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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$\Phi(\cdot ,x):I_x \rightarrow M$ injective?

I don't understand we the map $\Phi(\cdot ,x):I_x \rightarrow M$ from the excerpt from below of the lecture notes of my professor has to be injectiv. (Here $M$ denotes the domain of the function on ...
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9 views

A question on periodic points and recurrent points

I have been reading the book "dynamical systems and semisimple groups an introduction". In this book, a point of a topological $G$-space $X$ is a periodic point if $G/G_x$ is compact, where $G$ is a ...
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30 views

Find the fundamental matrix of a system of ODEs?

To linearize a system, in one of the steps I am required to find the fundamental matrix $\Phi$(t) of a system such that $\Phi$(0)=I. The example system my professor used: $\dot{x} = x - y - x^3 - ...
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1answer
25 views

Global existence of a dynamical problem.

Prove that all the solutions to the system $$ \begin{cases} \dot x= e^{-y^2}\sin(x^n+y^n), \\ \dot y= x^n\sin(x^n+y^n), \end{cases} $$ where $n$ is a fixed natural number, are defined on ...
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15 views

Green's function to operator

I would like to understand how one can show that the Green's function in this table is a Green's function to the D'Alembert operator? I refer to the wikipedia page about Green's function
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9 views

How to check if a point is on the attractor?

Consider a dissipative hyperbolic dynamical system defined on a set with a (strange) attractor. Given a point X on the phase-space, how do I (algorithmically) check if it is on the attractor? For ...
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2answers
40 views

How to find the derivative of the flow of an autonomous differential equation with respect to $x$

Ok, may be this is a silly question but consider the following. Let $\dot x=f(x)$ be an autonomous differential equation with $f$ having enough smoothness (Say $C^2$). Let $\xi:\mathbb ...
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2answers
46 views

1-dimensional foliation on a surface

Is it possible to find a 1-dimensional nonsingular foliation on an orientable surface with one boundary component such that lines of the foliation are transverse to the boundary?
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9 views

Backstepping analysis of multi-input system

Suppose I have a simple system that's like following: $\dot{x}_1 = A x_2 + Bx_3 \\ \dot{x}_2 = u_1 \\ \dot{x_3} = u_2$ I am familiar with a standard method of backstepping if there was only one ...
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1answer
79 views

Stability of nonlinear system given by $\dot{x} = f_1(x) + f_2(x)$

I have a nonlinear system $\dot{x} = f_1(x) + f_2(x)$ defined in a domain $U \subset \mathbb{R}^n$. I know that $x_0$ is an asymptotically stable and the only equilibrium point of the two systems ...
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1answer
20 views

Conditions for “forward” measure-preservation

A transformation $T$ being $\mu$-invariant is by definition a transformation satisfying $$\mu(T^{-1} E) = \mu(E)$$ for all measurable sets $E$. I was wondering what are sufficient conditions for being ...
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38 views

Let $\eta (x)=\int_0^\infty e^{at}\xi(\phi_t(x)) dt$ then $\eta$ is a $C^1$ function

Consider the following problem. Suppose that $a>0, r >0$ and $\xi:\mathbb R \to [o,\infty)$ is a $C^2$ which vanishes in the complement of the interval $(-r,r)$. Also suppose that ...
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31 views

Invariant Measure

Let $\dot{x}=u(x)$ a dynamical system ($x\in\Gamma$) with solution $x(t)=\Phi^t_u(y)$ and $\mu$ a $\Phi^t_u$-invariant measure on $\sigma_\Gamma$. I want to show that the smooth density $\rho=d\mu/dV$ ...
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0answers
17 views

Mixing System and density argument

A Mixing system is defined as a dynamical system $(\Omega,\phi^t,\mu)$ for which the following relations holds $$ 1)\qquad\lim_{t\rightarrow\infty} \mu(\phi^{-t}(A)\cap B)=\mu(A)\mu(B); $$ $$ ...
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1answer
62 views

Determining whether the origin is an attracting fixed point for a scalar system

I have been asked to determine and prove the attraction properties of a continuous-time dynamical system, generated by the ODE \begin{equation} \frac{dx}{dt} =-x \end{equation} which gives the system ...
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1answer
34 views

Every solution of the system is attracted to the center manifold

I am trying to solve the following problem. Determine a center manifold for the rest point at the origin of the system \begin{align} \dot x &=-xy \\ \dot y&= -y+x^2-2y^2 \end{align} a) ...
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2answers
365 views

How do I find transfer function of a discrete-time system when its state-space form is given?

