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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Why is the winding number 1 for the travelling wave solutions?

In this document, on page 7 below, it is said that for the travelling wave solutions $$ u_k^t=U_L(t\pm k-m) $$ with $L\geq e+r+1$ and $$ U_L(j)=\begin{cases}j, & 0\leq j\leq e+r\\0, & e+r+1\...
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23 views

Polar coordinates for vector field to find sticking flow

I am currently working on an impacting system which is basically just a spring damper and a circular enclosure. Because of the rotational symmetry of the problem I need the vector field in polar ...
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22 views

What is this bifurcation of a fixed point of a two-dimensional diffeomorphism with two parameters?

Suppose I have a diffeomorphism of a plane, $$\bar{x} = F(x,s,t)$$ where $x \in \mathbb{R}^{2}$ and $s \in [a,b] \subset \mathbb{R}$ and $t \in I_{2} \subset{ \mathbb{R}}$ are parameters. Suppose ...
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81 views

Proof that $\sum_{i=1}^\infty 2^i (\mathbf{1}_{A_i})^*(x)$ is finite almost surely [on hold]

Let $(X, \mathcal{A}, m, T ) $ be a dynamical system and $f$ satisfying $\int |f| \ln^+ \ln^+ |f| {\rm d}m <\infty$. Put $A_i=\{x: 2^{i}\le f(x)< 2^{i+1}\}$ for $i\ge 2$ and $A_1= \{x:f(x)<4\...
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20 views

Why'' half-orbits'' of minimal $\mathbb{Z}$- action on compact Hausdorff space are still dense?

We say an action of $\mathbb{Z}$ on a compact Housdorff space $X$ minimal if every orbit of the action is dense in $X$. We assume the action is free and $X$ has no isolated points. Then in this case,...
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33 views

Good closed form approximation for iterates of $x^2+(1-x^2)x$

Let $f(x) := x^2+(1-x^2)x$. Is there a nice nontrivial closed form approximation $g_n(x)$ over $[0,1]$ for the $n$-fold composition $f^{\circ n}(x)$? Obviously near $0$ we have that $f^{\circ n}(x) = ...
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2answers
49 views

What does 'dissipative PDE' means?

Can you give me an idea what is meant with dissipative partial differential equations? I am no phycist (and do not know the difference between initial energy to final energy), but wikipedia told me ...
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34 views

How to pass from an ODEs system to reactions?

I have the following system: $$\frac{dx}{dt}=a_1+\frac{b_1x^n}{K_1^n+x^n}-gxy-d_1x,$$ $$\frac{dy}{dt}=a_2+\frac{b_2x^m}{K_2^m+x^m}-d_2y,$$ where $a_1,a_2,b_1,b_2,K_1,K_2,g,d_1,d_2,n,m$ are real ...
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26 views

time-$t$ flow map of vector field

I met the following expression in some article "... where $\Phi_t$ is the time-1 flow map of the Hamiltonian vector field produced by the Hamiltonian function $H$ = ..." I haven't met any explicit ...
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16 views

Computational Methods for Determining Stability

For a nonlinear system $\dot{x} =f(x,\alpha)$ where $\alpha$ is a parameter, with fixed points $x^*$ such that $f(x^*,\alpha) = 0$, what methods are there for computationally determining the stability ...
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23 views

Analysing the modes of a signal with Laplace transform

If I have a linear dynamical system (assume continuous time for the time being) I can create the transfer function, let's say: $$\frac{1}{(s+a_1)(s+a_2)}$$ and the pole-zero map (this one is for e.g. ...
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35 views

Clarification on asymptotically stability of dynamical systems

I'm wondering if someone can provide a clarification between 2 seemingly opposing definitions from reputable sources on dynamical systems! My Russian textbook, "Dynamical Systems I: Ordinary ...
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28 views

An integral involving a dirac

Let $x(t)$ a process with a differentiable trajectory. How do you understand $\int_0^t |\dot{x}(s)| \delta_{x(s) = 0} d s$?
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187 views

Solving one Systems Equations, Research Level Questions?!

