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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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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37 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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18 views

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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28 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
37 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
38 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
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? ...
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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 ...
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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 ...
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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 ...
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26 views

Existence and uniqueness of solutions for a system of first order PDEs

Which results can be applied and which conditions are needed, to ensure the existence and uniqueness of the solutions of the first order of PDEs: A$\dfrac{\partial}{\partial ...
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39 views

Example of a dynamical system which has an $\omega$-limit which is a cylinder of closed orbits

I have been studying dynamical systems and have recently come accross the following theorem: Suppose $n=3$. Let $L$ be a compact limit set which contains no equilibrium. Then: $L$ is either a ...
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26 views

Dynamical Systems: Disease model, what happens to variable $m$?

In the diagram it shows that people can die from other causes at a rate $m$, however in the equations the $m$ and the variable $M_a$ disappear. Is there a mathematical reason for this to happen?
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38 views

how to get the probability $p=1−(1−1/N)^{Tma_i}$ of extracting at least one ball in the urn

I am reading supplementary information of the paper Activity driven modeling of dynamic networks. It analogys the number of out degree of a activity node by Polya urns problem: it will equal to ...
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1answer
25 views

Periodic point of dynamical system

Hi please help me someone with the proof: We have a function $f:\mathbb{R}\longrightarrow\mathbb{R}$ continous and invertible, discrete dynamical system is given by $x_{n+1}=f(x_n)$ (a): prove that ...
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43 views

Volume Contraction

I need to determine if this system exhibits volume contraction: $\dot x =yz-x-x^3$ $\dot y =xz-y-y^3$ $\dot z =xy-z-z^3$ My approach is to just calculate the divergence of the vector field F: ...
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1answer
38 views

Non-atomic, ergodic measure which is left and right shift invariant.

Given a one-sided shift space, say $X = \prod\limits_{n=1}^\infty \mathbb Z_2$. Denote the left shift by $T$: $T(x_1 x_2 x_3\cdots) = x_2x_3 \cdots$. There are lots of examples of $T$-invariant ...
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What is the significance of studying the steady state behaviors of a system? What information do steady state models provide us?

For example, http://www.jameslovelock.org/page31.html In this 1983 paper by Lovelock and Watson modeling Daisyworld, in equations (10) through (14), the paper considers the non zero steady state ...
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121 views

Uniform unboundedness of linear operators

Question: Suppose that $(T_k)_{k=1}^{\infty}$ is a sequence of invertible linear operators on $\mathbb{R}^n$. Suppose that $\forall x \in \mathbb{R}^{n}\setminus \{0\}$, we have $$\lim_{k\to\infty} ...
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3answers
40 views

Reference Request: Parameter Dependent Center Manifold Theorem for ODEs

Suppose we have an $n$-dimensional first order ODE of the form $\frac{dx}{dt}= f_{\mu}(x)$ with $\mu \in \mathbb{R}^k$ a parameter and with an equilibrium at $x=0$ $(f_{\mu}(0) =0)$. For fixed $\mu$ ...
4
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1answer
136 views

Relationship between Möbius transformations and flows/vector fields

I've noticed that the pictures illustrating the effect of Möbius transformations on the Riemann sphere (after stereographic projection to the plane) resemble the phase portrait of a vector field. For ...
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1answer
12 views

Definition of measure-preserving

In the definition of measure-preserving dynamical system, the crucial equation is $$ \mu (T^{-1} (A)) = \mu (A ) . $$ Why is it not the seemingly more natural $$ \mu (T (A)) = \mu (A ) ? $$
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1answer
15 views

Proving global exponential stability of a perturbed system

Consider the system $$\dot{x} = \left(A + \frac{1}{2\varepsilon}BB^TP\right)x + Dg(t,y),\quad y=Cx,$$ where $g(t,y)$ is continuously differentiable and satisfies $$\Vert g(t,y)\Vert_2 \le k\Vert ...
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23 views

Eigenvalues and speed of convergence

Consider some dynamical system \begin{align} \dot x = f^1(x,y)\\ \dot y = f^2(x,y)\\ \end{align} There exists a fixed point at $E = (\tilde x, \tilde y)$, i.e. $f^1(\tilde x, \tilde y) = f^2(\tilde ...
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1answer
32 views

Tent Transformation $x\mapsto3\min(x,1-x)$

Suppose I have a Tent Transformation which is defined by: \begin{align*}T(x)=\begin{cases}3x&\text{if $x\le\dfrac12$,}\\3(1-x)&\text{if $x\ge\dfrac12$.}\end{cases}\end{align*} After noticing ...
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0answers
33 views

Why should $\phi'$ and $\phi''$ be $\mathcal O(1)$?

As Strogatz writes in his book Nonlinear Dynamics And Chaos (p. 64) There are often several ways to nondimensionalize an equation, and the best choice might not be clear at first. Therefore we ...
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26 views

Poincare' map to find periodic solution.

Consider the equation $\dot{y} = (acos(t) + b)y - y^3$ $a > 0, b>0$. I know that I need to recast the equation as a first order system $\dot{y} = (acos(x) +b)y - y^3, \dot{x} = 1$. Also, we are ...
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1answer
28 views

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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47 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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0answers
50 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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1answer
13 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
22 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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0answers
25 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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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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36 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
35 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
74 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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32 views

Classifying the Trajectories of Pendulum

The equation of the pendulum is: $$\ddot{\theta}+\frac{g}{l}\sin\theta$$ After some manipulation, we get $$H=\frac{\dot{\theta}^{2}}{2}-\frac{g}{l}\cos\theta=\mathrm{positive\ constant}$$ ...
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23 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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32 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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35 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
55 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
37 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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0answers
16 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
18 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?