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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15 views

Sampling a matrix of an AR model

Let us consider a dynamic system $x_t = A x_{t-1}+v_t$ where $v_t$ is multivariate normal noise with zero mean, i.e. $v_t\sim\mathcal{N}(0,\Sigma)$ and $A$ is a matrix. As far as I know, for some $A$, ...
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0answers
47 views

Topological entropy of isometric extension

L.s., This is a homework question some of my fellow students and I are having great difficulty with. Let $Y,Z$ be compact metric spaces, $X = Y \times Z$, and $\pi$ the projection to $Y$. Denote $h$ ...
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1answer
52 views

Alpha and Omega limit sets (dynamical systems)

What are all the possible $α$- and $ω$-limit sets of points for the linear system of equations $x'=Ax$ given by the following matrices $A$: I guess I don't really know how to get started, Could I ...
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0answers
51 views

Alpha and omega limit sets [duplicate]

What are all the possible $α$- and $ω$-limit sets of points for the linear system of equations $x'=Ax$ given by the following matrices $A$: I guess I don't really know how to get started, Could I ...
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2answers
25 views

System of equations, limit points

This is a worked out example in my book, but I am having a little trouble understanding it: Consider the system of equations: $$x'=y+x(1-x^2-y^2)$$ $$y'=-x+y(1-x^2-y^2)$$ The orbits and limit sets ...
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38 views

Jacobian Matrix in dynamical systems

Can someone explain what exactly the Jacobian matrix is (specifically in its application to dynamical systems) and maybe give an example of how to compute it? It really confuses me...and I haven't ...
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18 views

Nullclines and regions of positive value

To get an idea how the solutions of an autonomous ODE', i.e. $$\dot{x}=f(x,y) \\ \dot{y}=g(x,y)$$behave, one approach is to sketch the nullclines and then picking for each nullcline points $(x_0,y_0)$ ...
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2answers
34 views

Preserving orbits by multiplication with a non-vanishing function

I'm reading through some notes from a past course of mine, where a system of ODE's of the form$$ \begin{array}{c} x'=h(x,y)f(x,y)\\ y'=h(x,y)g(x,y) \end{array} $$ appears, such that ...
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82 views

Translational invariance and zero eigenvalue

Page 2 (506), line 18 of http://www-personal.umich.edu/~orosz/articles/NonlinScipublished.pdf says that "The presence of translational symmetry in the nonlinear equations gives rise to a relevant ...
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34 views

dynamical systems classifying an equilibrium point

Consider the differential equation $$x''+(2a)x'-(b^2/2)x+x^3=0 \tag{1}$$ where $a,b >0$ are constants. (i) Write differential equation (1) as a first order dynamical system. (ii) Determine the ...
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1answer
19 views

Uniqueness of homeomorphism

In the theory of dynamical systems, the Hartman–Grobman theorem states that there is a homeomorphism of a neighborhood which conjugates the original system and its linearization. A problem bothers me: ...
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93 views

limit cycle of an ODE system

Consider a planar ODE system $z'=f(z)$ with $z=(x,y)$, $$ f(x,y)=(xy^2-x-y,y^3+x-y). $$ Using polar coordinates, one can get $$ r'=r(r^2\sin^2\theta-1),\quad \theta'=\cos 2\theta. $$ With ...
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2answers
31 views

Stability based on phase portrait

I wonder how can I find the largest delta in which if initial condition is inside circle (How do I specify radius of that circle) bounded by delta then the solution is always within the dashed circle ...
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0answers
64 views

Differential equation for a fish population

Consider the differential equation $\dot {x}=rx(1-x/K)-H$ for $x\ge\ 0$ which models a population of fish that is governed by the logistic equation and a number of fish H caught and removed every unit ...
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2answers
46 views

Stability of fixed points for a differential equation

Consider the differential equation $x'=x^2-9$ I am pretty lost on this problem.. a. find the stability type of each fixed point To find the fixed points, I set this equal to $0$, right? Would ...
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1answer
27 views

Stability of Linear Systems

for the following matrices A, classify the stability of the linear systems x=Ax as asymptotically stable, L-stable (but not asymptotically stable) or unstable and indicate whether it is a stable node, ...
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26 views

Modelling a transfer function for plant/system empirically

In an attempt to learn about PID controllers, I'm designing a small desktop thermal control system. I have a power resistor mounted to a heatsink, with a thermister placed nearby to measure ...
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1answer
139 views

Verifying if system has periodic solutions

Given the following system $\dot{x} = y$ $\dot{y} = y(9-x^2-2y^2) - x$ verify whether it has periodic solutions and if so are they attracting or repelling. I thought: The critical points or fixed ...
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1answer
62 views

