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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Calculating this derivative

I'm currently having trouble to verify this fact on page 90 in Robert L. Devaneys "An Introduction to Chaotical Dynamical Systems": Let $F(x,\lambda) = f_\lambda (x)$. Assume $f_\lambda (x) = 0$ for ...
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46 views

Confusion about Poincaré-Bendixson Theorem

The two following theorems appear to be contradictory to me. I'm sure I must have overlooked something significant here. The Poincaré-Bendixson II(a) says that if $R$ is a Type I invariant region, ...
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78 views

For what value of $v_0$ is the solution periodic?

A solution of the second-order differential equation $$ x''+x-x^3=0 $$ satisfies the initial condition $x(0)=0$ and $x'(0)=v_0$. For what value of $v_0$ is the solution periodic? I have tried ...
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42 views

The behavior of the trajectory of the phase portrait

For the plane autonomous system $$ x' = ax+by $$ $$ y' = cx+dy $$ If the solution to this system is, say, $ \binom{x}{y}= c_{1}\binom{1}{1}e^{-5t} + c_{2}\binom{1}{2}e^{-t} $, then it is ...
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59 views

Elementary properties of gradient systems

Consider $x_0\in\mathbb{R}^n$ and a $C^{1,1}$ function $f:\mathbb{R}^n\rightarrow\mathbb{R}$ (that is, a differentiable function whose gradient is Lipschitz function). Consider the system $$ \begin{...
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30 views

References for gradient systems

I am interested in the gradient system $$\dot{x}(t)=-\nabla f(x(t))$$ where $f:\mathbb{R}^n \to \mathbb{R}$ is a $C^{1,1}$ function (that is, a differentiable function whose gradient is Lipschitz ...
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24 views

Find all the equilibrium solutions of $x' = \cos(x^2)$ and determine stability.

One can determine equilibrium solutions of autonomous EDO's setting the derivative equal to zero. So, in this case, all the equilibrium solutions take the form $x^* = \pm \sqrt{\frac{\pi}{2} + k\pi}$, ...
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41 views

which is the relationship between infinite set and the orbits of their points?

I have been studying the proof of the following theorem: Theorem: Let's suppose that $X$ is some metric space and $X$ is a infinite set. If $f:X\to X$ is transitive and has dense periodic points then ...
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60 views

An analytic method to prove a specific curve is closed

In my study of Hamiltonian dynamics I have come across a Hamiltonian dynamic system with a solution curve I know to be closed via computer and via intuition but I require a rigorous way to prove this, ...
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20 views

Trouble understanding definition of an attracting set

From Wiggins' book, "Let $\cal{M}$ be a trapping region. Then $A=\cap_{t>0}\phi(t,\cal{M})$ is called an attracting set". Then he gives an example: $\dot{x}=x-x^3$ $\dot{y}=-y$, and claims that ...
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45 views

Find an energy functional for the nonlinear viscous oscillator $x' = v$, $v' =-b(v)v-k(x)x$, $t>0$ [closed]

Consider the nonlinear viscous oscillator $$\begin{cases} x' = v\\ v' =-b(v)v-k(x)x,\quad t>0, \\ \end{cases}$$ where $(x,v)$ is the position and velocity of the oscillator. Here $b : \mathbb R\...
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24 views

A specific problem finding homoclenic and heteroclenic orbits for a velocity field

In my study of dynamical systems I just met this specific problem: If a 2D autonomous system is governed by the Hamiltonian $ H(x,y)= \frac{A}{k}sin(kx)sin(\pi y) $ where $ A,k $ are non zero ...
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21 views

A criterion for area preserving dynamical system

In my investigation of dynamical systems I was met with this seemingly easy question I could not find an answer to: If we have a two dimensional system of autonomous ODEs viewed as a 2D dynamical ...
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1answer
71 views

Find the flow for the following dynamical system

I have the following dynamical system: $\dot{x_1}= -x_2 + (x_1(1-(x_1^2+x_2^2)^2))$ , $ \dot{x_2}= x_1 + (x_2(1-(x_1^2+x_2^2)^2))$, $\dot{x_3}= \epsilon x_3$ . I am required to work out the flow ...
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30 views

What it this theorem saying? - Regions of state space for which the flow eventually exists…

We have been given the following theorem to define regions in the state space for which the flow eventually exists. In questions, we use it to show that all trajectories eventually enter a bounded ...
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47 views

Poincare first return map, stability and bifurcations

Let $X= \mathbb R^3$ and consider the autonomous dynamical system $$\dot{x_1} = -x_2 + x_1 (1 - (x_1^2 +x_2^2)^2), \qquad{} \dot{x_2} = x_1 + x_2(1-(x_1^2 +x_2^2)^2), \qquad{} \dot{x_3}= \epsilon x_3$...
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30 views

Transversality: what is wrong with this counter example to persistence for small perturbations?

