Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [dynamical-systems]

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 models of diverse fields such as classical mechanics, economics, traffic modelling, population dynamics, and biological feedback.

0
votes
1answer
12 views

Dimension of Manifold in Dynamical System On a Plane

I've been reading about dynamical systems on a plane and the stable and unstable manifolds that can exist there. As part of this I was reading the definition of a manifold (that a manifold is a ...
2
votes
0answers
40 views

For $f_\lambda(x) = \lambda x (1-x) $, $1 < \lambda < 3$, show orbit of each $x \in (0,1)$ converges to $1 - \frac{1}{\lambda}$

Let $f_\lambda :[0,1] \to [0,1]$ be the logistic map with parameter $1 < \lambda < 3$, $$f_\lambda(x) = \lambda x (1-x).$$ The two fixed points of $f_\lambda$ are $0$ and $a = 1 - \frac{1}{\...
2
votes
0answers
46 views

Dynamical systems from an algebraic perspective

I am interested in learning more about dynamical systems. Most of my background is algebraic, specifically in group theory/geometric group theory. I was wondering if anyone knew of a reference that ...
2
votes
0answers
21 views

Why does the rotating wave approximation work?

Consider two coupled oscillators with position coordinates $X_a$ and $X_b$. In general, the motion is described by a system of coupled first order linear differential equations: $$ \frac{d}{dt} \begin{...
2
votes
1answer
50 views

Using a Lyapunov function to determine stability of equilibria

Given $$\left\{\begin{aligned} x' &= -x^3 + 7xy^2\\ y' &= -3x^2y+y^3\end{aligned}\right.$$ find $a, b > 0$ such that $L(x,y) = a x^2 + b y^2$ obeys $\frac{d}{dt}L \neq 0$ whenever ...
0
votes
0answers
15 views

Tent Map Eventually Fixed Point

show that if $\displaystyle x=\frac{k}{2^n}$ where "k" and "n" are positive integers and $0\leq x\leq1$ then "x" is an eventually fixed point of the tent map.
0
votes
1answer
29 views

Simultaneously bounding stable and unstable components

I am reading a passage from Perko's book about the Stable Manifold Theorem. Here is the problem: Let $\dot x = f(x)$ be a system where $f: E \subset \mathbb{R}^{n} \rightarrow \mathbb{R}^n$ (with $E$...
0
votes
1answer
46 views

Is there an invariant measure absolutely continuous wrt to the lebesgue measure for the map f

Let $f:[0,1]\rightarrow[0,1]$ where $f(x)=x/2$ $(1-x)$, and let $\lambda$ be the lebesgue measure on [0,1]. Is there a probability measure $\mu$ that is invariant and absolutely continuous wrt to the ...
0
votes
0answers
32 views

Phase, Isochrons, Isochrons map and Lift

at the moment i read the following paper: https://arxiv.org/pdf/1512.04436v1.pdf I have some questions about it and i hope someone can help me. On page 4/5 they introduce isochrons and the isochron ...
1
vote
0answers
16 views

Hyperbolic set- Dynamical systems

I already proved that if $Λ$ is an hyperbolic set for a diffeomorphism $f$, then it is an hyperbolic set for $f^2$. But is it true that if $Λ$ is an hyperbolic set for $f^2$ it is also hyperbolic for $...
1
vote
1answer
30 views

Existence of periodic solution for non-autonomous system

"Consider the n-dimensional system $x^{'}=f(x)+g(t)$ where $x^{T}f(x)\leq -k|x|^{2}, k>0$, for all $x$ and $|g(t)|\leq M$ for all t. Show that this equation has a w-periodic solution if $g$ is w-...
0
votes
0answers
11 views

On transfer function of a two degrees of freedom mimo control system

For a mimo control system that is shown below: with the below assumptions: for obtaining the generalized plant after formulating it with using of general control configuration in the form of below: ...
0
votes
0answers
29 views

