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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58 views

Can the system $\partial_x f(x,y) = \dot{y}$, $\partial_y f(x,y) = \dot{x}$ be related to some Hamiltonian system?

If one has found some function $f(x,y): \partial_x f = \dot{y}, \partial_y f = \dot{x}$, is there a simple transformation or change of variables that results in Hamilton's equations $\partial_p H = ...
3
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0answers
122 views

Stability of linear systems with singular state matrix

Given a linear time invariant system $\dot X(t) = AX(t)$ where $X \in {R^{n \times 1}}$ and $A \in {R^{n \times n}}$ is a singular matrix ($A$ has at least one zero eigenvalue). How can I study the ...
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0answers
45 views

Looking for articles on postcritically finite rational maps in Russian or French

I'm looking for articles on postcritically finite rational maps. I found a few articles in English, but I can't find any in Russian or French.
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1answer
709 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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1answer
40 views

In linear iteration, find values for a,b that cause different outcomes

When iterating the equation $X_n=AX_0 + B$ for some initial value $X_0$, I need to find concrete values for $A$ and $B$ that: 1) cause the series to converge for some initial value $X_0$ and diverge ...
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2answers
149 views

M-set interior point probability on the real axis

For the real axis, the Mandelbrot set consists of points from $[-2,0.25]$. Some of these points are in the interior of the m-set, and some are on the boundary. Those points in the interior are ...
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1answer
90 views

How are definitions of chaos related?

Chaotic systems can be defined in many ways. One definition is that the system has a positive Lyapunov exponent, that is, two trajectories starting near each other will diverge exponentially quickly. ...
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1answer
74 views

Are there any mathematical/physical concepts or theories for dealing with a matrix in which the values are changing in a certain way?

As a matter of fact, my application scenario is a recommender system in which the interests/preferences of the users change. I have such a global user-interest matrix: the rows are the records of many ...
2
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2answers
258 views

Gradient System, Conservative System Confusion

I'm reading Strogatz book on Nonlinear Dynamics and am a bit confused about the distinction between Conservative Systems and Gradient Systems. On page 160, it is claimed that Conservative Systems ...
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1answer
173 views

If $P$ is an invertible transition probability matrix, does $P^{-1}[i,j]$ have any interesting meaning?

Suppose we have a Markov chain transition probability matrix $P$ that is invertible, i.e., $P^{-1}$ exists. Question: Does there exist a meaningful interpretation of the $(i,j)$ entry in $P^{-1}$? ...
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1answer
223 views

How does chaos arise in Hamiltonian systems?

I have a question about how chaos arises in Hamiltonian systems. I've been taking a upper year undergrad class on dynamics and it has focused on chaos a lot. So far however we have only seen the more ...
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0answers
68 views

What is the relation between singular point for a function and the one in a vector field?

What is the difference between sigular point for a function and the one in a vector field? Is the derivative or divergence at the singular point must be infinity? By the way, what is the relation ...
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222 views

Dynamical Systems

Would someone care to explain the basic theory of dynamical systems? i.e. an explanation of the following definition of a dynamical system: A dynamical system is a tuple $(T,M,\Phi)$ where $T$ is a ...
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1answer
197 views

What kind of polygonal surface has an interior angle > 360°?

Consider this polygon as the setting for a dynamical billiard: When it's drawn in the plane, the polygon intersects itself; it is non-simple. However, I don't want to embed the polygon in the ...
4
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1answer
121 views

Trying to prove a matrix is always convergent.

I have a matrix $Z$ of the form $Z = \left[Q^{-1}-Q^{-1}A^T\left(AQ^{-1}A^T\right)^{-1}AQ^{-1}\right]\Phi$ where, $\Phi$ is a diagonal matrix of real non-negative values. $\Theta$ (not ...
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1answer
313 views

Are all hyperbolic points/orbits unstable?

My understanding of hyperbolic points (correct me if I'm wrong) is that there must be an unstable and stable manifold in the neighborhood of the hyperbolic point. So essentially the hyperbolic point ...
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1answer
169 views

What is the shape of external rays landing on fixed points in case of quadratic discrete dynamical system?

In case of parabolic discrete dynamical system based on the complex quadratic polynomial fc(z) = z^2 + c some external rays land on alfa fixed point. Hera are 34 external rays landing on fixed ...
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2answers
124 views

How does the singularity of a system matrix affect the system's stability?