I read this and this Wikipedia pages, but both of them are explaining continuous-time systems. My question is about discrete-time case. For example, given the state-space equations of the second ...
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1answer
29 views

Dynamical system with no point of period 3

This is a repost of http://math.stackexchange.com/questions/732343/period-3dynamical-systems. I posted an answer to that question. Someone voted that answer down so that the Community bot would delete ...
3
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1answer
39 views

Do Anosov flows exist on two dimensional compact manifolds?

Question: Let $M$ denote a two-dimensional, smooth, compact Riemannian manifold (without boundary). If $\phi:\mathbb{R}\times M \rightarrow M$ is a smooth flow, then prove that $\phi$ is not an ...
4
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1answer
53 views

The system $\dot{x}=x^2$, $\dot y=-y$, has infinitely many (local) center manifolds

Consider the system, \begin{align} \dot{x}&=x^2 \\ \dot y&=-y \end{align} I am trying to show that this system has infinitely many local center manifolds. Here is what I have done so far: ...
4
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1answer
45 views

Period doubling bifurcation in a quadratic map

I am attempting the find $\mu$ for which the map $$x_{n+1} = \mu + x_n^2$$ undergoes a period doubling bifurcation. I understand that finding the fixed points of the map is the first step towards ...
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26 views

Question on extension of cocycles

Given a countable discrete group $G$ and suppose $G$ acts on a compact metrizable abelian group $Y$ with normalized Haar measure $\mu$, measure preserving, let $\mathbb{T}$ denotes the unite circle. ...
3
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1answer
82 views

Number of limit cycles: Counterexample of the extended Bendixson-Dulac criterion?

The problem concerns the number of limit cycles in the vector field of coupled differential equations (ODEs) in two dimensions, i.e. $$ \ \dot{x} = X(x,y)\\ \dot{y} = Y(x,y) $$ Specifically, let $$ \ ...
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1answer
170 views

Bounding error when iterating a function

If I am iterating some function $f$ that goes to infinity as x goes to infinity with error $o(g(x))$, for example, is there anyway to bound the error? To be more specific, if I have some sequence ...
2
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1answer
38 views

Lyapunov Exponents for independent-nonidentically distributed matrices?

My question is highlighted in bold at the end. $\mathrm{\underline{Background}}$ Consider a product of i.i.d. $d\times d$ random matrices $A_{i}$ (with $\mathbb{E}\log\left\Vert A_{i}\right\Vert ...
4
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1answer
41 views

One-parameter group for spheres

Let $S^1$ be the unit sphere $x_1^2+x_2^2=1$ in $\mathbb{R}^2$ and let $X=S^1\times S^1\in\mathbb{R}^4$ with defining equations $f_1=x_1^2+x_2^2-1=0, f_2=x_3^2+x_4^2-1=0$. The vector field ...
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0answers
42 views

Uniqueness of solution for a system of differential equations

A friend of mine working on Auction Theory needs to establish uniqueness of solution (up to initial and boundary conditions) of a system of differential equations of the form $$ ...
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0answers
28 views

Square a linear ODE

Assuming that I have a linear ODE without any singularities over the complex numbers $$\sum_{k=0}^{n} g_i(x) y^{(k)}(x)=0.$$ Now I substitute $\sqrt{f}:=y$ into this differential equation and square ...
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25 views

Is mixing preserved under a measure theoretic-isomorphism of dynamical systems? [closed]

Show that mixing is an invariant of measure theoretic-isomorphism. I think that if I have two probability preserving transformation isomorphic together and one of them is mixing, I should prove ...
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28 views

Diagonalize Complex ODE

I'm trying to solve for the dynamics of one coordinate of a coupled system of linear differential equations with complex coefficients. Physically, a number of single-pole harmonic oscillators with ...
4
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1answer
73 views

Interpretation of generalized eigenvector in orbits

First of all, this is my fourth question about dynamical systems in a week, sorry for that. Considering a linear bidimensional dynamical (autonomous) system, the orbits can be plotted in the phase ...
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2answers
44 views

Derivative of a time derivative

I wasn't really sure how to phrase the title. Suppose I have a purely time-dependent function $x(t)$, and I want to know its time derivative $\dot{x}:=\dfrac{dx}{dt}$. Then I ask how the time ...
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0answers
18 views

Attracting basin is simply connected

How to prove that the immediate attracting basin of a (finite) attracting periodic point is simply connected? It's a question from Devaney's An Introduction to Chaotic Dynamic System and a hint is ...
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0answers
16 views

Blow up in holomorphic dynamics

Someone could explain the concept of blow up used in holomorphic dynamics? Especifaclly in the context of iteratation of holomorphics functions. This concept could be taken to some of the deformation ...
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2answers
37 views

What is a Parabolic Fixed Point?

