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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229 views

Show that Bellman-Gronwall's inequality

I'm trying to prove the theorem general of the Bellman-Gronwall's inequality: Assume that $u(t)$ be real valued non - negative continuous function, and such that $$u(t)\le u(\tau ...
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1answer
148 views

An ergodic theorem on the circle

Let $S^1$ be a circle (i.e. a closed $1$-dim. manifold) and let $F$ be a non-vanishing smooth vector field on $S^1$. Denote by $(t,x) \mapsto \Phi_t^x$ the flow generated by $F$. I want to show ...
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31 views

Is there a 2D 3-colorstate mobile automaton that grows like $ln^{0,5}(t)$?

Define an integer function $f(t)$ for an integer $t>25$ such that $|f(f(t)) - ln(t)| < \sqrt {ln(t)}+2$. Define $L(X(t))$ as the number of nonwhite states at iteration $t$ of mobile automaton ...
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108 views

Is there a generalization of the ODE Comparison Theorem to n dimensional systems such as this one?

Is the following theorem true? If so, under what conditions? If not, why not? For any finite set of points $S$, let $conv(S)$ denote the convex hull of $S$. Let $f:\mathbb{R}^{n+1} \to \mathbb{R}^n ...
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1answer
84 views

Showing the limit of a flow must be an equilibrium point under certain restrictions.

I'm stumped on how to approach this one: Consider the autonomous ODE $\dot{x} = f(x)$, $x \in \mathbb{R}^n$ with initial condition $x(0) = x_0$ and $f : \mathbb{R}^n \rightarrow \mathbb{R}^n$ (at ...
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98 views

Generic invertibility of a Gramian-like matrix

Motivation: Consider the cascade arising from connecting the system $\dot{x}_1 = A_1 x_1 + B_1 u_1$ with the system $\dot{x}_2 = A_2 x_2$, $y_2 = C_2 x_2$ according to $u_1 = y_2$, namely: $\dot{x} = ...
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1answer
148 views

Maps with every point being periodic

Does there exists a characterization of continuous maps $f:[0,1]\rightarrow [0,1]$ with every point $x\in [0,1]$ being periodic (i.e. if for every $x\in [0,1]$ there exists $n\in\mathbb{N}$ such that ...
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2answers
147 views

Prove that $x \equiv 0$ of $\dot{x}(t)=a(t)x$ is Uniformly Asymptotically Stable

I have a problem: Consider the scalar equation: $$\dot{x}(t)=a(t)x \tag{I}$$ where $a(t) \in C(\mathbb{R}^+)$. Prove that $x \equiv 0$ of $(I)$ is Uniformly Asymptotically Stable iff ...
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1answer
49 views

Properties for internal stability of a discrete-time system

These are two parts of a larger proof I'm working on, can't figure how i) implies ii) though. Dynamic system: $x_{(k+1)} = Ax_{k}, x(0)=x_0$ Where $A \in \mathbb{R}^{n\times n} $ is a real ...
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2answers
214 views

Dynamical systems and differential equations reviews/surveys?

I would be very glad if someone could point me to modern reviews/surveys on these topics. To be concrete, I'll provide some examples: S. Smale, Differentiable dynamical systems D. V. Anosov, On the ...
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2answers
96 views

A continuous function that when iterated, becomes eventually constant

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function, and let $c$ be a number. Suppose that for all $x \in \mathbb{R}$, there exists $N_x > 0$ such that $f^n(x) = c$ for all $n \geq ...
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1answer
156 views

There is non-trivial function satisfy the given condition?

Let $f:[0,1]\to\Bbb{R}$ to be a function satisfying that $$ f(x)=\begin{cases} \frac{f(2x)}{2} &\text{if }x<1/2 \\ \frac{f(2x-1)}{2}+\frac{1}{2} & \text{if } x\ge1/2\end{cases} \qquad ...
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2answers
56 views

Long run behavior of a dynamical system

I have to deal with a dynamical system that looks as follows, with $H$ being the initial state (parameters next to arrows denote transition probabilities between the states $H$, $L_1$ and $L_2$): I ...
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1answer
98 views

Parametrization of level sets of a smooth function

Let $H:\mathbb{R}^2\rightarrow\mathbb{R}$ be given by $H(q,p)=p^2/2+3q^2/2$ (single-well potential). This function has a critical point at $(0,0)$. Define $T:\mathbb{R}^+\rightarrow \mathbb{R}$ by, ...
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2answers
285 views

Show that System $(I)$ is stable iff $X(t)$ is bounded.