We have three equations as follows: $A_p=F \cos(\alpha+ \phi) - \mu N^{'}_{S_1} - \mu N_{S_1} - W \sin \theta = m ( \ddot x - r \ddot \theta (\sin (\gamma+ \phi)) )$ $B_p=F \cos(\alpha+ \phi) - \mu ...
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1answer
53 views

On the Liouville-Arnold theorem

A system is completely integrable (in the Liouville sense) if there exist $n$ Poisson commuting first integrals. The Liouville-Arnold theorem, anyway, requires additional topological conditions to ...
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1answer
20 views

Travelling wave ansatz

Consider $$ u_t=u_{xx}+f(u)-w,~~w_t=\varepsilon (u-\gamma w).~~(*) $$ Consider the ansatz $$ (u(x,t),w(x,t))=(u(\xi),w(\xi)),~~\xi=x+ct, c\in\mathbb{R}. $$ Putting this ansatz into $(*)$, it is said ...
2
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1answer
122 views

Finding a specific improper integral on a solution path to a 2 dimensional system of ODEs

In my study of dynamical systems I was recently met with this system of ODEs: $ \dot{x}=\frac{\sinh{(y)}}{\cosh{(y)}+A\cos{(x)}} $ $ \dot{y}=\frac{A\sin{(x)}}{\cosh{(y)}+A\cos{(x)}} $ for a ...
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1answer
52 views

Solving an improper integral contour integral, calculated via Wolfram but in need of analytic derivation possibly

In my studies of dynamical systems I have just encountered this supposedly tough looking improper integral, which is (not really relevant for my predicament) the Melnikov function, with the integral ...
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1answer
164 views

Existence of a homeomorphism that does not return much

Let $f:X\rightarrow X$ a homeomorphism where $X$ is a compact metric space. Fix $x\in X$, denote $O(f,x)=\{ f^n(x):n\in \mathbb{Z}\}$ the orbit of $f$ by $x$. For $m\in \mathbb{N}$ denote $O(f,x,m)...
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1answer
46 views

Why is cos and sin used in plotting this phase portrait

I'm plotting the phase portrait of the following system of ODE: $\frac{dx}{dt} = x + 3 y$ $\frac{dy}{dt} = -5 x + 2 y$ The code I have in matlab: ...
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1answer
46 views

Finding explicit solution to 2D system of ODE's

In my studies of dynamics, I came across this ODE system given by: $ \dot{x} = -A \frac{\pi}{k}\cos{(\pi y)}\sin{(kx)} $ $ \dot{y} = A\sin{(\pi y)}\cos{(kx)} $ where A and k are two non ...
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1answer
32 views

Energy surface of Hamiltonian system is compact

Suppose we have a Hamiltonian system in which the energy is conserved. (Specifically I am looking at a simple double pendulum.) Then we can consider the energy surface $M_e=\{(q,p)\in T^*M\mid H(q,p)=...
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20 views

How to calculate Floquet exponents for a Hamiltonian system?

Consider a periodic orbit $\bar{z}(T+t)=\bar{z}(t)$ with $z(t) =(x(t),y(t))$ of the Hamiltonian system \begin{equation*} \frac{dx_i}{dt} = \frac{\partial H}{\partial y_i} , \quad \frac{dy_i}{dt} = -\...
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1answer
36 views

Is there a method to quantify the return time to equilibrium?

I'd like to know how fast (or slow) a dynamical system will return to equilibrium, especially if it is near an unstable equilibrium. The question is vague, but that is intentional. I'm looking for ...
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0answers
31 views

Invariant measure of Poincare map

Let $M$ be a smooth manifold and let $v$ be a tangent vector field on $M$. Consider a system of ordinary differential equations $$ \dot x = v(x), $$ in local coordinates $x = (x_1, \ldots, x_n)$. ...
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0answers
21 views

Lyapunov exponent of multiple time series

I have multiple time series data that I have generated by varying the initial conditions infinitesimally. I now want to calculate the Lyapunov exponent to identify the sensitivity to initial ...
0
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1answer
37 views

