views

Can a gradient vector field with no equilibria point in every direction?

Suppose that $V:\mathbb{R}^n \to \mathbb{R}$ is a smooth function such that $\nabla V : \mathbb{R}^n \to \mathbb{R}^n$ has no equilibria (i.e. $\forall x \in \mathbb{R}^n : \nabla V (x) \not = 0$). ...
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1answer
70 views

The solutions of $y^{\prime \prime}+y=g$ are bounded

Suppose that $g$ is a continuous differentiable, increasing and bounded real function. How can one prove that the solutions of the differential equation $(E)$ $$y^{\prime \prime}+y=g$$ are bounded? ...
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1answer
18 views

Topological conjugation map for 1d flows

I am working on an example of 1d flow conjugation from class. For a vector field on $\mathbb{R}$, we have a flow $\phi_{t}(x)=xe^t$, with vector field $F(x)=x$ and another flow $\psi_{t}(x)=xe^{2t}$ ...
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77 views

straight-line solutions of dynamical system

Given the dynamical system $\dot x = f(x,y)$ and $\dot y = g(x,y)$ where f and g are homogeneous of degree n (i.e. $G(\alpha x,\alpha y)=\alpha^n G(x,y))$ show that straight line solutions of the ...
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1answer
50 views

Definition of the dynamical ball Bowen Walters

I'm learning continuous flows and I found this definition: Let $(X,d)$ be a compact metric space and $\phi:\mathbb{R}\times X\rightarrow X$ be a flow continuous. Denote by $\mathcal{H}$ the set of ...
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1answer
85 views

How to decide whether it is a system of Differential Algebraic Equations or a System of Ordinary Differential Equations?

I am struggling to name some of my dynamic models right. To be specific, I am not sure whether I should call it a system of Differential Algebraic Equations (DAEs) or a System of Ordinary Differential ...
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26 views

Stability of Extrema of Conserved Quantities

The following problem occurred on a homework assignment last week, and I wanted to know how to do the analysis to prove it in the nontrivial case. Consider a two dimensional system with ...
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2answers
51 views

How to write a system of equations as a dynamical system?

I am having a lot of trouble understanding how to move from a system of ODE's to a dynamical systems point of view (that will allow me to make a phase-plane analysis). Assume I want to write the ...
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138 views

Uniqueness of a periodic solution for nonlinear pendulum

I am working with the system of ODE's or second order differential equation representing the nonlinear pendulum with constant torque and damping. \begin{equation*} \theta'=v \end{equation*} ...
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1answer
41 views

time traveled between two points on phase portrait

Given the dynamical system: $\dot x = 1-x^2-2y^2-xy$ $\dot y = 2x^2+y^2+xy-1$ If the system starts at (1,0), show that it goes to(0,1) and find the time t at this point. Attempt: The Phase ...
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43 views

Deriving convergence region of iterative formula

A year ago I asked this question about fractal icons, however I didn't get any wiser yet. Now I am trying to understand the convergence of a simplified version of the fractal, to learn more about the ...
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56 views

finding the flow of a differential equation

Solve this differential equation $$x^{\prime}=\frac{-y}{\sqrt{x^2+y^2}},$$ $$y^{\prime}=\frac{x}{\sqrt{x^2+y^2}}$$ in $A=\{(x,y): x^2+y^2\in[1,4]\}$. Could help me get the general equation for the ...
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60 views

Show that every rational number $q\in\mathbb{Q}, q\in [0,1]$ has an eventually repeating ternary expansion.

Show that every rational number $q\in\mathbb{Q}, q\in [0,1]$ has an eventually repeating ternary expansion. Recall that $q$ is a rational number provided it can be written as $q=\frac{m}{n}$ where ...
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21 views

Why can every twist map be realized as the time-1 map of a time-dependent Hamiltonian?

if have problems getting my head around the following claim made by Moser in "Monotone twist mappings and the calculus of variations" and Gole in "Symplectic twist maps". Setting: Let $F : ...
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24 views

Question on the image of the parameter space under two non-commuting flows.