Dynamical Systems - Hyperbolic Theory

I already proved that if Λ is an hyperbolic set for a diffeomorphism $f$, then it is an hyperbolic set for $f^2$. But is it true that if Λ is an hyperbolic set for $f^2$ it is also hyperbolic for $f$, ...
1
vote
0answers
49 views

Lyaponov function in dynamic system (in polar cordonates)

Let the following system in $R^2$ \begin{equation} (S) \left\{ \begin{array}{l c } \overset{.}{\rho}=\rho(1-\rho) \\ \overset{.}{\theta}=\sin^2(\frac{\theta}{2}) \end{array} \right. \end{equation} ...
0
votes
1answer
16 views

Test if a function given as a non-integrable ode set is Bijective

Given that state space trajectories of an autonomous system do not cross, can I deduce that a mapping function f:(x,y)→(x',y') given by a solution of an ODE of an autonomous system is bijective? ...
3
votes
1answer
66 views

Requirement of Lyapunov Stability in Asymptotic Stability

In my Differential Equations course, we defined the equilibrium point $x_0$ of a dynamical system $\frac{dx}{dt} = f(x(t))$ (for $f$ defined on an open subset of $\mathbb R^n$, say $\mathbb R^n$ ...
1
vote
0answers
21 views

Revisit “example of an unstable fixed point for which the linearized dynamics are stable”

I am reading the following discussion: example of an unstable fixed point for which the linearized dynamics are stable The above discussion is for the vector field (continuous time). Is there an ...
0
votes
0answers
8 views

Why is it sufficient to check at one point on the orbit to determine the hyperbolicity?

In this Wikipedia entry on hyperbolic sets of dynamical systems, on the examples section, it was asserted that "more generally, a periodic orbit of $f$ with period $n$ is hyperbolic if and only if $Df^...
0
votes
0answers
12 views

Double Expectation over Two Samples of 2 Joint Variables

I have 2 random variables X and Y. These variables are joint in the sense that $y_t = f(x_t) + v_t$ with $v_t$ i.i.d zero mean and finite covariance. We have two independent samples (x,y) and $(\hat{x}...
2
votes
1answer
50 views

Fixed-point iterations for quadratic function $x\mapsto x^2-2$

Let $f(x)$ be $x^2-x-2$. I want to find the root using FPI in an interval where it will converge. I have chosen $g(x)=x^2-2$ and so $g'(x)=2x$. The convergence condition, $|g'(x)|<1$ is ...
-1
votes
0answers
20 views

Hyperbolic sets in dynamical systems [closed]

Let f be a diffeomorphism. If S is an Hyperbolic set for f^2 then is it and hyperbolic set for f?
1
vote
0answers
7 views

Moser and Smale-Birkhoff homoclinic theorems

I have found in J. Moser "Stable and Random Motion in Dynamical Systems" the theorem about the topological conjugacy to the Bernoulli shift on a symbol space, and then again very well summarized and ...
0
votes
1answer
23 views

How to compute the number of years needed for the population be $52,000$ using the concept of logistic growth?

To use the concept of logistic growth, it is needed to identify the maximum or the limit of population. In this problem, I can't identify what will be the maximum of the population should I use. The ...
-2
votes
0answers
30 views

Showing that rigid rotations are distinct. [closed]

Let $T_{\theta}: S^1 = \mathbb{R}/\mathbb{Z} \rightarrow S^1 = \mathbb{R}/\mathbb{Z} \\$ $x \bmod 1 \mapsto x + \theta \bmod 1$. Let $T$ be rigid. Let $\theta$ be irrational. Prove that $0,\ T_{\...
0
votes
2answers
24 views

Topological semiconjugacy preserves topological transitivity

Let (X, f ), where X is a compact metric space and f : X → X is a continuous function and let (Y, g) where Y is a compact metric space and g : Y → Y is a continuous function. Suppose they are ...
1
vote
0answers
19 views

Calculating the topological entropy of certain maps

I am currently going over some example questions for my dynamical systems module and I am wondering how to calculate the entropy of certain maps. The maps in question are $$T:[0,1] \to [0,1] \hspace{...
0
votes
1answer
36 views

Intuition of cocycles and there use in dynamical systems

I’ve come across several papers and lectures that use cocycles to talk about dynamics on a manifold. However, I haven’t come across an actual definition of what a cocycle is. Could someone give a ...
0
votes
0answers
62 views

Compare solutions in some moment of time for two equations by Lyapunov derivatives?

