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 ...

learn more… | top users | synonyms (1)

0
votes
1answer
18 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) ...
1
vote
1answer
23 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
votes
1answer
25 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 ...
3
votes
1answer
33 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
votes
1answer
48 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
votes
1answer
39 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 ...
3
votes
1answer
74 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 $$ \ ...
1
vote
0answers
41 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 $$ ...
0
votes
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 ...
-2
votes
0answers
25 views

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

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 ...
0
votes
0answers
27 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 ...
0
votes
1answer
146 views
+50

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 ...
0
votes
2answers
43 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 ...
0
votes
0answers
14 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 ...
1
vote
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 ...
0
votes
0answers
31 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 ...
4
votes
1answer
65 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 ...
1
vote
0answers
25 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 $\rho=d\mu/dV$ ...
0
votes
0answers
20 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 ...
2
votes
0answers
74 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 ...
4
votes
1answer
71 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 ...
1
vote
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 ...
1
vote
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 ...
0
votes
0answers
25 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. ...
1
vote
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 ...
7
votes
2answers
104 views

Rigorous mathematical treatments of engineering topics

I started out as an engineering student and got interested in mathematics. So after some point (Rigorous analysis and linear algebra, some real analysis, basic measure theory and topology etc.) I ...
7
votes
1answer
81 views

Who are the most influential cows in a herd of cattle?

You have a herd of cattle moving in different directions. The cows in the herd are more or less always moving, at different direction and in different velocities.When a cow bumps another cow it ...
2
votes
1answer
45 views

Link between the two definitions of a “hyperbolic point”

The common definition of a hyperbolic point for a flow of a vector field $f$ is a fixed point in which the eigenvalues of the Jacobian matrix of $f$ all have non-zero real parts ...
0
votes
2answers
42 views

Prove a limit with condition specified at infinity

Suppose that $$ \lim_{t\rightarrow \infty}\left(\dot{x}(t)+\gamma x(t)\right)=0,\quad \gamma>0. $$ How can I prove $$ \lim_{t\rightarrow \infty}x(t)=0~? $$ Please give a strict proof. Thanks!
2
votes
1answer
54 views

Limit of a Discrete Dynamical System

For the system defined below, the point by point evolution remains bounded for all $t$ so I could see that some sort of limit exists. However, the question is what sort of limit is it -- a single ...
1
vote
0answers
45 views

Baker's transformation: continuity, orbits of irrational and rational points

I've reading the Pugh's Analysis book and I have problems with one exercise. This says: The baker's transformation: a rectangle of dough is stretched to twice its length and folded back on itself. ...
0
votes
1answer
27 views

The largest real eigenvalue of a matrix is bigger than 1

I have a problem which is interesting: given a real matrix $A_{n\times n}$, when this matrix has a largest real eigenvalue which is strictly bigger than 1. If possible, can you give some conditions ...
1
vote
1answer
36 views

Existence of ergodic joining

Let $\underline{X}=(X,\mathcal{B},\mu,T)$ and $\underline{Y}=(Y,\mathcal{B},\mu,S)$ be ergodic measure preserving systems on Borel probability spaces. A joining of $\underline{X}$ and $\underline{Y}$ ...
2
votes
0answers
30 views

What is the solution to the system $\frac{df_n}{dt} = kf_{n-1}-(k+l)f_n+lf_{n+1}$?

I'm trying to solve the system $$ \begin{matrix} & \frac{df_1}{dt} = kf_1+lf_2 \\ & \vdots \\ & \frac{df_n}{dt} = kf_{n-1}-(k+l)f_n+lf_{n+1} \\ & \vdots \\ & \frac{df_N}{dt} = ...
2
votes
1answer
44 views

Prerequisite of Dynamical system and applied PDE

For the further research interest, I want to focus on the application of Dynamical systems and PDE in the field of robotics and neuroscience, particularly from a mathematical points of view. ...
0
votes
1answer
75 views

