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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Proving that $y(t)\to0$ given a dynamical system

Consider a nonlinear system of the form $$\dot{y}(t)=p(y(t)) + u(t)$$ where $$p(q) = a_kq^k+a_{k-1}q^{k-1}+\ldots+a_1q$$ $$u(t) = -\left(\alpha_ky(t)^k+\ldots+\alpha_1y(t)\right)-y(t)$$ with ...
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35 views

Solve the dynamical system in polar coordinates

I have the system (it is time dependent, this is a simplified notation): \begin{cases} x' = x - y - x^3 \\ y' = x + y - y^3 \\ \end{cases} I can't seem to solve it for r, $\theta$. (The change of ...
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17 views

Index of a curve is independent of the curve c?

Index of a curve C: $I_{C}$ is defined as the net number of counterclock wise revolutions made by the vector field as the vector field x moves once counterclockwise around the curve C. If C is a ...
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43 views

Idea behind Poincaré Bendixson theorem

The Poincaré Bendixson theorem states: If R is a closed bounded subset of $\mathbb{R}^{2}$ containing no fixed points and $\Psi_{t}\left ( x_{0} \right ) \in R$ for all $t\geq 0$, then, the ...
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8 views

Turing Instabilities

In the solution all partial derivatives are evaluated at the equilibrium point Why does the solution not talk about the fact that the determinant of the Jacobian Matrix=$f_ug_v-f_vg_u$ at the ...
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1answer
17 views

Logarithmic Spiral- N-gon

In the mice problem, also called the beetle problem, $n$ mice start at the corners of a regular $n$-gon of unit side length, each heading towards its closest neighboring mouse in a counterclockwise ...
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6 views

Particle locating/collision prediction in bounded (two-dimensional) environments [on hold]

I believe that many physics engines, particle simulators, and even video games use discrete-event simulation to determine where a particle or object is at any moment, and the direction in which it is ...
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23 views

Part of Picard–Lindelöf theorem proof

Say we have the sequence $$X_{n+1}(t) = X_0 + \int_{0}^{t} f(s, X_n(s)) ds \quad ,\ X_0(s) = X_0.$$ $f$ is continuous over I $\times$ U, I being an open interval in $\mathbb{R}$ and U an open set in ...
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20 views

Dynamical Systems problem

I have a problem that have been trying to solve but it's not going so good. I would like some guidelines on how to work myself around this problem: Two neighboring countries spy on each other and ...
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30 views

Question about n- expansive homeomorphism

Let $(X, d)$ be a compct metic space and $f$ be a homeomorphism on $X$ . Suppose $\Gamma_c(x)=\{y: d(f^{n}(x), f^{n}(y))<c \ , \forall n\in Z\}$ and for some $z\neq x$, $z\in \Gamma_c(x)$. ...
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1answer
28 views

Finding a function within a dynamical system using a lyapunov function

Consider the system for $(x_1(t),x_2(t))$ \begin{align} \dot{x}_1 &= x_1^2+x_1^3+x_2\\ \dot{x}_2 &= x_1^2+u \end{align} Find a function $u=\phi(x_1(t),x_2(t))$ so that if $$V(x) = ...
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2answers
67 views

Periodic solutions of $x'=x^2-1-\cos t$

Consider $x'=x^2-1-\cos t$. What can be said about the existence of periodic solutions for this equation? I'm not sure if periodic solutions exist, but if they do, they must have period equal to $ ...
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23 views

Separatrix and trajectories corresponding to solutions of an equation(pendulum).

The equation of the pendulum is: $\ddot{\theta}+\frac{g}{l}sin\theta$ After some manipulation, we get $H=\frac{\dot{\theta}^{2}}{2}-\frac{g}{l}Cos\theta$=positive constant Trajectories ...
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20 views

To prove the properties of Denjoy's Maps

We need to show that the Denjoy homeomorphism constructed may actually be made $C_1$. a)For each integer $n$,let $$l_n=\frac{1}{(|n|+1)((|n|+2)}.$$Show that $$\sum_{n=-\infty}^{\infty} l_n ...
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28 views

Regularity of the solutions of the infinite dimensional dynamical systems

Consider a densely defined unbounded operator $A:D(A)(\subset H)\to H$ which is infinitesimal generator of a strongly continuous semigroup $\mathbb{T_{t\ge0}}$ for the following dynamical system: ...
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33 views

Convergence of $\dot{x}(t) = -\alpha(t)x(t) + b\mathrm{e}^{-\lambda t}$

Let $x(t)\ge 0$ obey the following differential equation: $$ \dot{x}(t) = -\alpha(t)x(t) + b\mathrm{e}^{-\lambda t}, $$ where $b>0$, $\lambda>0$, $\alpha(t)\in\mathbb{R}$ is both lower- and ...
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1answer
52 views

Question about proof of rotation number of inverse map of circle homeomorphism

This question concerns a previous question, Rotation number of inverse maps on the circle. in which all the terminology and notation used below is defined. The question is given the rotation number ...
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1answer
33 views