Let $\phi^t_1$ and $\phi^s_2$ be two smooth flows on $\mathbb{R}^2$ defined for $t\in[-1,1]$ and $s\in [-1,1]$. Assume (1) $\phi^0_1\big((0,0)\big)=\phi_2^0\big((0,0)\big)=0$ (2) $(\phi^t_1)'|_0$ ...
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1answer
57 views

Change of variable in $\int_{r_{0}}^{r_{1}}\frac{dr}{r(1-r^{2})}=\int_{0}^{2\pi}dt $

In Strogatz's Book Nonlinear Dynamics and Chaos the example 8.7.1 we have the vector field $\dot{r}=r(1-r^2)$ , $\dot{\theta}=1$ given in polar coordinates. Let $r_0$ and $r_1$ points in the positive ...
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53 views

Geometric solution to non-linear differential equation

Let the non-linear differential equation be $\dot{x}= sin\left ( x \right )$ Plotting this on the graph, we have the standard sin curve with the exception that the horizontal axis is labelled $x$ and ...
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37 views

Flow defined by ode

Recall the phase portrait of the linear system $\dot{x}=Ax$ with $A = \begin{pmatrix} -1 & -3 \\ 0 & 2 \end{pmatrix} $ Describe $\phi_{t}(N_{\epsilon}(x_0))$ for $x_0=(-3,0)$ , ...
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1answer
53 views

How to investigate the behavior of a ODE system with periodically pulsed input.

Suppose we have the ODE system \begin{array}{ccc} \frac{dT}{dt} & = & f_{1}(T,C,F,D)\\ \frac{dC}{dt} & = & f_{2}(T,C,F,D)\\ \frac{dF}{dt} & = & f_{3}(T,C,F,D)\\ \frac{dD}{dt} ...
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23 views

Can I use calculus of variations in this dynamical system

I have a dynamical system of the form $$\dot{\underline{x}} = \underline{f}(\underline{x})$$ where $\underline{x} = \underline{x}(t) \in R^4$ and $\underline{f}$ is smooth. With one of the systems ...
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183 views

Is there a possibility to determine/ estimate the topological entropy?

By $E$, denote the set of excited states $E=\left\{1,2,\ldots,e\right\}$ and by $R$ the set of refractory states $R=\left\{e+1,e+2,\ldots,e+r\right\}$. By $0$, denote the equilibrium state. The ...
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1answer
22 views

Showing the ergodicity of a rotation on the unit sphere

Consider the rotation $R_\alpha(z) = \alpha z, R_\alpha : S^1 \to S^1$. Show that $R_\alpha$is ergodic with respect to Haar measure on $S^1$ $\iff$ $\alpha$ is not a root of unity. I don't know how ...
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1answer
58 views

Analyzing a trajectory

I have the discrete dynamical system $f(x) = 13xe^{-x}$ and want to know its stable period 4 orbit, its fixed points, and period 2 orbit. Of course also check stability of these things as well. I've ...
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1answer
49 views

Non-linear system equilibrium points but critical point

I have a question about critical points of nonlinear systems. So I am given the following prey-predator model $$f(x,y) = x' = 10(1-\frac{x}{k})x-\frac{10xy}{1+x}$$ $$g(x,y) = y' = ...
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1answer
71 views

Norm of matrix and its maximum eigenvalue

I've seen in some inequalities in the theory of ODEs that $\lVert Q \lVert \le \lambda_{max}(Q)$. What theorem from Linear Algebra is relevant here?
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1answer
41 views

Detectability of a particular system implies detectability of a subsystem. (Eigenvalue problem)

I am looking at the system \begin{eqnarray*} \begin{pmatrix} \dot{x}_1 \\ \dot{x}_2 \end{pmatrix} &=& \begin{pmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{pmatrix} \begin{pmatrix} ...
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1answer
34 views

Existence of measure(s) of maximal entropy, given a finite-to-one chaotic global attractor $A$ which is, moreover, the non-wandering set

Let $X$ be a compact metric space and let $f\colon X\to X$ define some dynamics, $f$ being continuous and finite-to-one on a global attractor $A\subset X$. Moreover, $A$ is the non-wandering set ...
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88 views

Determining stability of equilibria of a nonlinear pendulum with torque ode system

I am working with the system of ODE's or second order differential equation: \begin{equation*} \theta'=v \end{equation*} \begin{equation*} v'=-bv-\sin(\theta)+k \end{equation*} with $b,k>0$ for ...
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1answer
73 views