Induced sytem ergodic implies normal sytem ergodic

Okay, we consider a measure preserving system $(X, \mathcal F, \mu, T)$ and let $A \in\mathcal F$ be such that $\mu(A) > 0$ and $\mu ( \cup ^{\infty}_{n=1} T^{-n}A) = 1 $. Now I want to show that ...
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0answers
33 views

Dense orbits on the 2-torus

For $\alpha\in\mathbb{R}\setminus\mathbb{Q}$ and $f: T^2 \rightarrow T^2$ the 2-torus homeomorphism given by $f(x,y) = (x+\alpha, x+y)$. Why is $f$ topologically transitive. If the forward orbit of ...
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34 views

ODE and recurrence relation

I am trying to understand the following claim (I came across it while reading a paper): Consider the map (Standard/Arnold map) $T_{k}:(x,y)\mapsto(x+y+kf(x), y+kf(x))$, with ...
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1answer
35 views

Showing a second-order ordinary differential equation has periodic orbits

Is it possible to show that $x''(t)-x'(t)²+x(t)²-x(t)=0$ has at least a periodic orbit? I've made it a system by setting $y=x'$ and get $x'=y; y'=y²-x²+x$. I'm asking the question because I find a ...
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1answer
25 views

questions about transversal surfaces (curves) to a vector field

The following is an excerpt from Dynamical Systems by Shlomo Sternberg: By a transversal, $L$, to the vector field $V$ we mean a surface of codimension one which is nowhere tangent to $V$ . In ...
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1answer
41 views

α and ω possible limit sets of points

What are all the $α$ and $ω$ possible limit sets of points for: $$A=\begin{pmatrix}-4&-2\\3&-11 \end{pmatrix}.$$ I am not really sure what to do.. ...
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44 views

Solutions of nonlinear equations

Show that the differential equation $x^.=x^1/5$ with initial condition $x(0)=0$ has non unique solutions. Why does the theorem on uniqueness of solutions not apply? ...
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25 views

Lubrication Theory: Quick Question!

Basically, I'm modelling the flow of a "coating" process -- a fluid flow between a flat moving plane and a stationary cylinder, 2D, cartesian coordinates. Subscript 0 is the at the minimum height b/w ...
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1answer
34 views

If a linear ODE system has a solution that tends to zero, it also has an unbounded solution

$a:[0,\infty)\to \mathbb{R}$ is a continous and bounded and $$x'(t)\ =\left(\begin{matrix}0&1\\-a(t)&0\end{matrix}\right) \ x(t)$$ has a non-zero solution like $y(t)$ such that $\lim_{t ...
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1answer
60 views

show that all other solutions are bounded

Suppose $G(x)$ is a solution of the differential equation $$x'(t)\ =\left(\begin{matrix}-5&2\\-4&1\end{matrix}\right) \ x(t)+ \ f(t)$$ where $f(t)$ is a continous function and ...
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1answer
16 views

Unique equilibrium that is not an attractor

Fix an ODE system $\dot{x} = f(x)$ where $f : \mathbb{R}^n \to \mathbb{R}^n$ is locally Lipschitz. In the case where $x^\ast$ is a global attractor of $f$, it obviously holds true that $x^\ast$ is the ...
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16 views

system dynamics: overshoot

My HW problem asks to look at the system dynamics in MATLAB of a feedback system using state-space models in Simulink. The system has no input other than its own output times a feedback, K: $\dot{x} ...
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1answer
30 views

Local asymptotical stability for an ODE

Consider a system: \begin{align*} \frac{d}{dt} \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} = \begin{pmatrix} x_2 \\ a x_1 + b x_1^2 \end{pmatrix} \end{align*} with $a < 0$ and $b\ne0$. My question ...
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28 views

Complex Dynamics of the map

Consider a dynamical systems $$ Z_{n+1}=f(Z_n, Z_{n-1}), $$ where $f$ is a mapping from $\mathbf{C}^2$ to $\mathbf{C}$, defined as $f(z,w)=\dfrac{\alpha}{z}+\dfrac{\beta}{w}$. $\alpha$ and $\beta$ ...
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14 views

Solving system with non-symmetric, indefinite coefficient matrix

So I have a non-linear system that I have linearized using the small angle approximation. When I linearize, I noticed that the coefficient matrix is not symmetric and it has positive real, negative ...
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1answer
32 views

Show that the action variable is $J = \sqrt{A^2 + 2E} - A$

I have the hamiltonian: $$H = \frac{1}{2}p^2 + \frac{1}{2}A^2 \tan^2(q)$$ And I would like to show that the action variable is $J = \sqrt{A^2 + 2E} - A$, where $E$ is the energy. I'm having a ...
3
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1answer
56 views