Michaelis-Menten steady state hypothesis

In part $ii) $the part underlined in green suggests that we substitute an equation we get from when $v'=0$ to garner a solution of $s'$ for all time from the time when $v'=0$. However $v'$ does ...
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12 views

Constructing bifurcatino and Phase diagram [on hold]

I need to draw bifurcation diagram for cut of panel (c) in the following picture for c = 0.159 and c = 0.163. and also draw the phase space for each of the regions. diagram
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15 views

Euler explicit and semi-implicit

I am given a simple dynamic system with an initial condition: $a(t) = 0.9 - 0.1v(t)$ $v(0) = x(0) = 0$ I want to calculate $x(1)$ with a time step of $\Delta t = 1$ using Euler explicit and semi ...
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1answer
15 views

Theta-logistic equation

I can't comprehend any of the solution for iii). WHy for $\theta=1$ do we have linear dependence of the growth on the population size?
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1answer
17 views

Question about finding eigenvectors for differential equations?

I have a non linear system to analyse and sketch the phase portrait of. At one of the equilibria the Jacobian of the linearised system is given by $$\textbf {J}= \begin{pmatrix} 2 & 7\\ 7/2 ...
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1answer
27 views

How to find the straight line paths of saddle points for a nonlinear Hamiltonian system?

I have the system $$\dot{x}=y+2xy\\\dot{y}=-x+x^2-y^2$$ Which is Hamiltonian with $$H(x,y)=\frac{1}{2}x^2-\frac{1}{3}x^3+xy^2+\frac{1}{2}y^2$$ Now I want to plot the phase portrait for the system so ...
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1answer
17 views

Weak or strong Liapunov function

You are given the system $$\dot{x}=-x-xy^2; \dot{y}=2x^2y-x^2y^3$$ (a) What does the linearization about $x^*=(0,0)$ tell us about the local behavior. So $Df(x,y) = \begin{bmatrix} -1-y^2 ...
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26 views

Transform system to polar and sketch phase portrait. Show that $(0,0)$ is an unstable focus.

Transform the system $$x' = y - x(x^2+y^2-1)$$ $$y' = -x - y(x^2+y^2-1)$$ to polar coordinates, and sketch the phase portrait. Show that it has a unique limit cycle and that all ...
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26 views

How to use Newton's method for finding fixed points in Poincare maps.

As a homework I have to reproduce the numerical method given in the paper. Where there's the system $$ \dot{u}=f(u)+s(t)\\\\u=(u_1,u_2,u_3)\in\mathbb{R}^3$$ and ...
3
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0answers
21 views

How to keep symplecticity of a diffeomorphism after a coordinate rescaling and Taylor series expansion?

Let's take some, implicitly written, symplectic diffeomorphism $F$ of $\mathbb{R}^{2n}$, let it preserve the (possibly non-standard) symplectic form $\omega$. Suppose I introduce a small parameter ...
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1answer
17 views

omega-alpha limit set and manifold

Definition: The $\omega$-limit set $L_{\omega}\left ( x \right )$ of $x \in \mathbb{M}$ >is the set of $y \in \mathbb{M}$ which for each y there exists a strictly increasing unbounded ...
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1answer
18 views

Geometric intuition of an invariant set, positively invariant and negatively invariant

Definition: Invariant set A set $S \subseteq \mathbb{M}$ is invariant if $\Psi_{t}\left ( S \right )\subseteq S,\forall t$ -if $\forall x \in S$, $\Psi_{t}\left ( x \right )\subseteq ...
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1answer
24 views

Forms of functions in dynamical systems

I wanted to read some introductory material about dynamical systems since I might need a basic understanding of them in a related task. So, as far as I see, in a continuous time dynamical system, we ...
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2answers
70 views

Discrete one-dimensional 2-cycle system

Is it possible to classify all maps $x_{k+1} = f(x_k)$ that have the property that all orbits are period 2 cycles only? Also, how would I do it for period 3 system?
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1answer
31 views

Eigenvector for a non-linear system

Using the reversibility arguments alone, show that the system $\dot{x}=y$ $\dot{y}=x-x^{2}$ has a homoclinic orbit in the half-plane $x\leq 0$ This is a non-linear system. A ...
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1answer
27 views

About Expansive homeomorphism

Let $f:X\rightarrow X$ be a homeomorphism. $f$ is called an $c$-expansive homeomorphism, whenever for every $x\neq y$, there is an integer $n$ with $d(f^{n}(x), f^{n}(y)) >c$. Question. Is there ...
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1answer
33 views

Eigenvector of unstable and stable manifold of a non-linear system with non-linear center

Show that the system $\dot{x}=y-y^{3}$ $\dot{y}=-x-y^{2}$ has a non-linear center and plot the phase potrait. My attempt: The system is non-linear so we linearise it: The ...
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2answers
51 views

Basin of attraction an open set?