We have two differential equations $\dot x_1=f_1(x_1)$ and $\dot x_2=f_2(x_2)$ which are too complex to solve but we could show by the same Lyapunov function $V$ (and its derivatives) that equilibrium ...
1
vote
1answer
54 views

Evaluating stability using Jacobian in dimension 1

Generally, when we are considering a system of differential equations, to evaluate the stability of fixed points, we find the eigenvalues of the Jacobian to evaluate stability. I was wondering how ...
0
votes
0answers
15 views

Non-periodic orbit in Newton-Raphson example

Let $f(z)=\frac{z^2-1}{2z}$. I want to find a point $\xi\in\mathbb{R}$ such that the $f$-orbit of $\xi$ is non-periodic and$$ \lim_{n\to\infty} \frac1n \#\{t\leq n \,:\,f^t(\xi)<0 \}=\frac13. $$ ...
0
votes
1answer
36 views

Finding the Jacobian matrix & eigenvalues of a matrix

Suppose I have a 2D dynamical system with $$\frac{dx}{dt} = f(x, v), \hspace{5mm} \frac{dv}{dt} = g(x, v)$$ My Jacobian is then given by $\begin{pmatrix} f_x & f_v \\ g_x & g_v\end{pmatrix}...
3
votes
2answers
66 views

A challenging system of coupled recursive sequences

So I was runing through my old school drafts, and I've just come upon this challenging problem that years ago, one of my former math teacher had let for enthousiastic students to try. Consider the ...
1
vote
0answers
57 views

How to prove Asymptotic Stability in Lyapunov's Stability theorem

Lyapunov's Stability Theorem: Let $x = 0$ be an equilibrium point for the autonomous system $$\dot{x}(t) = f( x(t) ),$$ and $D \subset \mathbb{R} ^ n$ be a domain containing the equilibrium point, i.e....
0
votes
1answer
23 views

Book reference request for “Synchronisation in discrete dynamical system”

I'm looking for rigorous mathematical treatment of synchronisation in discrete dynamical system, in particular, stability/unstability of syncronisation. It would be great if anyone gives me suggestion ...
0
votes
0answers
23 views

Plot of the stable/ unstable manifold in case of complex eigenvectors of the Jacobian matrix.

I was trying to plot the unstable manifold of a fixed point of a two dimensional map $f(x,y)$. I have the fixed point say $(x^*,y^*)$. Next, I found the Jacobian matrix at the fixed point from which I ...
1
vote
1answer
30 views

Stability matrix - order of elements

I have two equations: $$\frac{dx}{dt} = v(k-ux) - \delta x, \hspace{3mm} \frac{dv}{dt} = v(r-px)$$ I want to calculate the Jacobean so I can analyse the stability of the fixed points. Does it ...
1
vote
0answers
19 views

Types of triangles admitting periodic billiard orbits

It is an open problem in dynamical systems if every triangle has a periodic billiard orbit. So far it has been proven that equilateral triangles, isosceles triangles, right triangles, and triangles ...
3
votes
2answers
28 views

Proportion of a compartment's mass which originates from another compartment

System & Objective I have the following open system (mass flow) with 2 compartments $A$ and $B$ and constant flow rates $a,b,\alpha,\beta,\gamma,\delta$. I would like an expression for "the ...
3
votes
0answers
54 views

Prove existence of a Heteroclinic Orbit

How would you go around proving that there exists a heteroclinic orbit between two equilibria ( in the problem I'm trying to solve, one is a stable node(say n) and the other is a saddle (say s))? I ...
1
vote
1answer
38 views

