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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30 views

Are these two kinds of definitions about expansivity equivalent?

Let $(X,d)$ is a metric space and $X$ has no isolated points, $T:X\rightarrow X$ is a continuous self-map. Def1. $T$ is strongly expansive if there exist $\varepsilon>0$, for any $x,y\in X$, ...
4
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1answer
44 views

Negative divergence implies convergent flow?

Suppose we have a differentiable vector field $X:\Omega\to\mathbb{R^n}$ defined on an open, bounded and simply connected region subset $\Omega$ of $\mathbb{R^n}$, and its divergence is negative ...
2
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0answers
21 views

Mixing process in statistics vs. mixing in classical ergodic theory texts

In dynamical systems a transformation $T$ is strongly mixing if $\lim_{n\rightarrow \infty} P(A \cap T^{-n} B) = P(A)P(B)$ (e.g., Patrick Billingsley's Ergodic Theory and Information) For stochastic ...
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0answers
19 views

Seeking for a discrete time vector Gronwall type inequality

I have a sequence of vectors $\{v_n\} (\ v_n\in \mathbb{R}^N,\ n\ge 0),$ evolving in such a way that they satisfy the following inequalities, $$\|v_{n+1}\|_2\le \left\|a_n+\sum_{0\le k\le ...
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1answer
23 views

Are these two kinds of definitions about sensitivity equivalent?

Let $(X,d)$ is a metric space, $T:X\rightarrow X$ is a continuous self-map. Def1. $T$ is strongly sensitive if there exist $\epsilon>0$, for any $x\in X$ and any positive number $\delta>0$, we ...
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63 views

Periodic orbits on centre manifold

I am interested in periodic orbits of mechanical systems of second-order dynamics with no damping, i.e. governed by an equation of the type \begin{equation}(1)\quad \ddot x + f(x)=0 \end{equation} ...
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35 views

Anosov system: original reference requested

In the book S. Wiggins, Introduction to applied nonlinear dynamical systems and chaos. Vol. 2. Springer Science $\&$ Business Media, (2003). the author recalls a system: $$\begin{cases} ...
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1answer
59 views

Elegant approach to prove the convergence of this recursive sequence

Suppose $$S_1=1,S_{n+1}=S_n+\frac1{S_n}-\sqrt 2$$ Prove that $S_n$ converges. I was hinted to observe $S_{2k+1}$ and $S_{2k+2}$ respectively, so I tried calculating $$S_{n+2}-S_n=\frac{(1-\sqrt 2 ...
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1answer
58 views

Does it make sense to think of a non-constant solution to $\frac{dx}{dt}=0$ (A steady state solution)? [closed]

For instance if $\displaystyle\frac{dx}{dt}=x-t$, then $\displaystyle\frac{dx}{dt}=0$ at $x=t$
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13 views

Noncausal dynamical system

The differential equation $$a_ny(t)^{(n)} + \dots + a_0y(t)^{(0)} = b_mu(t)^{(m)} + \dots + b_0u(t)^{(0)} $$ with $a_i,b_i \in \mathbb{R}$ and $y,u:\mathbb{R}\to\mathbb{R}$ describes a ...
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32 views

Modelling food in population dynamics

I understand that this model of food means that the amount of food available decreases as the population increases, however I do not understand the two parts underlined in green. How do these agree ...
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110 views

Population dynamics

I don't understand why we make the three assumptions underlined above.
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26 views

Ergodic means and Birkhoff theorem

Let's consider the following map $$F(x, y) = \lim_{n \to \infty}{\frac{1}{n} \sum_{k=0}^{n-1}{f(\{x + ky \})}}$$ and $f(x) = x(1-x)$. I would like to evaluate the value of $F(x, y)$ for arbitrary ...
4
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0answers
32 views

a weak notion of flow in a metric space

I am seeing the definition of flow in a metric space : $f:M\times \mathbb{R}\rightarrow M$ is one flow if $M$ is metric space, $f$ is continuous and $f(x,t+s)=f(f(x,t),s)$ Note that the condition is ...
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15 views

Asymptotical stability

Population is gouverned by the difference equation: $y_{n+1}=(1-a)y_ne^{3-(1-a)y_n}$ $0<a<1;$ For what values of $a$ does the population have an asymptotically stable positive equilibrium? I ...
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1answer
17 views

Discrete population models

Consider the model: $y_{n+1}=ry_n(1-\frac{y_n}{k}); r>0$ a)Show that $y_{n+1}<0$ if and only if $y_n>k$. b)Show that $y_{n+1}>k$ is possible with $0<y_n<k$ only for $r>4$. ...
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1answer
33 views

Recommendation of a good source on Lyapunov theorem in dynamical systems

As part of my research I wish to read a full proof of Lyapunov's classic theorem on dynamical systems that for an analytic planar vector field where all Lyapunov/focal values are zero, the local phase ...
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1answer
48 views

