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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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 $(X,\mathcal{M},\mu,T)$ ...
0
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0answers
38 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 ...
2
votes
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 $r_2 = \frac{3}{2}$ ...
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votes
1answer
51 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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3answers
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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1answer
74 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
votes
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 ...
0
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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 ...
0
votes
2answers
28 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 \mathbb{...
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0answers
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 ...
0
votes
2answers
60 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 ...
1
vote
1answer
50 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 ...
0
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0answers
15 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 ergodic....
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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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0answers
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 $\boldsymbol F^{-1}(\boldsymbol f)$,...
0
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1answer
17 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 ...
1
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1answer
47 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 ...
1
vote
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 $M\gt2^...
1
vote
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 ...
0
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1answer
22 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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0answers
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 ...
0
votes
1answer
44 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}\\ \dot{...
0
votes
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
54 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 ...
0
votes
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 = ...
0
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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: $$f(x,y)=(x,y) $$ $$\...
1
vote
1answer
47 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 \...
10
votes
2answers
173 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 = \frac{8}{3}\\...
0
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0answers
57 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 ...
0
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1answer
53 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 ...
1
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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 m\{\frac{1}{...
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votes
1answer
41 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 $-\frac{1}{4}&...
0
votes
1answer
31 views

Prove that the iteration of $\sin(x)$ goes to zero as $n$ goes to $\infty$ [duplicate]

Basically let $S(x)=\sin(x)$ such that $S^2(x)=\sin(\sin(x))$ and $S^3(x)=\sin(\sin(\sin(x)))$ and so on until $S^n(x)=\sin(\sin(\ldots\sin(x)\ldots))$ Prove that $S^n(x)\rightarrow 0$ as $n\...
0
votes
3answers
42 views

Positive invariance of a set under a system of ODEs

Given the system of ODEs, $$x'=x(1-x-y)$$ $$y'=y(x-1),$$ $Q=\{(x,y):x\ge 0, y\ge 0\}$, and $S=(x,y)\in Q:x+y\le k$, $k>1$, I need to show that $S$ is invariant under this system of ODEs. Attempted ...
0
votes
1answer
38 views

Invariance of sets for systems of ODEs

Given the system of ODEs $$x' = x(1-y)$$ $$y'=y(x-1),$$ let the set $Q=\{(x,y):x\ge 0, y\ge 0\}$. Explain why $Q$ is invariant for this system of ODEs. My explanation: If $x > 1$ and $y<1$ then ...
2
votes
1answer
38 views

Why c>1/4 is not in Mandelbrot set

As title: $f_c(x)=x^2+c$ I got to the step: $f_c(x)>x$ (for all x) But what's next? How to show that after k iterations, $f^k_c \to \infty$ as $k \to \infty$ Thanks,
2
votes
1answer
32 views

Solve a system of second order differential equations

I have a system of second order differential equations which is $$m_1x_1''=-k_1x_1-k_2(x_1-x_2)\\m_2x_2''=-k_2(x_2-x_1)$$ where $(x_1(0),x_1'(0),x_1(0),x_2'(0))=(1,0,2,0)$ and $(m_1,m_2,k_1,k_2)=(1,...
0
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0answers
21 views

Stability of LTI systems under saturation

Consider the saturation function $$ \sigma(u)=\max(\min(u,1),-1) $$ for $u\in\mathbb{R}$. With slight abuse of notation, if $u\in\mathbb{R}^n$ let $\sigma(u)$ also denote the same function applied ...
0
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1answer
50 views

behavior of the Linear system of an ODE model

I am working on a predator-prey model and the linearization about and equilibrium point $(0,e_2)$ has Jacobian matrix as follows $$\mathcal{J} = \begin{pmatrix} 0 & 0\\ b& -b \end{pmatrix},$$...
0
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1answer
16 views

How to show that $x_{n+1} - x_* = M(x_n) (x_n - x_*)$ dynamics converges?

I am trying to find conditions of convergence (or non-convergence) of a system that behaves in the following manner (quasi-linear since the matrix is not stationary): $$ x_{n+1} - x_* = M(x_n) (x_n - ...
1
vote
1answer
28 views

Provinf Orbits are eventually fixed or eventually prime-2-periodic

Please I need help with this question: Let $S = \{a,b,c,d\}$ be a finite set and suppose that $f \colon S \to S$ has the property that: $$f(a) = f(b) = f(c) = d$$ Prove that each of the orbits of ...
1
vote
1answer
15 views

Entropy of factor map

Let $A$ and $B$ be two compact metric spaces with $B\subset A$- Moreover, let $T\colon A\to A$ continuous and let $S\colon A\to B$ a continuous surjection with $S\circ T=T\circ S$. Moreover assume ...
1
vote
1answer
50 views

Can the root locus of a minimum phase plant become unstable?

I have a discrete system for which the root locus equation is given as: $$A(z) + K\cdot B(z) = 0$$ They are such that $A(0) = 1, B(0) > 0$, and $K>0$. $\frac B A$ is minimum phase and a ...
0
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0answers
25 views

Center manifold reduction

This a verbatim copy of an example on center manifold reduction on nonlinear dynamical system I found on some lecture note: Consider the system \begin{align*} \dot{x}&=x^2y-x^5\\ \dot{y}&=-y+...
1
vote
0answers
32 views

Perburb the Monodromy of Lefschetz fibration over a disk

Suppose that $\pi:E \to D$ is a 4-dimensional Lefschetz fibration over a disk,(more general, Lefschetz fibration over a surface with boundary ) and let $\Omega$ be a closed 2-form on $E$ such that it ...
2
votes
2answers
38 views

What is the period of the harmonic oscillator?

I am trying to find the elapsed time $T$ (or transit time) over one cycle of the harmonic oscillator $$\ddot{x} + \omega^2x=0.$$ I worked this out to be \begin{align} T &= \int_{R}^{-R} \...
1
vote
0answers
45 views

Hartman-Grobman Theorem - Necessary?

The Hartman-Grobman theorem states, in layman's terms, that a nonlinear system and the corresponding liniarized system behave similarly around a hyperbolic equilibrium point (in terms of vector fields ...
1
vote
2answers
48 views

Stability criteria for linear systems with auxiliary variables

Classical texts for control theory show the linear system $\dot x=A \,x$, is stable if the real parts of the eigenvalues are negative. Does the same criteria apply for a system of the following form:...
0
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0answers
14 views

What is an example of a stochastic nonlinear dynamic system with 2 separated stable orbits

I have some social science data to which I would like to fit a stochastic difference or differential equation in two variables. (I observe the system only at discrete intervals). This system that has ...