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

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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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
43 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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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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72 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 ...
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0answers
29 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
21 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 ...
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18 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 ...
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1answer
25 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?
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19 views

Using Linear Kalman Filters with a Nonlinear System?

Can you answer these questions I have about using linear Kalman filters and extended Kalman filters with a nonlinear system? 1. Does using a linear Kalman filter mean that I must have a ...
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1answer
36 views

Eigenvector for a non-linear system

Using the reversibility arguments alone, show that the system $\dot{x}=y$ $\dot{y}=x-x^{2}$ has a homoclinic orbit in the half-plane $x\leq 0$ This is a non-linear system. A ...
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1answer
32 views

About Expansive homeomorphism

Let $f:X\rightarrow X$ be a homeomorphism. $f$ is called an $c$-expansive homeomorphism, whenever for every $x\neq y$, there is an integer $n$ with $d(f^{n}(x), f^{n}(y)) >c$. Question. Is there ...
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50 views

Eigenvector of unstable and stable manifold of a non-linear system with non-linear center

Show that the system $\dot{x}=y-y^{3}$ $\dot{y}=-x-y^{2}$ has a non-linear center and plot the phase potrait. My attempt: The system is non-linear so we linearise it: The ...
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2answers
57 views

Basin of attraction an open set?

Definition: The basin of attraction is the defined as the set of all initial conditions $x_{0}$ such that $x(t$) tends to an attracting fixed point $x^{\ast}$ as time $t$ tends to $\infty$. ...
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283 views

Is there an easy way to see that this simple recurrence is 9-periodic?

In a colloquium talk yesterday, Robert Bryant pointed out that for all initial values $a_0, a_1 \in \mathbb{R}$, the sequence generated by the recurrence relation $$ a_{n+1} = |a_n| - a_{n-1} $$ turns ...
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1answer
21 views

Measure preserving ergodic map commutes with complementation?

This is probably trivial (in which case I apologize), but it's late and I would really like a quick proof/counterexample for this (for a different problem that I'm doing): if $(X,\mathcal{M},\mu,T)$ ...
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0answers
34 views

Devaney's definition of chaos

I'm reading Banks, et. al. paper On Devaney's Definition of Chaos. In it, they say "It is not difficult to find transitive examples for which sensitivity is not preserved under conjugation." I'm ...
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1answer
46 views

Existence of periodic orbit of the ODE system $\dot{r}=r-r^3 \cos^2(\theta),\,\dot{\theta}= 1$

Consider the system of ODEs (in polar coordinates): $$\dot{r}=r-r^3 \cos^2(\theta)$$ $$\dot{\theta}= 1$$ If we take $r_1 = \frac{1}{4}$ then $\dot{r}> 0$, and if we take $r_2 = ...
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1answer
50 views

If $f$ and $g$ are bounded, then every solution of the autonomous system of differential equations is defined for $t \in \mathbb R$.

Consider the system of autonomous differential equations (autonomous system of differential equations?) $$x' = f(x,y)$$ $$y' = g(x,y)$$ where $x=x(t)$ and $y=y(t)$ Let $f$ and $g$ have first ...
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41 views

Is the closure of the set of all irrational rotation maps on $S^1$ dense in $Homeo(S^1)$?

I study about rotation maps on circle, and I have a question. Let $Homeo(S^1)$ be the set of all circle homeomorphisms with sup-metric $d(f,g)= \sup \{ d(f(x),g(x)| x \in S^1 \}$, and rotation map ...
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72 views

Find the fixed points of the system, and sketch the trajectories of the system

I am given the following system: $$x' = [(x-1)^2 + y^2]y$$ $$y' = -[(x-1)^2 + y^2]x \tag{*}$$ where $x = x(t), y = y(t)$. I am supposed to Find the fixed points of the system, and ...
2
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1answer
60 views

Long term Behavior of Dynamical System

Given the following dynamical system: $ \dot x = -6x^2+yz+x-1 $ $ \dot y = 4xz-3y^2+y-2 $ $ \dot z = 9xy-2z^2+z-3 $ What can you say about its long term behavior? Attempt: First, finding the ...
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0answers
30 views

Prove any $n>2$ DE doesn't hold Poincaré-Bendixson theorem.