Let $M$ and $N$ be differentiable manifolds in $\mathbb{R}^{n}$, and let $p \in \mathbb{R}^{n}$. We say that $M$ and $N$ are transversal at $p$ if $$T_{p} M + T_{p}N = \mathbb{R}^{n}.$$ By dimension ...
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23 views

Discrete Dynamical System - determine what the model predicts will be the long-term distibution

If I have the following matrix: $$X_{n+1}\begin{pmatrix}1&0\\ 0&0.2\end{pmatrix}X_n$$ and if I also have the following initial state vector: $$X_0=\begin{pmatrix}5\\ 7\end{pmatrix}$$ What ...
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25 views

FitzHugh–Nagumo system with diffusion

I was studying the FitzHugh-Nagumo model with diffusion and I quite do not understand the meaning of it. If we consider the system without diffusion, \begin{equation}\label{FHN}\begin{cases} \dot{u}=...
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65 views

Find the limit cycles of $\dot r =r(1-r^2)$, $\dot \theta=1$

Apparently $r(t)=1$ is a limit cycle for the above system. Can anyone please explain why? Thank you
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19 views

Proving Conservative Forces are Path Independent

The transition from the 3rd equation to the 4th (the 1st being the one next to W=...) confuses me, what happens to the nabla, why does the derivative of the trajectory function become just the ...
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1answer
26 views

How to Show that Lorenz equations are invariant?

I am struggling a little bit with this question. I know that that the Lorenz equations are: \begin{align} \dot{x} &= \sigma(y-x)\\ \dot{y} &= rx - y- xz\\ \dot{z} &= xy - bz \end{align} ...
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34 views

About periodic trajectories of a Hamiltonian system

Consider a Hamiltonian system with Hamiltoniana $H (\mathbf{q}, \mathbf{p})$, where $H$ doesn't depend on time $t$. It is known that in some domain of phase space the trajectory of system are peiodic. ...
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23 views

Show for a sufficiently large C the ellipsoid $rx^2 + σy^2 + σ(z − 2r) ^2 = C$ is the boundary of a trapping region for the Lorenz equations?

We're told that geometrically, inflow means that the dot product of the normal vector and the flow is negative which means $∇f [\dot{x}, \dot{y}, \dot{z}]^T$ is negative for all points on the boundary....
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33 views

Does anyone have nice explanation about the theory? [closed]

I have hard time interpreting the Floquet theory. Does anyone have nice explanation about the theory?
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42 views

Numerical methods to solve Differential-Algebraic-Equations

I am new to the topic of differential-algebraic-equations: $ \dot x = f(x,u,c) $ $0=g(x,u,c) $ where $u$ are control variables and $c$ algebraic variables.In my first literature study i found two ...
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2answers
46 views

Solutions of autonomous system $\dot{x} = f(x)$ if $f\circ T = -T\circ f$ for some nonsingular matrix $T$

Having an autonomous system $\dot{x} = f(x)$ with general solution $\phi(t, \xi)$. If $T$ is an $m \times m$ nonsingular matrix such that $f(Tx) = -Tf(x)$ for all $x\in \mathbb{R}^m$ prove $\phi(t, T\...
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29 views

How to find periodic solutions for dynamical system?

I have the Hamiltonian system given by $$H=\frac{1}{2}x^2-\frac{1}{3}x^3+xy^2+\frac{1}{2}y^2$$ Using computer software I managed to plot the dynamical system in the phase plane. I am aware that the ...
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38 views

Homoclinic orbits of cubic potential

I found in Carles Simo's 'Hamiltonian Systems with Three or More Degrees of Freedom', among other references, that the homoclinic orbit for the cubic potential $\frac{y^{2}}{2}+\frac{x^{3}}{3}-\frac{...
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35 views

What is the meaning of “smeared limit cycle”?

I'm reading the paper Phase dynamics of coupled oscillators reconstructed from data by Kralemann et. al. (2008), which is about representing phenomena that exhibit a stable limit cycle (i.e. non-...
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113 views

Poincaré surface of section of the kicked rotator

Wikipedia's article about the kicked rotator says that it's Hamiltonian is \begin{equation} H(p,q,t)=\frac{1}{2}p^2+K\cos(q)\sum_{n=-\infty}^{\infty}\delta(t-n) \end{equation} and it's Poincaré ...
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26 views

Pushforward of Liouville measure on configuration space

Say I have $M_1$ a 3-dimensional compact Riemannian manifold, and $M=M_1^{n}$ the product manifold representing n particles on $M_1$. I can identify $TM$ with $T^*M$ via the metric $g$ and then the ...
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30 views

How to determine this system of ODE's?

I'm facing this problem: "Suppose you have this system of ODE's: $\begin{pmatrix} \dot y (t)\\ \dot x (t) \end{pmatrix} = \begin{pmatrix} a & b\\ c & d \end{pmatrix} \begin{pmatrix} y (t)\\ ...
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1answer
41 views

Differential equations question: Follow-up on Dynamical systems?

