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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4
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1answer
73 views

Positively invariant neightbourhood using Lyapunov function

Given the nonlinear system of ODEs, $$x_1'=-x_1-x_2$$ $$x_2'=2x_1-x_2^3$$ I need to use the quadratic Lyapunov function $V(x) = x^TQx$ (where $Q$ is a positive definite matrix such that ...
0
votes
1answer
23 views

Dynamic real-time system problem

I am struggling with a systems theory problem, the task is as follows: u(t) -> H(s) -> y(t) H(s) being the transfer function $$ H(s) = H(s) = \frac{s+1}{s(s+2)^{2}} $$ $$ u(t) = e^{-5t} $$ So ...
0
votes
2answers
25 views

Show $x_k = F^k(x_0)$, with any seed $x_0$, satisfies $\lim_{k\to\infty} x_k=\bar{x}$

Consider the dynamical system $F(x) = ax+b$ where $a \neq 1$ and $b\in \mathbb{R}$. Show $x_k = F^k(x_0)$, with any seed $x_0$, satisfies $\lim_{k\to\infty} x_k=\bar{x}$ I found $\bar{x}$ to be ...
0
votes
2answers
98 views

Showing system contains peroidic orbit

Consider the IVP for the $2$-dimensional dynamical system ($X=[0, \infty )^2$) $$\dot{x_1}=a-x_1-\frac{4x_1x_2}{1+x_1^2}$$ $$\dot{x_2}=bx_1 \bigg( 1- \frac{x_2}{1+x_1^2} \bigg)$$ for all $t \in I$, ...
0
votes
1answer
23 views

Why does the logistic map $x_{n+1}=rx_n(1-x_n)$ become unstable when $\lvert\frac{df(x^*)}{dx}\rvert=1$?

I'm having trouble understanding why the logistic map becomes unstable when $$\lvert df(x^*)/dx\rvert=1,$$ where $x^*$ are the fixed points of $f(x)=rx(1-x)=x$. I have read that it can be seen from ...
1
vote
1answer
32 views

Does any analytic function from the unit disk to a compact subset of itself have a fixed point?

I was reading a proof by Beardon of the Wolff-Denjoy Theorem in Complex Dynamics. In the proof, a family of maps $$f_\epsilon=(1-\epsilon)f(z)$$ is used, where $f(z)$ is an analytic map from the unit ...
1
vote
1answer
25 views

Wandering set definition

I've seen two apparently different definitions and am wondering which is correct. A set $W$ is wandering if $\{T^{-k}W; k\in \mathbb{N}_0\}$ (resp. $\{T^{k}W; k\in \mathbb{N}_0\}$) are pairwise ...
0
votes
0answers
26 views

Range of parameter values for a stability of a fixed point for this 2d map

So I am trying to do a linear stability analysis for a very simple 2d discrete system: \begin{equation} \begin{aligned} x_{n+1} &= y_{n}\\ y_{n+1} &= -\frac{x_{n}}{2} + ay_{n} + y_{n}^{3} ...
0
votes
0answers
46 views

SIR-Model in Maple

This is my assignment I've been stuck on for a good part of the day. I'm confused because example's I've looked at would do part 3 before part 1 because they use substitution of $$u=S/N$$ $$v=I/N $$ ...
16
votes
1answer
89 views

For what kind of infinite subset A of $\mathbb Z$ and irrational number $\alpha$, is $\{e^{k\alpha \pi i}: k\in A \}$ dense in $S^1 $?

