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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45 views

Correct determination of local stable and unstable manifold?

Consider the system $$ \dot{x}=-x+xy,\qquad\dot{y}=y+x^2. $$ The task is to determine the equations for the local stable and local unstable manifold $W_{\text{loc}}^{s}$ and ...
2
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2answers
75 views

Simple, stable $n$-body orbits in the plane with some fixed bodies allowed

I'm working on a visual simulator for the $n$-body problem in the plane (here). The goal is to show how complex behavior can arise from the simple inverse-square law of gravity. To that end, I want ...
2
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0answers
39 views

Name of bifurcation that causes eigenvalues to switch sign in a saddle?

What is the name for a bifurcation where the signs of the eigenvalues switch? E.g. Given a 4-dimensional saddle (two positive, two negative real eigenvalues), as I bifurcate a parameter two ...
2
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1answer
39 views

Compute local equations of stable/ unstable manifolds

Today, the tutor of our dynamical system course said that in the exam one part will be to determine equilibria and to compute the local equations of stable and unstable manifold. I do not know what ...
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1answer
43 views

How does the solution of a differential equation on a manifold yield a map?

In: "A solution $x^μ(λ)$ is a map from $\mathbb{R} → M$": Why is $x^μ(λ)$ considered a map and why does it go from $\mathbb{R} → M$? I can't seem to illustrate this in my mind. In:"If the manifold ...
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1answer
53 views

Solving system of ordinary linear differential equation

I want to solve the following system of differential equations $$\dot x_0(t)=-ax_0(t)+\mu x_1(t)$$ $$\dot x_n(t)=(\lambda(n+1)+a)x_{n-1}(t)-(a+(\lambda+\mu)n)x_n(t)+\mu(n+1)x_{n+1} \qquad \forall ...
0
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1answer
21 views

The limit of minimal points is still minimal?

$X$ is a (compact) metric space, $T:X\rightarrow X$ is a continuous self-map. Let $x\in X$, $E\subseteq X$, $E$ is said to be $T$-invariant if $TE\subseteq E$. $E$ is called a minimal subset of ...
0
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2answers
80 views

Initial value problem for a system of ODEs with two parameters

Consider the following initial value problem for the autonomous system of ODEs \begin{equation}%%\label{eqn: } \begin{cases} %\vspace{3mm} x'(t)=y(t),\; t>0,\\ %\vspace{3mm} ...
1
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1answer
29 views

Feigenbaum attractor is not an attractor?

I am reading about Feigenbaum attractor (FA) and am getting very confused with something that is described in some books. It is written that FA is not an attractor because in its neighbourhood however ...
2
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1answer
35 views

Discrete systems with complicated basin boundaries?

I am trying to come up with the strategy to write my Master's thesis in mathematics. At the moment it is as follows: Finding a (preferably) discrete dynamical system that possesses at least 3 ...
2
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0answers
30 views

Feigenbaum attractor is uncountable set?

How to prove that Feigenbaum attractor (appearing at the accumulation point in logistic map) is an uncountable set? (I am not a mathematician but know how to prove that a Cantor set is uncountable.)
3
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104 views

Is it possible to solve such a system?

I have the following two equations: $$P_t = \frac{t-1}{t}P_{t-1} + \frac{1}{t}Q_{t-1}$$ $$Q_t = \frac{1}{t} + \frac{t-1}{t}Q_{t-1} - \frac{1}{t}P_{t-1}$$ with $P_0 = 0$ and $Q_0 = 0$. As time goes ...
0
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0answers
20 views

Topologically transitive, pointwise minimal systems

Let $T$ be a group, and let $(X, T)$ be a flow, i.e. $X$ is a compact Hausdorff space and $T$ acts on $X$ by homeomorphisms. A flow $X$ is called topologically transitive if for any open $U, ...
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57 views

Decide whether the systems are topologically equivalent

Consider the systems $$ \begin{cases}\dot{x}=-y-2x-4x^3\\\dot{y}=-2y-x\end{cases}\text{ and }\begin{cases}\dot{x}=-x-7y\\\dot{y}=-2y+10x\end{cases}. $$ Decide whether they are ...
2
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16 views

Is there a mixing condition to get the decay property I want?

Let $(X,\mu)$ be a probability measure space and $T:X\to X$ an ergodic invertible measure preserving transformation. Consider a measurable set $A\subset X$ with $0<\mu(A)<1$ For each $N$ define ...
1
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29 views

Bifurcation points, and nonlinear stability analysis

I am studying a system of the form $$Q\ddot{x}+F = 0$$ where $Q$ is an $n\times n$ matrix, with entries depending on the dependent variables $x_1,x_2,..,x_n$, while F is an $n\times 1$ vector, and ...
1
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1answer
51 views

Application of Poincaré-Bendixon Theorem

Consider the planar system in polar corrdinates: $$ \frac{dr}{dt}=r-r^3+r^2\sin\phi,\qquad\frac{d\phi}{dt}=1+\frac{1}{2}r\cos\phi. $$ Show that it has at least one periodic orbit. ...
4
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1answer
100 views

A rational orbit that's provably dense in the reals?