I know the definitions of hyperbolic and elliptic fixed point (or equilibrium). However, when I google I find references to 'parabolic elliptic points' but not a proper definition.
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32 views

Parabolic PDEs and Gradient Systems

Apologize in advance for the length of this question, I need some help in clearing some things up that I haven't quite got my head around yet. It seems to be easy to find things out about finite ...
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2answers
295 views

Closed form solution of this second order linear difference equation?

$$ y(k + 2) - 3y(k + 1) +2y(k) = 2^k + k $$ Transform into a system of $n$ first order equations (Step 1) $$\begin{align} x_1(k) &= y(k)\\ x_2(k) &= y(k + 1) \end{align}$$ It follows that ...
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1answer
288 views

stable, center and unstable manifolds

‎let ‎‎$‎f:M‎\rightarrow ‎M‎$ ‎be a‎ ‎‎‎‎diffeomrphism and‎‎‎ ‎$ ‎\Lambda‎ $ a‎‎ ‎hyperbolic ‎set. ‎We ‎can ‎give a‎ ‎characterization ‎of ‎(local) ‎stable ‎and ‎unstable ‎manifolds ‎by‎: ‎‎for ...
5
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2answers
194 views

Dynamics of the repetitions of $f(z) = z^{2} +\frac{1}{4}$

This is probably a classic and most likely an easy question, but I was not able to find an answer. If we have the iteration $z_{n+1}=g(z_n)$, where $$g(z) = z + a z^{2},$$ with $a\ne 0$, then using a ...
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0answers
57 views

Integrability of 1-forms and Stokes' Theorem

Let $\alpha$ be a $1$-form defined on a manifold $M$ and $\Delta = ker (\alpha)$. The classical theorem of Frobenius says that $\Delta$ is integrable if $\alpha \wedge d\alpha =0$ i.e if $d\alpha$ is ...
4
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1answer
66 views

Notational issues on differential equations

I am studying dynamical systems and I have some trouble in understanding the notation used for differential equations. For example when I read $$\overset{..}{x}=F(x),$$ how should I interpret ...
6
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2answers
268 views

Transition time in a Lotka-Volterra system

I am working with a set of real-valued ordinary differential equations based on the Lotka-Volterra competition equations: $$\begin{align} \dot{a_1} & = a_1 \left( 1 - a_1 - 2 a_2 \right) \\ ...
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0answers
21 views

Factor and isomorphism

As we all known, if two dynamical systems are isomorphic, then we can say these two systems are the same. If one dynamical system $S_1$is a factor of the second dynamical systems $S$, then we say the ...
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1answer
22 views

Unique solution to a arbitrary non-linear system under monotonicity assumptions

I have a map $f:\mathbb{R}^n\times\mathbb{R}^m \to \mathbb{R}^n$ of two arguments $x, y$, which has a following properties: The jacobian matrix of $f$ wrt to the first argument $\frac{\partial ...
2
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0answers
78 views

Quaternion conversion

We have a normalized orthogonal co-ordinate frame travelling through the curve as in figure 1 below, from one end to other. Let us call starting end as A and ending end as B. What we know is initial ...
3
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153 views

Center Manifold Theorem

Let $f:M \rightarrow M$ be partially hyperbolic diffeomorphism of $M$ with the usual definition that at each $p$ tangent space splits to $Df$ invariant subspaces: $T_pM = E^s_p + E^c_p + E^u_p$ with ...
2
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1answer
112 views

Properties of this Sturm-Liouville problem.

Given the ODE $$f''(x) + \left(\alpha_1 \cos(x) + \cos^2(x) - \lambda \right) f(x)= 0,$$ where $\theta \in [-\pi,\pi]$, $||f||_{L^2} < \infty$. I was wondering whether there is anything we ...
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0answers
45 views

Central manifold theorem => Stable/unstable manifold?

I'm a bit confused why we always separate the stable/unstable manifold theorem and the central manifold theorem. The stable/unstable manifold theorem applies to a hyperbolic point ...
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1answer
512 views

Prove that the tent map has exactly nine 6-cycles.

Prove that the tent map T(x)= {2x if 0<=x<=1/2 and 2-2x if 1/2
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3answers
512 views

Eigenvalue problem for ODE with singular coefficients, $-(1-x^2) y'' + py'+qy=\lambda y$

(I did not change anything, I just rewrote the ODE in a simpler form): I started with an ODE (first ODE) : $-(1-x^2)y''(x) +x y'(x) - \left( \alpha x + \gamma x^2 \right) y(x) = \lambda y(x),$ ...