How can I build a differential equation to show that Poincaré-Bendixson theorem doesn't hold for $n≥3$ ? Is it easy to take any D.E with $n=3$ and prove it? More specifically, can you give me a ...
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2answers
24 views

Simple eigenvalue of Koopman operator

Let $T : X \to X$ be a measure-preserving transformation and $U_T : L^2(X, \mu) \to L^2(X, \mu)$ , $(U_T f) (x) = f(Tx).$ What does it mean a $\bf{simple}$ eigenvalue of $U_T$? $\lambda \in ...
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16 views

Reference request: monotone and strongly monotone with respect to derivatives

Recall, let $H$ be a real Hilbert space. A mapping $F:H \rightarrow H$ is said to be monotone if $$ \langle F(u)-F(v), u-v\rangle\geq 0, \quad \forall u,v\in H; $$ strongly monotone if there exists ...
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2answers
59 views

Dynamics of a three dimensional system

I have a dynamical system in three dimensions given by: $\dot x = (1-x^2-y^2-z^2)x+xz-y$ $\dot y = (1-x^2-y^2-z^2)y+yz+x$ $\dot z = (1-x^2-y^2-z^2)z-x^2-y^2$ I analyzed the system by first finding ...
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1answer
49 views

Suppose the period of γn is λn. If there are points Xn ∈ γn such that Xn → X ∈ γ , prove that λn → λ.

I was wondering if someone could help me with an exercise from Hirsch, Smale, and Devaney, Differential Equations, Dynamical Systems, and an Introduction to Chaos. Let γ be a closed orbit of a ...
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0answers
14 views

If $1$ is a simple eigenvalue of $T$, then $T$ is an ergodic measure-preserving transformation

Let $T : X \to X$ be a measure-preserving transformaton and $U_T : L^2 (X, \mu) \to (X, \mu)$, $$(U_T f) (x) = f(Tx).$$ I have to show that if $1$ is a simple eigenvalue of $U_T$, then $T$ is ...
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0answers
20 views

Hamilton principle/dynamics teaching in earlier stages.

In finding dynamic motion of particles we use laws of conservation of energy and momentum. It is found the dynamics formulation using action integral $$ \int (T-V)\, dt $$ builds ODEs for dynamic ...
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19 views

Proof that $M_ {\boldsymbol f}$ has a neighbourood diffeomorphic to the product $T^n\times D^n$

I'm reading Arnold's proof of Liouville's theorem and got stuck with the following problem in subsection §50, A. Here the manifold $M_{\boldsymbol f}$ is defined as $\boldsymbol F^{-1}(\boldsymbol ...
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1answer
16 views

Estimation for entropy

Let $T\colon X\to X$ be continuous and $X$ compact and $K\subset X$ compact. By $s_n(2^{-k},K,T)$ denote the maximal cardinality of any $(n,2^{-k})$ separated subset of $K$. Suppose, we know for ...
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1answer
46 views

Constructing a Poincare Map

I need to construct a Poincare Map of the following dynamical system: $\dot x = x-(x+y)(x^2+y^2)$ and $\dot y = y + (x-y)(x^2+y^2)$ I changed the system to polar coordinates which gives me: $\dot ...
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1answer
12 views

Let $\{ y_k \}$ that satisfies $ y_k\le {2^k\over M}y_{k-1}^\beta$ , then $\lim_{k\to \infty}y_k=0$.

Let be a sequence $\{ y_k \}^\infty _{k=0} \subset (0,\infty) $ that satisfies $$ y_k\le {2^k\over M}y_{k-1}^\beta , $$ where $k=1,2,...$, and $\beta\gt 1$ , $M\gt0$. Prove that if ...
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1answer
43 views

Topics in Geometry and Dynamical Systems

Dynamical Systems/Fractal Geometry and Differential Geometry/Topology are really interesting areas of study. My question is whether there is any direct connection between them? In other words, are ...
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1answer
35 views

configuration spaces in mathematics and in physics

On the Wekipedia website Configuration space , there are two configuration spaces defined. One is Configuration spaces in physics, the other is Configuration spaces in mathematics. Question. Do ...
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1answer
21 views

Different ergodic probability measures are mutually singular

Can someone, please, give me a hint on how to demonstrate that different ergodic probability measures are mutually singular? Thank you!
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20 views

Dynamical system in a square

I am considering a problem that is asking me to explore a deceptively simple dynamical system and discover some of surprising properties. I want to consider the motion of four particles A,B,C and D in ...
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1answer
40 views

Stability of nonlinear ODE linearized on a periodic solution.