Solving for Center Manifold with Parameter

I have a system of ODEs given by $$\frac{dX}{d\tau}=\beta X\left(1 - \frac{X+Y}{N}\right)$$ $$\frac{dY}{d\tau}= Y\left(1 - \frac{X+Y}{N}\right)$$ where $\beta $ is a parameter. How should I ...
0
votes
0answers
30 views

Careers in Mathematical Modeling of Non-Linear Dynamical Systems

What are some industrial uses of nonlinear dynamical systems and the subsequent mathematical models and simulations? Further, what particular industries and specific companies make use of such ...
0
votes
1answer
33 views

periodicity of an interval exchange transformation(IET)

Let $T$ be an IET. That is, $T:[0,1] \rightarrow [0,1]$ is a piecewise orientation-preserving isometry. Let $D$ be the set of points whose entire forward iterates are well-defined. I have the ...
1
vote
1answer
39 views

Fixed Matrices over finite field by a map

Consider a set $M_n$ of all possible square matrices of dimension $n$ over a finite field $F_q$. Clearly the cardinality of the set $M_n$ is $q^{n^2}$. Let us consider a map $f:M_n$ $\longrightarrow$ ...
3
votes
1answer
37 views

Definition of strong mixing and definition of measure-preserving

I have a few questions regarding measure-preserving dynamical systems $(X,\mathcal{A},\mu,T)$. 1) The definition of measure preserving is always stated as $$\mu(T^{-1}B)=\mu(B),$$ for all $B$. I ...
3
votes
0answers
100 views

Intersections of two exponential curves in a plane

I am struggling to show that two exponential curves in $\mathbb{R}^n$ do not overlap except finite distinct points. Could anyone help me? Let $A\in\mathbb{R}^{n\times n}$ and ...
9
votes
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) : $-(1-x^2)y''(x) +x y'(x) - \left( \alpha x + \gamma x^2 \right) y(x) = \lambda y(x),$ ...
5
votes
2answers
76 views

System of non-linear ODE's

do you have any suggestions to solve analytically the Non-linear ODE system $\dot x=18 x^2 y-3p x^2+6p xy$ $\dot y=18 x^2 y-6p xy $ where $p$ is a real constant. Thank you very much cheers
8
votes
3answers
213 views

Determining the maximum value for the solution of this delay differential equation?

I am working on the following delay differential equation $$\frac{df}{dt}=f-f^3-\alpha f(t-\delta)\tag{1},$$ where $\frac{1}{2}\leq\alpha\leq 1$ and $\delta\geq 1$. I know that there are three ...
3
votes
0answers
54 views

Boundedness of solutions of Difference equation

Consider a second order difference equation in complex plane, \begin{equation} z_{n+1}=\frac{\alpha + \beta z_{n}}{1+z_{n-1}},\qquad n=0,1,\ldots \end{equation} where the parameters $\alpha, ~\beta$ ...
1
vote
1answer
35 views

Partial derivation of a population kinetic's equation

In reviewing my biophysics' course on population kinetics I am stuck in finding which equation was actually used to derive from. It uses an example to "explain" the analytical method, in order to ...
2
votes
0answers
37 views

Stability properties of discretization of ODE

I am trying to find some conditions which guarantee that a continuous time dynamical system and it's discretization have the same behavior with regard to equillibrium points. Specifically that if the ...
0
votes
0answers
22 views

How to generate a Poincare section for discrete particle trajectory?

I'm a novice when it comes to generating Poincare sections, and I can't seem to get it right. I have a particle moving in a 3D periodic field, and I wish to generate a Poincare section of its ...
2
votes
1answer
33 views

Poincaré Recurrence Theorem (measure theory version)

I had a look on the proof of the following Recurrence Theorem of Poincaré: Let $(\Omega,\Sigma,T,m)$ be a conservative dynamical system in measure theory for which the function $T^{-1}$ ...
2
votes
1answer
43 views

Differential Equation Examples for different type of critical point

For a linear system $X'=AX$, there are only limited types of critical points according to the eigen values of $A$. When I want to considering non-linear dynamical system in $\mathbb{R}^2$ and ...