A “counterexample” on Takens' embedding theorem for phase space contruction

Background: Consider the 1-dimensional submanifold $y=\sin{x}$ for $x\in\mathbb{R}$, it is a dynamical system with the flow/dynamics function $f(t,(x,y))=(x+t,\sin(x+t))$. Now consider its projection ...
1
vote
0answers
17 views

Lyapunov dimension of the associated dynamical system of a self-similar IFS

The following construction is sometimes called the associated dynamical system to a given IFS: Suppose $X$ is a nonempty compact subset of $\mathbb{R}^n$ and $(X,S,p)$ is a contractive IFS with ...
0
votes
1answer
20 views

Time period of a mathematical pendulum

Consider the mathematical pendulum $$\dot{\theta}=\omega$$ $$\dot{\omega}=-\frac{g}{L}sin\left(\theta\right)$$ How can one prove that it is impossible that the time period $T$ depends only on the ...
0
votes
0answers
32 views

Showing the equilibrium point to be globally exponentially stable using Lyapunov indirect method.

We have the system $\ddot{q} + \dot{q} + g(\dot{q},q) + q = 0, \forall t \geq 0$. $x = \begin{bmatrix} x_{1}\\ x_{2}\end{bmatrix} = \begin{bmatrix} q\\ \dot{q}\end{bmatrix}$ $\dot{x} = Ax + h(x)$ ...
0
votes
1answer
64 views

How to linearize a kinematic bicycle model?

I have the following system: $$\begin{aligned} x(k+1) &= x(k) + T_sv\cos(\phi(k) + \beta(k)) \\ y(k+1) &= y(k) + T_sv\sin(\phi(k) + \beta(k)) \\ \phi(k+1) &= \phi(k) + \frac{T_sv}{l}\sin(\...
1
vote
0answers
23 views

Infinite periodic orbits

Given a billiard cycling around a polygon, reflecting off each edge perfectly with no loss of energy, we know that there exist periodic orbits in specific classes of polygons (such as equilateral ...
1
vote
1answer
14 views

If a set is a dense or relatively dense subset of a topological group

Suppose that $X$ is a topological space, and $T$ is a topological group which continuously acts on $X$ on the right. We call the pair $(X,T)$ a (right) transformation group. We know that $(X,\mathbb ...
2
votes
0answers
22 views

How can interpret the sensitivity analysis of an ordinary differential equation?

Suppose we have the following system of differential equations: $$ \dot{x_1} = - c_1x_1x_2 + c_2x_3 -c_3x_1 +c_4x_2 $$ $$ \dot{x_2} = - c_1x_1x_2 + c_2x_3 + c_3x_1 -c_4x_2 $$ $$ \dot{x_3} = ...
0
votes
1answer
12 views

$\mathcal{H}_\infty$ norm of a system is a lower bound of the $\mathcal{L}_1$ norm?

For a stable causal SISO LTI system $G(s)$, let $H = \|G(s)\|_{\mathcal{H}_\infty}$, and let $\omega^*$ be the frequency at which this is achieved$^\dagger$. The output of the system to an input of $\...
2
votes
0answers
25 views

Autonomous dynamical system on $\mathbb{R}^2$. Slow decay with superposed but also decaying oscillation.

I am looking at the following autonomous dynamical system $\dot{\mathbf x}(t)=\mathbf{f}(\mathbf{x}(t))$ on $\mathbb R^2$, with $\mathbf x=(x,y)$ $\dot x=-3x\sin(y)^2,$ $\dot y=-\frac{1}{x}-\frac{3}{...
1
vote
1answer
54 views

Poincaré-Bendixson Theorem and Limit Cycle:find the trapping region

$\dfrac{dx_1}{dt} = x_2 - x_1^3 + x_1$ $\dfrac{dx_2}{dt} = -x_1 - x_2^3 + x_2$ Hi, I have been given the non-linear system as above and I need to show if there exists limit cycle in the annular ...