I'm reading a paper about the effects of pulsed immunotherapy in cancer treatment. They have system$$ \begin{array}{ccc} \dot{E} & = & cT-\mu_{2}E+\frac{p_{1}EI_{L}}{g_{1}+I_{L}}+s_{1}\\ ...
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1answer
18 views

Show that the following functions are flows on the spaces indicated

Show that the following functions are flows on the spaces indicated. Find the vector field for each flow $$\phi_t(x)=\frac{x+\tanh t}{1+ x \tanh t}, x \in [-1,1]$$ Solution so far So $\phi_0(x)= ...
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1answer
52 views

How to convert dynamical system to polar coordinates? [closed]

I have a dynamical system on the plane given by $$\dot{x}=-y+x\left(1-\sqrt{x^2+y^2}\right)\\ \\ \dot{y}=y+x\left(1-\sqrt{x^2+y^2}\right)$$ I want to convert this into polar coordinates as it will be ...
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0answers
25 views

Long-time behavior of a diffusion-like equation

I would like to known the long-time behavior of $\mathbf{u}_{t}(n)$ obeying the equation $ \mathbf{u}_{t+1}(n) = T_l\mathbf{u}_{t}(n-1) + T_0\mathbf{u}_{t}(n) + T_r\mathbf{u}_{t}(n+1) $ where $T_l = ...
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0answers
24 views

Determine the stability of the equilibria

I'm given a discrete dynamical system in which $f$ is given by $$f(x,y)=(1+y-ax^2,bx). a,b \in \mathbb{R}, b\neq 1$$ I'm asked to find the equilibria which I have done below: $$f(x,y)=(x,y) $$ ...
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1answer
46 views

Bifurcation Problem

I am trying to classify the type of bifurcation for the dynamical system given by: $\dot x = x^2+y^2-2my$ $\dot y= mx-y$ with m as a varying parameter The fixed points are at (0,0) and ($2m^2 ...
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2answers
166 views

How to know whether an Ordinary Differential Equation is Chaotic?

Assuming we have an ordinary differential equation (ODE) such as Lorenz system: $$ \dot x=\sigma(y-x)\\ \dot y=\gamma x-y-xz\\ \dot z=xy-bz $$ where $$ \sigma = 10\\ \gamma = 28\\ b = ...
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0answers
53 views

A question about the “state-transition-matrix” of a physical system,

Here's a simple but potential research problem that I am learning about. Let's say I am studying a physical system that is governed by N objects. At each time, each object is either "active" and ...
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1answer
52 views

2D Bifurcation Classification

Given the system with m as a varying parameter: $\dot x = mx^2-y$ and $\dot y = m+y - x$ Determine any bifurcations that occur Attempt: x nullcline $y=mx^2$ y nullcline $y=x-m$ Fixed ...
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0answers
56 views

Find imaginary part of complex expression

Given the system of ODEs, $$x'=x^3-3xy^2$$ $$y'=3x^2 y-y^3,$$ it can be shown that the system may be written as $z'=z^3$, where $z=x+iy$. However, I don't seem to get how to show that $\Im ...
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1answer
40 views

Compactness and positive invariance of set under flow of ODEs

Given a system of ODEs, $$x'=y$$ $$y'=x-x^3-y$$ $$x(0)=x_0$$ $$y(0)=y_0,$$ also given a set $S=\{(x,y):V(x,y)\le k, x>0\}$, $V(x,y)=-\frac{x^2}{2}+\frac{x^4}{4}+\frac{y^2}{2}$, where ...