What to call the eigenvalues that don't correspond to a conserved quantity

Consider your favorite continuous time nicely behaved dynamical system. If we linearize at an equilibrium and the largest eigenvalue is negative, then it's stable. Now let's add to that system the ...
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29 views

The zero point $(0,0)$ of $x'=x+y,y'=xy-bx-y$ is stable when $b>1$

Proof when $b>1$, the zero point $(0,0)$ of ODE $\left\{\begin{align}&x'=x+y\\&y'=xy-bx-y\end{align}\right.$ is stable. I couldn't find a proper Lyapunov V function. Let $v=x'$ we has 1st ...
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19 views

Maximum number of limit cycles of a system

Is there any theorem giving the maximum number of limit cycles that a system can have? More specifically, my system is $\dot{x} = -r(x+c)(x+y+z)$ $\dot{y} = b_1(a_1 x + (1-a_1)(z-y))$ $\dot{z} = ...
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0answers
28 views

Determining the global stability of a 3-dimensional system

I am trying to determine the global stability of the following system $\dot{x} = -r(x+c)(x+y+z)$ $\dot{y} = b_1(a_1 x + (1-a_1)(z-y))$ $\dot{z} = b_2(a_2 x + (1-a_2)(y-z))$ where $b_1,b_2,r\in(0,+\...
1
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1answer
32 views

Prove the invertibility of $X^T X$ when $X$ is a (rectangular) Toeplitz-like matrix.

In order to use a minimum squares estimator over some discrete dynamic system parameters, it is necessary to prove that the product $X^T X$ is invertible. Consider the following $N$ by $n+1$ matrix $X$...
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36 views

What is the difference between an autonomous system and a dynamical system?

Wikipedia says, Autonomous systems are closely related to dynamical systems. Any autonomous system can be transformed into a dynamical system and, using very weak assumptions, a dynamical system ...
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7 views

Examples of multiply-connected compact configuration spaces

I'm a looking for examples of dynamical systems that have multiply-connected compact configuration spaces. Since I'm not a 100% sure about the correct terminology for the systems (I am sure about the ...
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0answers
12 views

Finding the local stable/unstable set (which is not a manifold) in a 2D discrete dynamical system

This is an example that I found in a paper: http://www.sciencedirect.com/science/article/pii/0022039673900776 The author considered a discrete dynamical system given by the following map: $$ f(x,y)=(...
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31 views

How to find a suitable function for Dulac's criteria in this example?

I have a system of odes $\dot{\mathbf{x}} = \mathbf{f(x)}$ where $\mathbf{x} \in \mathbb{R}^{2}$ and $\mathbf{f(x)}$ is defined below: $$\dot{x} = x- y - x^{3}, \qquad \dot{y} = x+y-y^{3}$$ I would ...
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55 views

Finding a parametrization of the solutions of $\frac{dx}{dt}=\frac{\sinh y}{\cosh y+A\cos x}$, $\frac{dy}{dt}=\frac{A\sin x}{\cosh y+A\cos x}$

I am trying desperately to find a parametrization for the following: $\frac{dx}{dt}=\frac{\sinh y}{\cosh y+A\cos x}$ $\frac{dy}{dt}=\frac{A\sin x}{\cosh y+A\cos x}$ I tried to devide the equation ...
4
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0answers
78 views

$\{\{(2^n+3^m)\alpha\}:n,m\in\mathbb{N}\}$ is dense in [0,1]??

Let $\alpha$ be an irrational real number. I wonder whether $\{\{(2^n+3^m)\alpha\}:n,m\in\mathbb{N}\}$ is dense in [0,1] in which $\{x\}$ means the fractional part of x. This is equivalent to the ...
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28 views

Projection of periodic trajectories

Let $(\bar x(t),\bar u(t)),\, t\in [0,1]$ solution of $$ \left \{ \begin{array}{l} \dot x_1 (t) = u(t)\, f(x(t)) \\ \dot x_2 (t) = u(t)\\ x_1(0) = 0, x_1(1)=1 \\ x_2(0) = x_2(1) \end{array} \right. $$ ...
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33 views

Reference request: how to check whether a set is invariant for a second order dynamical system?