Definition: The basin of attraction is the defined as the set of all initial conditions $x_{0}$ such that $x(t$) tends to an attracting fixed point $x^{\ast}$ as time $t$ tends to $\infty$. ...
24
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1answer
269 views

Is there an easy way to see that this simple recurrence is 9-periodic?

In a colloquium talk yesterday, Robert Bryant pointed out that for all initial values $a_0, a_1 \in \mathbb{R}$, the sequence generated by the recurrence relation $$ a_{n+1} = |a_n| - a_{n-1} $$ turns ...
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1answer
21 views

Measure preserving ergodic map commutes with complementation?

This is probably trivial (in which case I apologize), but it's late and I would really like a quick proof/counterexample for this (for a different problem that I'm doing): if $(X,\mathcal{M},\mu,T)$ ...
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31 views

Devaney's definition of chaos

I'm reading Banks, et. al. paper On Devaney's Definition of Chaos. In it, they say "It is not difficult to find transitive examples for which sensitivity is not preserved under conjugation." I'm ...
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1answer
44 views

Existence of periodic orbit of the ODE system $\dot{r}=r-r^3 \cos^2(\theta),\,\dot{\theta}= 1$

Consider the system of ODEs (in polar coordinates): $$\dot{r}=r-r^3 \cos^2(\theta)$$ $$\dot{\theta}= 1$$ If we take $r_1 = \frac{1}{4}$ then $\dot{r}> 0$, and if we take $r_2 = ...
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1answer
46 views

If $f$ and $g$ are bounded, then every solution of the autonomous system of differential equations is defined for $t \in \mathbb R$.

Consider the system of autonomous differential equations (autonomous system of differential equations?) $$x' = f(x,y)$$ $$y' = g(x,y)$$ where $x=x(t)$ and $y=y(t)$ Let $f$ and $g$ have first ...
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41 views

Is the closure of the set of all irrational rotation maps on $S^1$ dense in $Homeo(S^1)$?

I study about rotation maps on circle, and I have a question. Let $Homeo(S^1)$ be the set of all circle homeomorphisms with sup-metric $d(f,g)= \sup \{ d(f(x),g(x)| x \in S^1 \}$, and rotation map ...
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1answer
71 views

Find the fixed points of the system, and sketch the trajectories of the system

I am given the following system: $$x' = [(x-1)^2 + y^2]y$$ $$y' = -[(x-1)^2 + y^2]x \tag{*}$$ where $x = x(t), y = y(t)$. I am supposed to Find the fixed points of the system, and ...
2
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1answer
55 views

Long term Behavior of Dynamical System

Given the following dynamical system: $ \dot x = -6x^2+yz+x-1 $ $ \dot y = 4xz-3y^2+y-2 $ $ \dot z = 9xy-2z^2+z-3 $ What can you say about its long term behavior? Attempt: First, finding the ...
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26 views

Prove any $n>2$ DE doesn't hold Poincaré-Bendixson theorem.

How can I build a differential equation to show that Poincaré-Bendixson theorem doesn't hold for $n≥3$ ? Is it easy to take any D.E with $n=3$ and prove it? More specifically, can you give me a ...
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1answer
19 views

Simple eigenvalue of Koopman operator

Let $T : X \to X$ be a measure-preserving transformation and $U_T : L^2(X, \mu) \to L^2(X, \mu)$ , $(U_T f) (x) = f(Tx).$ What does it mean a $\bf{simple}$ eigenvalue of $U_T$? $\lambda \in ...
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16 views

Reference request: monotone and strongly monotone with respect to derivatives

Recall, let $H$ be a real Hilbert space. A mapping $F:H \rightarrow H$ is said to be monotone if $$ \langle F(u)-F(v), u-v\rangle\geq 0, \quad \forall u,v\in H; $$ strongly monotone if there exists ...
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2answers
55 views

Dynamics of a three dimensional system

I have a dynamical system in three dimensions given by: $\dot x = (1-x^2-y^2-z^2)x+xz-y$ $\dot y = (1-x^2-y^2-z^2)y+yz+x$ $\dot z = (1-x^2-y^2-z^2)z-x^2-y^2$ I analyzed the system by first finding ...
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1answer
46 views

Suppose the period of γn is λn. If there are points Xn ∈ γn such that Xn → X ∈ γ , prove that λn → λ.

I was wondering if someone could help me with an exercise from Hirsch, Smale, and Devaney, Differential Equations, Dynamical Systems, and an Introduction to Chaos. Let γ be a closed orbit of a ...
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11 views

If $1$ is a simple eigenvalue of $T$, then $T$ is an ergodic measure-preserving transformation

Let $T : X \to X$ be a measure-preserving transformaton and $U_T : L^2 (X, \mu) \to (X, \mu)$, $$(U_T f) (x) = f(Tx).$$ I have to show that if $1$ is a simple eigenvalue of $U_T$, then $T$ is ...
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18 views

Hamilton principle/dynamics teaching in earlier stages.

In finding dynamic motion of particles we use laws of conservation of energy and momentum. It is found the dynamics formulation using action integral $$ \int (T-V)\, dt $$ builds ODEs for dynamic ...