There is a well-known result saying that $\{e^{k\alpha \pi i}: k\in \mathbb Z \}$ is dense in $S^1$. By density, we can select an infinite subset $A$ of $\mathbb Z$ such that $\{e^{k\alpha \pi i}: ...
0
votes
1answer
18 views

regarding solutions to differential equations as vector fields

my dynamical system/ ODE textbook says: "if we consider the solutions of the autonomous system $x'=f(x)$, $x(0)=x_0$, for $f\in C^k(M,\mathbb{R}^n)$ and open $M\in\mathbb{R}^n$, we can regard such ...
2
votes
1answer
48 views

Finding the stable period $4$ orbit of a trajectory

I am told to find the stable period $4$ orbit of $f(x) = 13xe^{-x}$ for my discrete dynamical system through direct numerical iteration. However, I am a bit confused on what is meant by direct ...
2
votes
1answer
35 views

Range of values which satisfy this inequality

Consider the following inequality: $$|f(a)| = \left|\frac{1}{2}(a \pm \sqrt{a^{2}-2})\right| \leq 1$$ I got this inequality while doing stability analysis of a fixed point of a certain discrete ...
0
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0answers
179 views

What's the probability distribution of a deterministic signal or how to marginalize dynamical systems? (functional integrals in probability theory)

In many signal processing calculations, the (prior) probability distribution of the theoretical signal (not the signal + noise) is required. In random signal theory, this distribution is typically a ...
0
votes
1answer
32 views

Relation between minimal and distal homeomorphism on compact metric space

I would like to compare minimal homeomorphisms and distal homeomorphisms on a compact metric space. $f$ is said to be minimal if $\overline{Orb(x)}=X$ for all $x \in X$. And $f$ is said to be distal ...
0
votes
1answer
63 views

How to prove that expansivity is a topological property in a compact space?

A homeomorphism $f:X\rightarrow X$ is called expansive if there exists a constant $\delta>0$ such that if $d(f^n(x),f^n(y))<\delta$ for all $n\in\mathbb{Z}$ then $x=y$. How to prove that if $X$ ...
4
votes
1answer
115 views

Does $A$ commute with $e^{\int A \: dt}$

I have been studying the linear system of the form: $$D_tX = AX + \textbf{b}$$ Where $A$ is not necessarily constant Suppose we aim to find an integrating factor $M$ such that: $$M[D_tX - AX] = ...
2
votes
1answer
55 views

Stable, periodic solution for $\dot x +x=f(t)$

I'm having trouble on this problem from Strogatz. Given a $T$-periodic, smooth function $f(t)$, is true that $\dot x +x=f(t)$ necessarily has a stable, $T$-periodic solution $x(t)$? I must either ...
0
votes
0answers
44 views

Signs of a Neimark-Sacker bifurcation?

This bifurcation diagram looks to me like it could just be a lot of little pitchforks. Is there something visual/graphical about it that gives it away as a Neimark-Sacker bifurcation?
-1
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1answer
20 views

Difference of DAE and ODE [closed]

What are the differences between ODE and DAE? Is every ODE a DAE? Please explain with examples
0
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1answer
46 views

Dynamical system problem involving iteration, periodicity

Test is coming up very soon, and I just couldn't get hold of this one suggested problem!! Consider the dynamical defined by iterating the function $f(x)=1-x^2$, that is, for a given initial point ...
0
votes
2answers
30 views

What is the 'stability' of a critical point of a (1 dimensional) dynamical system?

What is meant by the term 'stability' when describing a critical point of a (1-dimensional) dynamical system? Suppose $x_{0}$ is a critical point of such a system. Does the stability of $x_{0}$ ...
0
votes
1answer
35 views

steady state or equilibrium

I need to find the equilibrium point of the following system of equation: $ds/dt = u - asi - bsw - us$ $di/dt = asi + bsw - (u + g)i$ $dw/dt = e(i - w)$ I found that $$s = \frac{u}{ai + bw + ...
1
vote
1answer
79 views

How to show that all trajectories of the system approach a unique periodic solution?