I am looking for some examples where invariant set is proved for second order systems For a example, consider the Van Der Pol equation: $$\dot x_1 = x_2$$ $$\dot x_2 = -x_1 + 0.5(1-x_1^2)x_2$$ In ...
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1answer
53 views

Using the Baire Category Theorem to prove $\mu$ is trivial

Suppose we have a probability measure space $(X,\mathcal A,\mu,T)$ where $T$ is measure-preserving. Then if for every $A,B\in\mathcal A$ we have $\mu\left(A\cap T^{-n}B\right)=\mu(A)\mu(B)$ for all $n\...
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1answer
26 views

Kinematics Mechanics Find the length of the belt and the speed of the rack. [closed]

For A. I know that the formula for belt is simply $L=\frac{Pi(D_a+D_b)}{2}+2C+\frac{(D_b-D_a)^2}{4C}$ Which gives me $L= 116.99"$ since C is equal to $50$ For B. However I'm stuck and can't get the ...
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32 views

unstable manifold in 2d dynamical systems

I have 2d dynamical non-linear system $\dot{x} = f(x)$ with $x\in \mathbb{R}^{2}$ with exactly two stable attractors $x_{1} = ( a\quad b)^{T}$ and $x_{2} = (c\quad d)^{T}$ and one saddle point (One ...
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1answer
134 views

How to prove $(\{2^n3^m\alpha\})_{m,n\in\mathbb{N}}$ is dense in [0,1]?

$\forall \alpha\in [0,1]\setminus\mathbb{Q}$, how to prove $(\{2^n3^m\alpha\})_{m,n\in\mathbb{N}}$ is dense in [0,1]? $\{x\}$ is the fractional part of x. Any hint would be appreciated!
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31 views

Hartman-Grobman Theorem in One Dimension

Here's a (possibly) related question Dynamics question from Devaney which is the one dimensional form of Hartman-Grobman theorem on the real line. Let $p$ be a hyperbolic fixed point for $f\in C^...
3
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107 views

generating functions for catastrophe theory

I am studying Thom's theorem in catastrophe theory and am having a hard time understanding what the "generating functions" actually do. How exactly are they used to classify generic caustics? The ...
2
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1answer
43 views

Stability of a line of equilibria

I'm working with a nonlinear autonomous system $x'=f(x)$. This system stays in $\mathbb{R}^n_+$ whenever it begins there, and it has a ray of equilibria, i.e. there is a positive vector $x_0$ so that ...
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0answers
28 views

Linear structural stability for maps on the real line

This might be a basic question, but I think I'm completely missing the idea of this. The question comes from Devaney's An Introduction to Chaotic Dynamical Systems (p.g. 59, Ex. 11) We define the ...
2
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1answer
74 views

Analytical Solution to Coupled Nonlinear ODEs

I am looking to solve several coupled nonlinear ODEs like this one: $\hspace{20mm} \frac{d x(t)}{dt} = C_1 \cdot x(t) + C_2 \cdot y(t) + C_3\cdot (x(t)^2 + y(t)^2) x(t),$ $\hspace{20mm} \frac{d y(t)...
0
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0answers
17 views

Time-periodic vector field vs. time-periodic Hamiltonian

Let $X(t,x,y): \mathbb{R} \times \mathbb{R}^2 \rightarrow \mathbb{R}^2$ be a time-periodic Hamiltonian vector field, i.e. $X(t+1,x,y) = X(t,x,y)$ for all $t$. Let's say that $X$ is generated by a ...
0
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1answer
23 views

Neighborhood of inclusion in space of Lipschitz maps is 1-1

Let $B \subset \mathbb{R}^n$ be the closed unit ball. Let $i(x) = x$ denote the inclusion map. Let $\|\cdot\|$ be any norm. Given $f:B\to \mathbb{R}^n$, define the sup norm $\|f\|_\infty:=\sup_{x \in ...