What can be said about system stability, given a singular system matrix below? \begin{align} A = \begin{bmatrix} 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 ...
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4answers
195 views

Enlightening Mathematical Models

What is your favourite Mathematical Model? What features make it intuitive or elegant? This question is largely inspired by an example and a desire to find other's like it. Suppose we have two ...
3
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1answer
74 views

Uniform Hyperbolicity Decay Estimate

This question has been posted on math overflow with little interest, so I am posting it here. http://mathoverflow.net/questions/139010/uniform-hyperbolicity-decay-estimate I have been trying to ...
3
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2answers
574 views

dynamical systems and invariant sets

I have basic questions to understand the invariant sets of dynamical systems. Let me define a dynamical system $\left\{ {T, X, \phi^{t}}\right\}$. Here an orbit with a starting value $x_{0}$ is ...
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2answers
1k views

Questions about uncoupling dynamical systems and phase plane portraits of the uncoupled systems.

I have found all equilibria, studied their nature, and have been able to make a parametric plot of the following non-linear system along a time axis: $$r'(t)=i-l.r(t)-\text{ux}. r(t). x(t)-\text{uy}. ...
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2answers
120 views

A function that is not contractive with respect to any metric

I am struggling with this homework question with is related to iterated function system and fixed point theory. The question is: Let $\Delta \in R^2$ be a filled non-degenerate triangle with ...
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2answers
621 views

Difference between equilibrium point and Unique Equilibrium Point?

Is there any difference between unique equilibrium point and equilibrium point? If yes, please tell me what it is and How is it used to solve a dynamical system. You can consider a dynamical system ...
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1answer
91 views

Possible to make a flow that forms horseshoes on a 2-dimensional manifold?

It it possible to have a flow $\phi(t,x)$ on a 2-dimensional manifold where for some $t > 0$, the map $g(x) := \phi(t,x)$ creates a horseshoe? By $\phi(t,x)$ I mean the solution to the ODE ...
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1answer
190 views

Global stable manifold always an embedded submanifold? Typo or misreading?

I was reading Brin and Stuck's Introdroduction to Dynamical Systems (link to pdf of book can be found by googling "Brin and Stuck's Introdroduction to Dynamical Systems"), and I came across on page ...
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2answers
139 views

“Constrained” numerical solutions of ODEs with conservation laws?

Hi know little about numerical methods and I was considering the following problem that possibly has standard solution in the literature. Suppose you have an ODE for wich we already know that it must ...
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5answers
333 views

Is Collatz' conjecture the only stable solution of its type?

The Collatz Conjecture is well known with the sequence $$f(n) = \begin{cases} n/2 &;\text{if } n \equiv 0 \pmod{2}\\ k\,n+1 &; \text{if } n\equiv 1 \pmod{2} \end{cases}$$ and $k=3$; the ...
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1answer
225 views

Bounded sets of isolated points in compact metric spaces

Context and definitions: Say $(X,d)$ is a compact metric space, with $f: X \rightarrow X$ continuous. For each $n \in \mathbb{N}$, the metric $d_{n}(x,y) = \max_{0 \leq k \leq ...
2
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4answers
297 views

Topological Dynamics: closure of forward orbit vs. $\omega$ limit set

Let $f: X \rightarrow X$ be continuous, where $X$ is a topological space. This forms a topological dynamical system. For $x \in X$, define $\omega (x) = \cap_{n \in \mathbb{N}} \overline{\cup_{i \geq ...
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1answer
94 views

Describe attractors of a finite family of contraction mappings

The question is to describe the attractor of iterated function system $\mathcal{F}=\{R^2,f_1,f_2\},$ where $f_1,f_2$ are the two affine transformations$\begin{bmatrix} 0 & 0.8\\ -0.5&0 ...
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1answer
181 views

Integrable systems and algebraic geometry

Could you recommend a good book to study integrable systems? I want it to be from the point of view of algebraic geometry.
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1answer
288 views

Constructing a non-linear system with prerequisites about the nature of its critical points.