Consider an overdamped linear oscillator forced by a square wave. The system can be nondimensionalized to $\dot{x}+x=F(t)$, where $F(t)$ is a square wave of period $T$: $F(t)=\begin{cases} +A, & ...
-1
votes
1answer
33 views

A bunch of statements on pitchfork bifurcations I want refuted or verified

Here is how I understand conditions for a pitchfork bifurcation, intuitively and mathematically: I am mostly referring to the graph of x vs. a, where a is the parameter that varies and leads to ...
0
votes
1answer
76 views

Computing the period via elliptical integral

I am having trouble showing that the period (T) of this system can be expressed in terms of an elliptical integral. Given the dynamical system governed by the differential equation below: ...
3
votes
1answer
58 views

On the $\omega$-limit set of a trajectory converging to a submanifold

Let $x\in M$, where $M$ is an $m$-dimensional smooth manifold. Let $N$ be an $n$-dimensional smooth and compact manifold so that $n<m$. Let $X:M\to TM$ be a smooth vector field and denote by ...
1
vote
0answers
36 views

General solution to DE

Consider $y^{''}+y^{'}-2y=0$. Find the general solution to this differential equation. Convert the second order equation to a system of linear equations corresponds to the general solution of the ...
2
votes
1answer
26 views

Rotation of the torus $T^2$ by irrational numbers linearly dependent over $\mathbb Z$

It is known that the rotation $x \to x + \alpha$ of $S^1 = \mathbb R / \mathbb Z$ with irrational $\alpha$ is ergodic and, in particular, $\alpha n$, $n = 1, 2,\dots$, are dense in $S^1$. In two ...
2
votes
0answers
51 views

Analyzing a discrete dynamical system for stable orbits

I have a discrete dynamical system $x_{n+1} = f(x_n)$ where $f(x,a) = \frac{2}{x} + \frac{x}{4} + a$ where $a = -1.5, -1.75, \cdots$ is a varying parameter. I wish to determine if a stable periodic ...
2
votes
1answer
59 views

Fixed point of a dynamical system

What does a fixed point mean in a autonomous dynamical system, I mean I know the definition of it, but I keep hearing that if a dynamical system starts at a fixed point then it will remain there, why ...
0
votes
1answer
16 views

Show that $F(\cdot)$ maps the interval $[\bar{x} - \delta, \bar{x} + \delta]$ to the interval $[\bar{x} - |a|\delta, \bar{x} + |a|\delta]$

Suppose $F(x) = ax+b$ with $|a|<1$, and so $\bar{x} = \frac{b}{1-a}$ is its fixed point. For $\delta > 0$ show that $F(\cdot)$ maps the interval $[\bar{x} - \delta, \bar{x} + \delta]$ to the ...
1
vote
1answer
66 views

Existence and uniqueness for $y' = \sqrt y+1$

Given that $y' = \sqrt{y}+1$, $y(0)=0$, $x\in [0,1] =: I$, how does one show that this ODE has a unique solution on I? I was thinking that one might be able to show that this ODE satisfies a Lipschitz ...
0
votes
1answer
29 views

What is a Turing point?

What is "the Turing point associated with the bifurcation of spatially uniform solutions"? I'm primarily concerned with what a Turing point is - Google is convinced that I mean a turning point.
0
votes
1answer
39 views

Confusion over Bifurcation Diagram [closed]

I have the function $f(x,a) = \frac{2}{x} + 0.75x + a$ and want to create a bifurcation diagram of this function as $a$ varies. An $\textit{equilibrium}$ point (I thought) of the trajectory made by ...
1
vote
1answer
37 views

Limits of irrational rotations of the circle.

We consider the unit circle, and identify it with $X=\mathbb{R}/\mathbb{Z}$. Let $\alpha$ be irrational, and consider the $\mathbb{Z}$-action on the space $X$ given by $T(x)=x+\alpha \mod 1$ ...
0
votes
1answer
22 views

Infinite product of maps not having pseudo orbit tracing property

Show that infinite product of maps need not have pseudo orbit tracing property , where each map has pseudo orbit tracing property.
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2answers
102 views

Initial conditions that converge to an unstable equilibrium

Consider the discrete-time dynamical system \begin{align} x_{k+1}=T(x_k), \end{align} where $k\in\mathbb{N}$, $x_k\in\mathbb{R}^n$, $x_0$ is the initial condition, and $T:\mathbb{R}^n\to\mathbb{R}^n$ ...
0
votes
0answers
32 views

Prove that if non-homogeniety of ODE is bounded then its solution is also bounded

