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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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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1answer
14 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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1answer
14 views

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

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 ...
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12 views

Use Osgood's criterion and comparison principle to show that the equation $p'={\alpha}p^a+{\beta}p^b$ has global and local solutions. [on hold]

Fix $\alpha , \beta \in (0,\infty)$ . Use Osgood's criterion to show that the equation $$p'={\alpha}p^a+{\beta}p^b,\quad t>0,$$ has at most one nonnegative solution if $a,b \ge 1$. Also, prove ...
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14 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 ...
4
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1answer
20 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
63 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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27 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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35 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 ...
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18 views

$\Sigma_2^+$ is complete given metric $d$

Let $\Sigma_2^+$ be the set of functions $f : \mathbb{N} \to \{1,2\}$. I wish to show that this set endowed with the metric $$ d(s,t) = \sum_{j = 0}^\infty \frac{\vert s_j - t_j \vert}{3^j} $$ is a ...
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1answer
29 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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2answers
22 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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0answers
21 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} ...
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1answer
58 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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1answer
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
24 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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1answer
23 views

Second order nonlinear ODE system [closed]

How to solve a second order nonlinear ODE system like this $u''=\frac{1}{Du}(u-av-u^2v)\\ v''=\frac{1}{Dv}(-b+av+u^2v)$ where $Du,Dv,a,b>0$ are constants. I don't have the slightest idea of how ...
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0answers
31 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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20 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 ...
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32 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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40 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, ...
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1answer
25 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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1answer
37 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 ...
3
votes
1answer
33 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. ...
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110 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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24 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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26 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
39 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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30 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
45 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 ...
2
votes
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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1answer
18 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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2answers
32 views

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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1answer
29 views

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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1answer
18 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 ...
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0answers
36 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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Ergodic measure for action of $SO_2$ on lattice

Let $X:= \Gamma/PSL_2(\Bbb R)$ and for each $x \in X$ define $\phi_x(g):= xg^{-1}$ for $g \in SO_2$. Then the induced measure $(\phi_x)_*m_{SO_2}$ is ergodic for the $SO_2$ action and is a factor of ...
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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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25 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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0answers
27 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
34 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
37 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 + ...
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1answer
34 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? ...
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1answer
27 views

Understanding shift to polar coordinates in the newtonian central force system of ODE's

This is from Hirsch, Smale and Devaney chapter 13. The larger context is moving towards blowing up the singularity at the origin of the system. The second order ODE is defined, $X:t\rightarrow ...
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29 views

For a nondecreasing map, if $\xi(a) < \eta(a)$, then $\xi(t) < \eta(t)$ for all $t \in [a,b]$.

I am studying the following theorem from Morris Hirsch's second paper on systems of differential equations which are competitive or cooperative: Let $V \subset \mathbb{R}^n$ be on open set and ...
2
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1answer
20 views

Showing uniform convergence to origin in 3rd quadrant for $x(t)=\frac{1}{\frac{1}{x_0}-t}$ as $t\ \rightarrow \infty$

I want to show that for the system $\dot{x}=x^2, \dot{y}=y^2$,any solutions starting in the 3rd quadrant not including 0, converge uniformly to the origin. For an initial point $(x_0,y_0)$, (note both ...
0
votes
1answer
32 views

Projection of measure with bowen - walters metric.

Given $X$ a compact metric space, $f:X\to X$ be a homeomorphism and consider the quotient space $Y^{1,f}=(X\times [0,1])/\sim$, where $(x,1)\sim(f(x),0)$ for all $x\in X$. Let $d^{1,f}$ be the ...
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1answer
36 views

Showing that a family of metrics induce all the same topology on special sequence space

Let $X = \{0,1\}$ and consider the discrete metric $$ d(x,y) := \left\{ \begin{array}{ll} 0 & x = y \\ 1 & x \ne y. \end{array}\right. $$ Now consider $X^{\mathbb N_0}$, the set of all ...