An exercise from the book I am reading is: "Construct a non-linear system that has four critical points:two saddle points, one stable focus, and one unstable focus." I have tried many systems. I ...
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3answers
187 views

Extending a map of manifolds continuously

Let $M$ and $N$ be manifolds, and $A \subset M$ compact. Let $f:A \rightarrow N$ be a continuous mapping. Show there exists an open neighborhood $U$ containing $A$ and continuous extension $g:U ...
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0answers
225 views

Nondimensionalize Pendulum Equation

Equations for the dynamics of a pendulum: $A=-a\sin(\theta)-bv$ $\theta(0)=\theta_0$, $\;v(0)=\omega_0$ where $\theta$ is the angle of the pendulum with respect to it's natural resting point, $v$ ...
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2answers
134 views

Closed loop stability

Regarding the Lyapunov stability, we check if a nonlinear system stays near the equilibrium point or approaches to e.p. as time goes to infinity, when it is disturbed. Let's assume that we have a ...
2
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1answer
532 views

Stability of nonlinear system with borderline linearization

I have the following nonlinear system: \begin{align} ...
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0answers
61 views

Could A Dynamic System Approximate To Spherical Harmonics?

Spherical harmonics describes electron orbitals in the Schrodinger equation. However the possibility remains that electron orbitals are dynamic systems of interacting particles that merely approximate ...
4
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1answer
281 views

The Denjoy Theorem

I'm currently studying Denjoy's theorem, which says the following: Theorem If $f$ is a diffeomorphism of $S^1$ with a irrational number rotation $ \rho$ and the variation of $f^{'}$ (denoted by ...
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1answer
107 views

Finding Transfer functions for linearised systems

I'm using Nise for my control systems class. Finding a linearised system is all gravy baby, but when it comes to finding the transfer function Nise does some stuff which confounds me: See page 6/7, ...
6
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2answers
284 views

How to figure out the starting point for this Mandelbrot?

My answer for another math stackexchange question, asked by Gottfried, involved observing Mandelbrot bifurcation for the iterated function in question, $f(z)\mapsto z-\log_b(z)$. In particular, for ...
5
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1answer
105 views

Matrix differential equation and closed orbits

everyone. I am asking for a reference for the nonexistence of closed orbits (periodic orbits) of Matrix differential equations of the form \begin{equation} v\prime = M(v)\cdot v \end{equation} where ...
0
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1answer
60 views

Matrix expansion does not decrease norms

Given a block matrix $A = \left[ {\begin{array}{*{20}{c}} {{A_{11}}}&{{A_{12}}}\\ {{A_{12}}}&{{A_{22}}} \end{array}} \right]$, where $A \in {R^{N \times N}}$, it is true that the euclidean ...
4
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1answer
221 views

Finding the jacobian of a differential system with a piecewise function

My system: $$\frac{\mathrm{dx} }{\mathrm{d} t}=-ax^2+y^2-\gamma z$$ $$\frac{\mathrm{dy} }{\mathrm{d} t}=- h(y)-\beta y $$ $$\frac{\mathrm{dz} }{\mathrm{d} t}=x+h(y)-\beta z $$ where $h$ is the ...
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0answers
94 views

Question on O.D.E

Given three parameters $L,a$ and $\alpha$, we consider the differential equation : $$(E)\qquad x''+\alpha x' +a x + \sin x =L, \ t\geq0$$ Assume that $a>0$ and $\alpha\geq 0$ We consider the ...
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1answer
148 views

Liapunov functions

I would really like to see some very simple worked out or with some well pointed hints on these guys. i have two textbooks that outline the idea behind them, but both give only one example that are ...
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0answers
66 views

Bounded solutions of ODE

I have this ODE $(E)... x''+\alpha x'+ax+\sin x =L , t\geq 0$ we suppose that $a>0$and $\alpha \geq 0$ , and that there existe un constant $C>0$ such that $\frac{a}{4}x^2+\frac{y^2}{2}\leq ...
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3answers
330 views

Textbook Recommendation: Topological Dynamics

I need to take credits satisfying a topology requirement, and can structure it myself. My field of study is dynamical systems, can someone recommend a textbook that handles differential ...
4
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3answers
256 views

Iterates of $f_b(x) = x - \log_b(x) $ - for $\log(b) \approx 0.399$: convergence to accumulation points or chaos?

In a previous question I arrived at the family of functions (depending on a real parameter b for the base of exponentiation/logarithm): $$ f(x) = x - \log_b(x) \qquad \qquad b \gt 0$$ with the ...
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3answers
153 views

Generating Bifurcation Animations

https://en.wikipedia.org/wiki/File:Hopf-bif.gif Does anyone know how this animation was produced? I could make it by stitching together snapshots (what I'm doing) but this seems primitive, especially ...