Positively invariant neightbourhood using Lyapunov function

Given the nonlinear system of ODEs, $$x_1'=-x_1-x_2$$ $$x_2'=2x_1-x_2^3$$ I need to use the quadratic Lyapunov function $V(x) = x^TQx$ (where $Q$ is a positive definite matrix such that ...
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1answer
23 views

Dynamic real-time system problem

I am struggling with a systems theory problem, the task is as follows: u(t) -> H(s) -> y(t) H(s) being the transfer function $$ H(s) = H(s) = \frac{s+1}{s(s+2)^{2}} $$ $$ u(t) = e^{-5t} $$ So ...
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25 views

Show $x_k = F^k(x_0)$, with any seed $x_0$, satisfies $\lim_{k\to\infty} x_k=\bar{x}$

Consider the dynamical system $F(x) = ax+b$ where $a \neq 1$ and $b\in \mathbb{R}$. Show $x_k = F^k(x_0)$, with any seed $x_0$, satisfies $\lim_{k\to\infty} x_k=\bar{x}$ I found $\bar{x}$ to be ...
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99 views

Showing system contains peroidic orbit

Consider the IVP for the $2$-dimensional dynamical system ($X=[0, \infty )^2$) $$\dot{x_1}=a-x_1-\frac{4x_1x_2}{1+x_1^2}$$ $$\dot{x_2}=bx_1 \bigg( 1- \frac{x_2}{1+x_1^2} \bigg)$$ for all $t \in I$, ...
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1answer
24 views

Why does the logistic map $x_{n+1}=rx_n(1-x_n)$ become unstable when $\lvert\frac{df(x^*)}{dx}\rvert=1$?

I'm having trouble understanding why the logistic map becomes unstable when $$\lvert df(x^*)/dx\rvert=1,$$ where $x^*$ are the fixed points of $f(x)=rx(1-x)=x$. I have read that it can be seen from ...
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1answer
36 views

Does any analytic function from the unit disk to a compact subset of itself have a fixed point?

I was reading a proof by Beardon of the Wolff-Denjoy Theorem in Complex Dynamics. In the proof, a family of maps $$f_\epsilon=(1-\epsilon)f(z)$$ is used, where $f(z)$ is an analytic map from the unit ...
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1answer
25 views

Wandering set definition

I've seen two apparently different definitions and am wondering which is correct. A set $W$ is wandering if $\{T^{-k}W; k\in \mathbb{N}_0\}$ (resp. $\{T^{k}W; k\in \mathbb{N}_0\}$) are pairwise ...
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28 views

Range of parameter values for a stability of a fixed point for this 2d map

So I am trying to do a linear stability analysis for a very simple 2d discrete system: \begin{equation} \begin{aligned} x_{n+1} &= y_{n}\\ y_{n+1} &= -\frac{x_{n}}{2} + ay_{n} + y_{n}^{3} ...
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52 views

SIR-Model in Maple

This is my assignment I've been stuck on for a good part of the day. I'm confused because example's I've looked at would do part 3 before part 1 because they use substitution of $$u=S/N$$ $$v=I/N $$ ...
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90 views

For what kind of infinite subset A of $\mathbb Z$ and irrational number $\alpha$, is $\{e^{k\alpha \pi i}: k\in A \}$ dense in $S^1 $?

There is a well-known result saying that $\{e^{k\alpha \pi i}: k\in \mathbb Z \}$ is dense in $S^1$. By density, we can select an infinite subset $A$ of $\mathbb Z$ such that $\{e^{k\alpha \pi i}: ...
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1answer
18 views

regarding solutions to differential equations as vector fields

my dynamical system/ ODE textbook says: "if we consider the solutions of the autonomous system $x'=f(x)$, $x(0)=x_0$, for $f\in C^k(M,\mathbb{R}^n)$ and open $M\in\mathbb{R}^n$, we can regard such ...
2
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1answer
48 views

Finding the stable period $4$ orbit of a trajectory

I am told to find the stable period $4$ orbit of $f(x) = 13xe^{-x}$ for my discrete dynamical system through direct numerical iteration. However, I am a bit confused on what is meant by direct ...
2
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1answer
35 views

Range of values which satisfy this inequality

Consider the following inequality: $$|f(a)| = \left|\frac{1}{2}(a \pm \sqrt{a^{2}-2})\right| \leq 1$$ I got this inequality while doing stability analysis of a fixed point of a certain discrete ...