Yesterday I asked a question on here. Unfortunately I closed off the page without fully signing up for my account so I could not comment on the answer I received, whilst the answer was very good there ...
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33 views

Time-$t$ map of a Hamiltonian flow: how to check twist property?

I would like to obtain a general formula to verify if a certain time-$t$ map of a Hamiltonian flow is twist. I have a Hamiltonian $1$ degree of freedom system $H=H(q(t),p(t))$, such that all orbits ...
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1answer
46 views

How to determine which initial conditions will make the solution of a Hamiltonian system periodic?

I have a Hamiltonian system given by: $$\dot{x}=x+y-x^2\\\dot{y}=2xy-y$$ I have found that the Hamiltonian function for the system is given by $$H(x,y)=xy+\frac{1}{2}y^2-x^2y$$ and I have managed ...
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17 views

Intermittent Coupled Oscillations

A forced harmonic oscillator has equation of motion x''+ 4x = F(t) where F(t) = pi^2 -t^2 for tpi I know how to find the solution i.e. we use initial conditions given (x=0,t=0,x'=0) and solve the ...
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1answer
27 views

How to rearrange this doubly infinite sum for a diffeomorphism using the existence of a first integral?

Let's take two diffeomorphisms $F,G: \mathbb{R}^{n} \mapsto \mathbb{R}^{n}$. Let $x \in \mathbb{R}^{n}$, and $x^{n} = F^{n}(x)$, where $n \in \mathbb{Z}$. Suppose that $F$ has a first integral, i.e. ...
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20 views

weak mixing dynamical system but not mixing

Are there examples which are weak-mixing but not mixing. Let $T: X \to X$ (with measure $\mu$ and events $\mathcal{B}$). There is mixing mixing means events become independent (eventually) $$ \...
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Reference recommendation in dynamical system

I need a good book that is self-study in dynamical system. I have the book "Geometric Theory of Dynamical Systems" by Jacob Palis but it is difficult and is not a self-study book. I need a book that ...
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Orbits and Flows ODE

Suppose $ω(Γ) = \{x^∗\}$ is a single element set where $Γ$ is an orbit of a locally Lipschitz vector field $f \colon E → \mathbb R^n$ with $E ⊂ \mathbb R^n$ is open. Question: Show that for $x ∈...
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20 views

Definition of period-$k$ orbit of a map

For $k>1$, a period-$k$ orbit of a map $F$, or $k$-cycle, is a set of $k$ distinct points $\{x_0,x_1,\ldots,x_{k-1}\}$, where $x_i=F^i(x_0)$. The part I do not understand in the above definition ...
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39 views

Periodic Orbit using Poincare Bendixson Theorem

Consider the system $$x' = −y + x(r^4 − 3r^2 + 1)$$ $$y' = x + y(r^4 − 3r^2 +1)$$ where $$r^2=x^2 + y^2$$ Question: Show that $r' < 0$ on the circle $r = 1$ and $r' > 0$ on the circle $r = 2$. ...
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15 views

Reduced crossed product $C(X)\rtimes_r G$ in terms of orbits and groupoids

Let $G$ be a discrete group, and let $X$ be a compact Hausdorff $G$-space. It can be shown that the reduced crossed product $C^*$-algebra $C(X)\rtimes_r G$ is isomorphic to the reduced $C^*$-algebra ...
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36 views

Finding bifurcation of trigonometric system

I'm really struggling to find the bifurcation(s) of the system $x'=x^2 + \cos(x+ \mu)$, $\mu \in [0,2\pi)$. I've tried substituting $y=\mu+x$, taylor expanding, and just about everything else I ...
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29 views

Applied nonlinear dynamics: the onset of chaos in biological cycles (reference request)

I have seen some applied research in the onset of chaos in the study of current regulation in the human heart and the transition into cardiac arrest. I would like to review any literature that exists ...
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1answer
39 views

Every fiber bundle with Cantor set fiber is the suspension of a homeomorphism of the Cantor set.

I've heard that every fiber bundle (over $\mathbb S^1$?) with Cantor set fiber is the suspension of a homeomorphism of the Cantor set. Can someone explain the intuition behind the fact? Is there a ...
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1answer
40 views

Mcgehee transformation, conversion to polar coordinates and blowing up the singularity

I am looking for any reference on the above topics as I am struggling to convert the below to polar coordinates in phase space: The system is: \begin{equation*} x''=\frac{-\mu x}{(\mu x^2 + y^2)^{...
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1answer
37 views

What is a dissipative system?

If one had a system: \begin{align} \dot{x} = f(x,y,z)\\ \dot{y} = g(x,y,z)\\ \dot{z}=h(x,y,z) \end{align} Where each function may have parameters. How would one know if the system is dissipative? ...