Convergence of $A^t$

Consider $$a_t = Aa_{t-1}$$ i.e. $$a_t = A^ta_0$$ for a matrix $A$ and an intial vector $a_0$. In wonder whether this process converges iff the process $$b_t = Ab_{t-1}+c$$ converges, starting ...
4
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2answers
126 views

Nature of a fixed point in dynamical system

I have the following system: $\dot{x}=y+x(x^4+2x^2y^2-4x^2+y^4-4y^2+4)(8-(x^2+y^2)^{\frac{3}{2}})^3$ $\dot{y}=-x+y(x^4+2x^2y^2-4x^2+y^4-4y^2+4)(8-(x^2+y^2)^{\frac{3}{2}})^3$ Using cylindrical ...
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2answers
42 views

Find a differential equation for a given solution

I've been facing a problem since days till now but I could not find out an answer. I have a Cauchy-problem like this: $x'(t)=sin(x(t))$ with initial condition $x(0)=\frac{\pi}{2}$. Now I define this ...
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0answers
18 views

Bifurcation diagram

Consider the logistic map $x_{n+1}=rx_n(1-x_n)$, whose bifurcation diagram is shown below for $2.4 < r < 4.0$: I need to find a particular value of $r$ so that "attracting $2^k$ periodic points ...
4
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1answer
167 views

Formal proof of Lyapunov stability

I was trying to solve the question of AeT. on the (local) Lyapunov stability of the origin (non-hyperbolic equilibrium) for the dynamical system $$\dot{x}=-4y+x^2\\\dot{y}=4x+y^2$$ The streamplot ...
12
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1answer
126 views

Is the point promised by Borsuk-Ulam stable under perturbation of the map?

The Borsuk-Ulam theorem says that, given a continuous map $f: S^n \to \Bbb R^n$, there is some point $x \in S^n$ with $f(x)=f(-x)$. There may, of course, be many such points (maybe $f$ is constant!) ...
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38 views

Find a system with only two saddle points such that there are no trajectories that connect them

I'm looking for an example of a dynamical system with only two fixed points, both saddles such that there are no trajectories that connect them. Is this even possible? A picture would be sufficient ...
6
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1answer
116 views

Finding Lyapunov function for a given system of differential equations

I am being introduced to the Lyapunov functions in order to determine the stability of a given system. I know that finding a Lyapunov function is not easy, so I would like to ask for any trick or hint ...
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0answers
26 views

Is time-1 map of a Hamiltonian vector field defined on a cylinder always twist?

Suppose I have a one degree of freedom analytic Hamiltonian $H(q,p)$ defined on a semi-infinite cylinder, i.e. $(q,p) \in \mathbb{T} \times \mathbb{R}^{+}$, such that all level sets $H(q,p)=c$ are ...
1
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1answer
40 views

Show that $f$ also has an orbit of period $2$.

Given $f : [\alpha,\beta] \to [\alpha,\beta]$ with an orbit of period four $\{a,b,c,d\}$ ($a<b<c<d$), and given also $f(a)=b$, $f(b)=c$, $f(c)=d$, $f(d)=a$, is there a way I can show that $f$ ...
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1answer
36 views

How to find solutions and to graph this second order DE?

1)find two non-zero solutions that aren't multiples of each other 2) sketch the direction field in the yv-plane 3) for each solution, plot both its solution curve in the yv-plane and its y(t) and v(t) ...
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19 views

Strong mixing property for endomorphisms of a finite set

Let's consider $X = \{1, 2, \ldots, n \}$. I would like to establish, how many of the maps $f: X \to X$ have the following strong mixing property: For a given triple $(X, \mathcal{B}, \mu)$, $T: X ...
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0answers
41 views

What are the equilibrium solutions of this first order linear system?

$$\frac{dx}{dt}=4x-7y-1$$ $$\frac{dy}{dt}=3x+6y-12$$ one part of my notes says if the determinant gives you some number other than zero then there is only one equilibrium solution (the origin). ...
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34 views

Prove that a contraction semigroup generates a omega-limit set for each point

I think this is a problem closely related to dynamical system... Let $T(t)$ be a $C^{0}$ semigroup of contractions on a Banach space $X$, and assume that the resolvent $R(\lambda,A)$ of the ...
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1answer
41 views

Are integro-differential equations considered dynamical systems?

A definition of the dynamical system is that: $\phi:R \times E \to E$ is a dynamical system where $\phi \in C^1$, $E$ open subset of $\mathbb{R}^n$, and if $\phi_t(x) = \phi(t,x)$, then ...
1
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1answer
25 views

Determine integral for $\dot{u} = v$ and $\dot{v} = 2-u^2 - v^2 - u$.