Iterating the map $\ \ x\ \mapsto\ x-\frac{1}{x},\ \ $ the orbit of initial point $2$ is "probably" dense in $\mathbb{R}$. Is there an explicit rational mapping together with an initial rational ...
0
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0answers
21 views

Bounded orbit is compact?

Maybe this is a foolish question but if we have a dynamical system, for example an ODE, and are talking about orbits (or trajectories), aren't bounded orbits $x_t$ automatically compact sets in ...
1
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1answer
35 views

'Continuity' necessary in proof dynamical systems?

The following is a (rough) translation of a statement and proof given during a course in dynamical systems. Let $D \subseteq \mathbb{R}^2$ be and open set, $f: D \to \mathbb{R}: (t, x) \mapsto ...
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85 views

Testing whether a particular set of measures borelianas is a set of Baire

Let $X$ compact metric space and $F:X\times \mathbb{R}\rightarrow X$ flow continuous ($F(x,t)=F_t(x)$). If $\delta>0$ we define $$\Lambda(x,\delta)=\bigcup_{h\in ...
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21 views

Construct a linear operator semigroup for $\frac{\partial u(t,a)}{\partial t}+\frac{\partial u(t,a)}{\partial a}=-\mu u(t,a)-kU(t-\tau)u(t-\tau,a)$

I want to use the linear operator semigourp theory for the following question.But I don't know how to construct the semigroup. $$\frac{\partial u(t,a)}{\partial t}+\frac{\partial u(t,a)}{\partial ...
3
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1answer
82 views

Prove that every solution of an ODE system converges to some point

Suppose $p(t)>2$ and is continuous for all $t\in\Bbb R$, $$x'=2y,\\ y'=-2x-p(t)y^3,$$ prove that for each solution $(x(t),y(t))$ there exists a point $(x^*,0)$ to which it converges. I ...
3
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0answers
26 views

Omega-limit set consists of one point. Does this mean the orbit tends to this point as $t$ grows?

The $\omega$-limit set is defined in this wikipedia article. My question is: If we have an orbit $x_t$ and the $\omega$-limit set of this orbit contains one element $w^*$, does this imply ...
7
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1answer
99 views

Near-integer solutions to $y=x(1+2\sqrt{2})$ given by sequence - why?

EDIT: I've asked the same basic question in its more progressed state. If that one gets answered, I'll probably accept the answer given below (although I'm uncertain of whether or not this is the ...
0
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1answer
21 views

B and C matrices for real modal representation of a 2x2 linear system with complex eigenvalues

What are the allowable transformations on the $B$ and $C$ matrices in a linear state-space system that preserve input-to-output behavior without changing the $A$ and $D$ matrices? I'm working with ...
1
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0answers
48 views

Must an expanding map be weakly expansive?

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 weakly expansive if there exist $\varepsilon>0$, for any $x,y\in X$, $x\neq ...
0
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30 views

An iterative function is also expanding?

Let $(X,d)$ is a metric space and $X$ has no isolated points, $T:X\rightarrow X$ is a continuous selfmap. $T$ is an expanding map if there exist a constant $c>1$ and a positive number ...
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17 views

Theorem on Variation of probability distribution in a flow

Let's assume I have a dynamical system $H(\mathbf{x},t) \in \mathbb{R}^n$, where the initial phase-space distribution at time $t_0$ in terms of $\mathbf{x}_0$ is gaussian. I define a flow ...
1
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0answers
36 views

An iterative function is also sensitive or expansive?

Let $(X,d)$ is a metric space and $X$ has no isolated points, $T:X\rightarrow X$ is a continuous self-map. $T$ is sensitive if there exist $\epsilon>0$, for any $x\in X$ and any positive number ...
1
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1answer
37 views

Must an expanding map be strongly expansive?

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 $x,y\in X$, ...
0
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1answer
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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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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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 ...
1
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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 ...
5
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1answer
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} ...
3
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1answer
58 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 ...
0
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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 $x=t$
0
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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 ...
1
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1answer
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 ...
3
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1answer
110 views

Population dynamics

I don't understand why we make the three assumptions underlined above.
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24 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
31 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 ...
0
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0answers
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 ...
0
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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 $r>4$. ...
2
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1answer
32 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 ...
1
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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
125 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 ...