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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41 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? ...
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28 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 $G:\...
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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 Bowen-...
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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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27 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 t}\pmb{v}(t,x)=B(t,x,\pmb{...
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40 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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41 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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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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46 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
39 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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30 views

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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122 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
47 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 $\mu$ ...
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145 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
19 views

Definition of measure-preserving: why inverse image?

In the definition of measure-preserving dynamical system, the crucial equation is $$ \mu \left(T^{-1} \left(A\right)\right) = \mu\left(A\right) . $$ Why is it not the seemingly more natural $$ \...
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16 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 y\...
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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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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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29 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
29 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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50 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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18 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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53 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 omega-...
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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
25 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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29 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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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) = \frac12(...
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2answers
76 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 $ 2\...
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0answers
34 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 <\infty$...
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33 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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38 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
56 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
38 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 not ...
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17 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?
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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
51 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
20 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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33 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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101 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 $s(t)=(0,0,\sum_{k=0}^{\infty}d\delta(t-...
3
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30 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
22 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 sequence ...
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1answer
19 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 S,\...
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1answer
27 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
75 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?