Please let me know if my proof is fine, or if you'd make some modifications. Consider the general system of ODEs $x'=Ax+b(t)$ (1), $x(0)=x_0$ where all eigenvalues of $A$ have negative real parts and ...
9
votes
0answers
60 views

Differential equations with dense solutions

Consider the differential equation $P(y',y'',y''',y'''')=0$ on $\mathbb R$, where $P(x,y,z,w)$ is the homogeneous polynomial of degree $7$ given by $$ ...
3
votes
0answers
67 views

Proving that $\lim\limits_{t\to\infty} e^{At}x_0 + \int\limits_0^\infty e^{A(t-s)}b(s)ds=\vec{0}$

Consider $x'=Ax+b(t)$, a system of differential equations. Given that $A$ has negative real parts in all its eigenvalues, and that $\lim\limits_{t\to\infty} b(t) = \vec{0}$, I need to prove that ...
4
votes
0answers
40 views

Is there a classical analog of Bloch's theorem?

In quantum mechanics, having a spatially periodic Hamiltonian imposes a lot of structure on solutions of Schrodinger's equation (e.g. band structure), primarily due to Bloch's theorem. In perfect ...
2
votes
0answers
47 views

Solution of $x'=Ax$ is not what it is supposed to be

Consider a system of ODEs $x'=\begin{bmatrix} 0 &1 \\ -1&0 \end{bmatrix}x$. Wolfram Alpha says that the solution is $x(t) = \begin{bmatrix} \cos t &\sin t \\ -\sin t& \cos t ...
1
vote
1answer
29 views

Definition of trajectory

I am writing something that involves comparing the solutions of many different differential equations, and I need precise definitions of the terms trajectory and solution curve. Given a dynamical ...
1
vote
1answer
22 views

Stable eigenspace of $x'=Ax$

Given the system $\bf{x'}=Ax$, where $\bf{A}$$=\begin{bmatrix} -2 &0 &0 \\ 2& 1 & 0\\ 0 &0 &1 \end{bmatrix}$, the solution is $x(t) = \begin{bmatrix} e^{-2t} & 0 ...
0
votes
1answer
51 views

$x'=\cos^5(x) +1$ has unique solution defined for all $t\in \mathbb{R}$

I would appreciate if someone could please give me a hint on how to do this problem. Or where to see some examples. Unfortunately, the sources that I have do not seem to actually explain it and show ...
0
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0answers
47 views

Writing 5-dimensional dynamical system as Hamiltonian system

I've got a 5-dimensional continuous dynamical system, i.e., $$ \dot{x}(t)=f(x,y,z,u,w)\\ \dot{y}(t)=g(x,y,z,u,w)\\ \dot{z}(t)=h(x,y,z,u,w)\\ \dot{u}(t)=q(x,y,z,u,w)\\ \dot{w}(t)=p(x,y,z,u,w) $$ Is ...
0
votes
1answer
12 views

Prove $\langle x_0\rangle$ has only finitely many elements if and only if there exists $k_1$ and $k_2$ with $k_1 < k_2$ so that $x_{k_1} = x_{k_2}$

Prove that the orbit $\langle x_0\rangle$ has only finitely many (distinct) elements if and only if there exists $k_1$ and $k_2$ with $k_1 < k_2$ so that $x_{k_1} = x_{k_2}$ I know this to be true ...
0
votes
1answer
24 views

Find $c, M > 0$ such that $\lvert e^{tA}x_0\lvert \le Me^{ct}\lvert x_0\lvert$

In a system of differential equations $x'=Ax$, where $A$ is a constant matrix, and the equation is a sink (all eigenvalues of $A$ have negative real parts), I need to find constants $c,M>0$ such ...
0
votes
2answers
45 views

Lyapunov Exponent sensitivity to initial conditions

I am plotting the Lyapunov exponent as a function of a parameter $r$ with an initial condition $x_0$. The equation looks like this: $$x_{n+1} =4rx_n (1-x_n)$$ When I try different initial conditions ...