I have a theorem: For a linear homogeneous system: $$\dfrac{dx}{dt}=A(t)x \tag{I}$$ Where $A(t)=(a_{ij}(t))_{n \times n} \in C(\mathbb{R}^+,\mathbb{R}^{n \times n})$ Suppose that $X(t)$ be the ...
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52 views

How is open set defined in linear map space

I got this statement in my homework: Prove that the invertible linear contractions are an open set in $Mat(2\times 2;\mathbb{R})$ I know what "invertible linear contraction" , "open set" and ...
2
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0answers
242 views

“is topologically mixing” vs. “is topologically transitive” in the defition of chaos

Wikipedia gives essentially "is topologically mixing and has dense periodic periodic orbits" as the definition of chaos, and this paper shows that its (the paper's) definition of chaos is equivalent ...
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1answer
114 views

Compute $e^{tA}$

When I do my homework (stability theory), I must use the knowledge to the matrix. But I don't remember it :(. Here's my problem: For the system of equations: $$\begin{cases} & \text{ } ...
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72 views

A problem about Cantor set and found when learning dynamical systems.

Consider the family of functions F(x)=$x^3 -\alpha$x, for $\alpha \gt 0$ Prove that if $\alpha$ is sufficiently large, then the set of points |$F^n(x)$| which do not tend to infinity is a Cantor ...
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1answer
2k views

Determining the bifurcation value(s) for a one-parameter family

Let's say we have a one parameter family: $$ \frac{dy}{dt} = y^2 + k $$ I want to find the bifurcation value. What does this mean? It seems like I need to set dy/dt = 0 and then solve for k, but ...
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841 views

When does gradient flow not converge?

I've been thinking about gradient flows in the context of Morse theory, where we take a differentiable-enough function $f$ on some space (for now let's say a compact Riemannian manifold $M$) and use ...
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1answer
161 views

Construct a Liapunov function for this system

Construct a Liapunov function for the system (Determine the stability of $x \equiv 0$): I have an example:$$\begin{cases} & \mathrm { } \dot{x}= -x^3+xy^2\\ & \mathrm { } \dot{y}= ...
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1answer
515 views

Determine the stability property of the critical point at the origin ($x=y=0$) for the following system.

Determine the stability property of the critical point at the origin ($x=y=0$) for the following system: I have an example: $$\begin{cases} & \mathrm{ } \dot{x}= \tan(y-x)\\ & \mathrm{ } ...
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1answer
47 views

Theory of periodic solution

I want to study the theory of the periodic solution of ordinary differential equations, but I don't know how to study. Could you give me some references?
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1answer
91 views

What does this dynamic system represent for?

I know systems like $$\frac{dx}{dt}=Sx$$ where $S$ is a symmetric matrix admit a solution that dialates along eigendirection of $S$. And systems like $$\frac{dx}{dt}=Ax$$ where $A$ is a ...
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1answer
79 views

Question on dynamical system

i have this exercise : we consider the following model : $$ \begin{cases} x'& = x(4-x-y)\\ y'&=y(2+2\alpha-y-\alpha x) \end{cases} $$ a) Find the critical point $P$ does not depend on ...
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57 views

Discrete vs Continuous Dynamics

Where does the problem arise concretely in considering a discrete process as a sampling of a continuous process? Can we use continuous methods of solving say differential equations and find the ...
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0answers
101 views

Meaning of quasiperiodicity in classical KAM

I'm learning about the classical KAM theorem, and I can't quite infer precisely what the term "quasi-periodic solution" means in the theorem's statement. I'm reading the following introductory note: ...
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1answer
170 views

Coupling in the circle map

I'm currently investigating Arnold tongues (areas in parameter space with rational rotation numbers $\rho$, ie. $\rho(\Omega, K) \in \mathbb{Q}$) arising when iterating the circle map $$ ...
4
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3answers
263 views

Discuss the convergence of $ \left \{ a_n \right\} $ where $ a_{n+1}=\frac{a_0}{2}+\frac{a_n^2}{2},n\geq 1 $

Let $$ a_{n+1} = \dfrac{a_0}{2} + \dfrac{a_n^2}{2} $$ where $ a_1 = \dfrac{a_0}{2} $ and $ n\geq 1 $ Discuss the convergence of $ \left\{a_n\right\} $
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53 views

How do I determine the distance of an object if the acceleration is constantly changing?

I am investigating a pendulum's movement and I am trying to create a proof, or an equation that describes the movement. My problem is that as the pendulum is falling the angle between the force of ...
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1answer
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
127 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
725 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 ...
2
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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
266 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 ...
2
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1answer
176 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
226 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 ...
4
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1answer
227 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
199 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
123 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 ...
2
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1answer
324 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
170 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 ...
2
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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 ...
3
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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 ...