I have the following system: $$ \begin{array}{ll} \dot{u} &= v\\ \dot{v} &= 2-u^2 - v^2 - u\end{array} $$ I need to determine an integral for this system, so a function $k(u,v)$ which is ...
6
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3answers
228 views

Dense set in the unit circle- reference needed

For $x \notin \pi\mathbb Q$, that is, a real $x$ that is not a rational multiple of $\pi$, consider the set $$\{(\cos nx,\sin nx):n = 0,1,2,...\}.$$ It is known that this set is dense in the unit ...
2
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0answers
21 views

Showing that points are in the Mandelbrot set

I am given ( a simplistic definition I think ) of Mandelbrot set: M- set of complex numbers $c \in \mathbb{C}$ s.t. the sequence $(z_n)$ is bounded where $z_0=0 , z_{n+1}=z_n^2+c $ Need to show ...
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0answers
15 views

What is the best way to get the directions of arrows in phase portraits?

Hi I would like to know the best way in which to find the direction of the trajectories in a phase portrait. Here is the system I am working with: ...
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1answer
29 views

2D system of ODEs and a constraint

I want to find functions $N:\mathbb{R} \to \mathbb{R}$ and $S:\mathbb{R} \to \mathbb{R}$ satisfying the following: $0 = -6N(x)^2 + 15 S(x)^2$ $\frac{dN}{dx} = \frac{12}{49} - 3 S(x)^2$ ...
0
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1answer
39 views

Properties of a differential equation

Question: Let $u,v \in C^1 ([0,1], \mathbb{R}^+)$ such that $u$ is increasing and $v$ is deacreasing and $u(0)=v(1)=0$. Suppose furthermore that the real function $0 \le x(t)\le 1$ satisfies ...
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0answers
37 views

Drawing the phase portrait

Let $\lambda >0$. Consider the non-linear system $$ \begin{cases}\dot{x}=x^2\\\dot{y}=-\lambda y\end{cases} $$ The aim is to get an idea of the phase portrait. There are some things I can see ...
2
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1answer
57 views

Image of a cube under the flow's action

Let's consider a system of ODEs: $$ \dot{x_{1}} = \sin{x_{2}}+x_{1}\\ \dot{x_{2}} = \cos{x_{3}}-2x_{2}+x_{1}\\ \dot{x_{3}}=\arctan{x_{1}}-x_{2}+x_{3}$$ I would like to find an image of the unit cube ...
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16 views

The Generalized Poincare-Bendixson Theorem2

Let f be a $C^{1}$ vector field in an open set $E\subset$ $R^{2}$ containing an annular region A with a smooth boundary. Suppose f has no zero in $\bar{A}$ ( the closure of A) and f is transverse to ...
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0answers
25 views

The Generalized Poincaré-Bendixson Theorem

Let f be a $C^1$ vector field in an open set $E \subseteq \mathbb R^2$ containing an annular region $A$ with a smooth boundary. Suppose $f$ has no zero in $\overline A$ ( the closureof A) and $f$ is ...
2
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0answers
45 views

Asymptotic analysis of non-linear ODE

I'm looking for references on the topic of asymptotic analysis of non linear ODE's of the sort $$ x'' + x'x = 0 $$ This specific case has an analytic solution (with some $\tanh(\cdots)$ involved) and ...
2
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1answer
43 views

to show that a set is positively invariant-dynamical system

consider the 2 differential equations : $${dS \over dt} = \lambda-\beta SI-\mu S+\theta I$$ $${dI \over dt} =\beta SI-(\mu +d)I-\theta I$$ In all papers that i read it is only mentioning that ...
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2answers
30 views

How is a state disturbance matrix constructed?

Consider the system: $\dot{x}$ = Ax + Bu y = Cx + Du Where x contains 4 states, we have 2 inputs $u = \begin{bmatrix}u_1\\u_2\end{bmatrix}$ and A, B, C & D are known. Now if 2 separate noise ...
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1answer
34 views

Example(s) of a symbolic dynamical system with proximal but not asymptotic points

Can anybody give me (an) example(s) of a symbolic dynamical system (preferably arising from a substitution) which has a pair of points which are proximal but not asymptotic? I would prefer to work ...
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0answers
27 views

Poincare-Bendixon Theorem on a unit disk

If applying the Poincare-Bendixson theorem to a region $D$ nearly all the text books I've read say that this region is generally an annulus. Would it be possible to apply the theorem to a unit disk ...
2
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0answers
169 views

Is $a_n = f(a + b c^n)$ when $a_0 = \ln(2)$ and $a_{n+1} = (e+1) a_n (a_n - 1)$?

Definition Let $a_n$ be a real sequence. Assume there exists a continuous real-periodic function $f(x)$ such that $f(n) = a_n$ And $f(x)$ has the period $t$ , where $t$ is An irrational real ...
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0answers
74 views

Show solution is an attracting center manifold.

Consider the small solutions of the following system \begin{align} \dot{x}&=\epsilon x-x^3+xy \\ \dot{y}&=-y+y^2-x^2 \\ \dot{\epsilon}&=0 \end{align} with $0<\epsilon\ll 1$. Show ...
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0answers
26 views

Continuous population models

I am trying to find all equilibria and determine which are asymptotically stable. \begin{align} x' &= rx\left(1 - \left(\frac xK\right)^\theta\right)\quad (0<\theta<1)\tag 1\\ x' &= ...