Hopf bifurcation and limit cycle

I am studying bifurcation and had a system like this: $$dx/dt=ux-y-x(x^2+y^2),$$ $$dy/dt=x+uy-y(x^2+y^2).$$ I want to determine whether a Hopf bifurcation would occur. I wrote the system into polar ...
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1answer
52 views

Nonlinear Dynamics and Chemical Reactions (Ivanova Reaction System)

I have a homework problem in which I'm given an Ivanova reaction system $X+Y \longrightarrow 2Y$, $Y+Z \longrightarrow 2Z$, $Z+X \longrightarrow 2X$, and I'm asked to write the mass-action ODEs ...
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1answer
28 views

Showing that the linear twist map is sensitive dependent

Choose $\Delta=\frac{1}{2}$ (I believe this value should work). let $\delta > 0$ and let $\textbf{x}_1=(x_1,y_1) \in X$. I assuming that $d$ is the Euclidean distance. Somehow I think we ...
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1answer
28 views

State space system gives different bode plot then transfer function matrix

I have a discrete state space system with matrices $A$,$B$,$C$ and $D$ with sampling period $T_s$. I can either create a state space system, sys1 = ss(A,B,C,D,Ts), ...
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38 views

Existence of Periodic Solution

I'm working with the system of equations below that represents a Pendulum with constant forcing. \begin{align*} \theta'&=v\\ v'&=-bv-\sin(\theta)+k \end{align*} Where $\theta$ gives the ...
4
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1answer
61 views

How to derive the hamiltonian from a non-classical lagrangian

For the non-classical lagrangian of a hydrogen atom: $$L = -mc^2 \sqrt{1-\frac{v^2}{c^2}} + \frac{e^2}{4 \pi \epsilon r}$$ We get that two conserved quantities are: $J = \gamma mr^2 \dot{\phi}$ and ...
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1answer
43 views

Reference Request: Discrete Dynamical Systems for Undergraduates

I am looking for a primer text in discrete dynamical systems for an undergraduate level of understanding in mathematics. I have taken introductory courses in numerical analysis and computational math, ...
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50 views

Asymptotic stability of $\dot{x}=-x^3$

So for an assignment I have to show that $\bar{x}=0$ is an asymptotically stable solution of $\dot{x}=-x^3 (\in\mathbb{R})$, using the definition of asymptotic stability (an equilibrium point $x^*$ is ...
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2answers
19 views

Index Theory: Can a closed curve around a single unstable fixed point have index $0$?

I know that a closed curve containing zero fixed points has index $0$. Is the converse also always true?
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1answer
51 views

Proving that $S=\bigcup_{j=0}^{2^k-1} S_{n-1+k}$ is a spanning set for the $2$-D Baker map

A set $S \subset X$ is a $(n,\epsilon)$-spanning set if $\forall x \in X$, $\exists y \in S $ such that $d_n(x,y)<\epsilon$. This is where we define $d_n(x,y)$ by $d_n(x,y)=\max_{0\leq k < ...
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Polar coordinate system of DE's to be written in cartesian form.

Suppose we have a system in polar coordinates: $\dot r = -r$ and $\dot \theta = \frac{1}{\ln{r}}$, we are asked to solve for $r(t)$ and $\theta(t)$ explicitly, so I just integrated both equations so ...
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1answer
44 views

Classify the fixed point at the origin of a dynamical system.

If we have a system $\dot x = -y+ax^3$ and $\dot y = x+ay^3$ I need to classify the fixed point at the origin for all real values of a. So I know we have to make the change of variables $ x = ...
3
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2answers
75 views

How to find a conserved quantity in this differential equation.

Consider the system: $$\ddot x = x^3 -x$$ What is the method to follow to find a conserved quantity for this system? So far what I have is: $\dot x = y$ and $\dot y = x^3 - x$ and I can find the ...
2
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2answers
73 views

Ergodic system has a.e. dense orbits

One more question: Let $X$ be a metric space with probability measure $\mu$ and $T\colon X \to X$ ergodic. $\Rightarrow f$.a.e. $x$ the orbit $O_x=\{T^n(x) : n \in Z\}$ is dense in $X$. So I have ...
2
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0answers
23 views

Fluid Flow: lubrication, integration, ODE

Basically, I'm modelling the flow of a "coating" process -- a fluid flow between a flat moving plane and a stationary cylinder, 2D, cartesian coordinates. Subscript 0 is the at the minimum height b/w ...
1
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0answers
46 views

recurrence of a dynamical system on a compact space

I have a question to an exercise which was already posted (but I'm not allowed to comment it). ...