# 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.

5,098 questions
Filter by
Sorted by
Tagged with
28 views

### Sketch the graph of the solution with various initial conditions.

Consider the following one-dimensional system $$\frac{dx}{dt}=x^2-1.$$ Then, using the phase portrait, sketch the graph of the solution $x(t)$ for various initial conditions. I just wanted to confirm ...
20 views

### Searching for unsolved problems in the field of stability

I have proposed an approach for constructing Lyapunov functions for autonomous systems in my Ph.D. thesis and find some useful examples. Now, I am searching for some another example in this field or ...
9 views

### Does linear discrete-time controllability imply stabilizability

Does linear discrete-time controllability imply stabilizability? I feel like it should, since controllability is the ability to steer from any state $x(0)$ to another state $x(1)$ in finite time and ...
11 views

### Learning Bifurcation Theory

I'm a physics graduate student. My interests are mainly statistical physics, so I usually deal with non-linear systems (both deterministic and stochastic). I did a dynamical system course, where we ...
7 views

### Convert plot offset along the y-axis to a sign function [-1;1]

I’m not sure that I clearly enough reflected the essence of a small problem in the title of the topic, but I’ll try to reveal its essence in the question itself. https://www.wolframalpha.com/input/......
19 views

### Why is the existence of periodic orbits (in triangular billiards) so hard to prove?

I don't get why proving that every triangular billiard has a periodic orbit should be that hard. I mostly understand the partial results on the matter, mainly Every triangle with rational angles ...
19 views

### Simulating measurement data from a dynamical system and its derivative

I am implementing the SINDy algorithm in Python. Discovering governing equations from data by sparse identification of nonlinear dynamical systems I have a question concerning the simulation of ...
30 views

### Are chaotic systems generally non-differentiable w.r.t. initial conditions?

There is some important background to go over before this question will make any kind of sense, so before calling me out on my poor understanding of dynamical systems, chaos, and differentiabilty, ...
30 views

### Stability of the trivial solution of a system of differential equations

I am trying to determine the stability properties of the equilbrium solution $(x,y) = (0,0)$ of the following system of ODEs: $$\dot x = x - y + kx(x^2+y^2), \\ \dot y = x - y + ky(x^2+y^2),$$ ...
28 views

### Evans and Murthy: if $\sum_{i=0}^r a_iA^ib=0 , a_i > 0 , i = 0,1,\dots,r$ then $x$ can be expressed as a linear combination of $A^ib$

In the article of Evans and Murthy (1977) the following lemma is given: If $A,b$ satisfy the relationship $$\sum_{i=0}^r a_iA^ib=0 \quad \quad a_i > 0 \quad i = 0,1,\dots,r$$ then any vector $x$ ...
24 views

30 views

### Dynamical systems described by coupled nonlinear differential equations

Suppose a dynamical system is described by two variables, $x$ and $y$, and they change over time according to the following two coupled differential equations: \begin{equation} \begin{split} &\...
29 views

### Convert first order ODE system into a complex ODE

Consider the first-order ODE system for the real 2D vector $X = (x,y)$ $$\frac{d}{dt} X(t) = (r + I) \nabla U = r \nabla U + I (\nabla U)$$ where $I$ is the $2\times 2$ rotation matrix of $\pi/2$ ...
54 views

32 views

### Is it possible to have function with a variable number of parameters?

I was wondering if there is a type of parameterized function where the number of parameters changes over time? How would you describe the derivative/properties of the number of parameters over time ...
85 views

### Limit points of $x_{n+1}=2\cos(x_n)$

Here is a question I am not able to answer. Find the limit points of the iterative sequence given by $x_0=2$ and $x_{n+1}=2\cos(x_n)$ If one relaces $2\cos$ by $\cos$ then it is a classical ...
16 views

### Topological conjugacy in the case of rational rotation numbers

Is every orientation-preserving homeomorphism on $\mathbb{R}/\mathbb{Z}$ with rational rotation number $\alpha$ topologically conjugate to rigid rotation $R_\alpha$ ?
71 views

### Stability with input same as with nonlinear function?

Assume there is a dynamical system $$\frac{d x(t)}{dt} = A \cdot x(t) + q(x(t))$$ and that $A$ is stable and that $q$ is a nonlinear and very complicated function. We only know $q$ is smooth and ...
134 views

### Dynamic Systems: Adjustment time to perturbations

I am new to the field of dynamic systems and have what I feel is a pretty basic question. If I have a simple dynamic system $\dot{x}=-kx$ with one stable equilbrium point, and I move my solution away ...
41 views

### Stommel model, 1961.

I'm doing some calculations from the article of Stommel $1961$: Thermohaline convection with two stable regimes of flow. At a certain point he write down a system of two nonlinear ODEs which has the ...
41 views

### Understanding Controllability Matrix

Consider \begin{equation*} \dot{x} = Ax + Bu,\quad x \in\mathbb{R}^n,\ u \in\mathbb{R}^m,\quad A \in\mathbb{R}^{n\times n},\ B \in \mathbb{R}^{n\times m} \end{equation*} \begin{equation*} \text{Rank}(...
23 views

### Immediate basin of attraction in a continuous function.

I had some trouble in order to prove Singer's Theorem by Devaney's book An Introduction to Chaotic Dynamical System 2ed. Suppose that $f$ is a differentiable ...
24 views

### Proof of Controllability Matrix

I have searched for the proof of the following theorem but wasn't satisfied with what I found, \begin{equation} \dot{x}(t) = Ax(t) + Bu(t),\quad x(t) \in \mathbb{R}^n \text{ and } u(t) \in \mathbb{R}^...
21 views

### Understanding hyperbolic dynamical systems

I am trying to understand uniformly hyperbolic dynamical systems from the definition given here. I understand Smale's horseshoe with expansion and contraction that is very clear to see, but I don't ...
42 views

### Analytic estimates of limit cycle parameters

Suppose we have a two-dimensional system of differential equations, say, the well-known Van der Pol oscillator: $$\dot{x}=y, \dot{y}=\mu (1-x^2)y-x$$ Everyone knows that the study of limit cycles ...
44 views

21 views

### Orbits of points under pseudo-Anosov diffeomorphisms

I have no intuition about orbits of pseudo-Anosov diffeomorphisms $\phi$ of closed surfaces $S$ of genus $>1.$ I understand that there are infinitely countably many periodic points, correct? What ...
33 views

### Determining the domain of stability of a dynamic system

Suppose I have the system: $\dot{x} = -x^3 - y^2$ $\dot{y} = xy - y^3$ ... and am asked to find the domain of stability of the system. Is my attempt and reasoning below deemed a correct approach? ...
30 views

The system $x'=Ax$ is an attractor. Let $h$ be defined by $$h(0)=0 \qquad h(x)=e^{t_x}e^{t_xA}x$$ where $t_x$ is the real number such that $q(e^{t_x A}x)=1$ and $q(x)=\int_0^{\infty}\langle e^{tA}x,e^... 0answers 13 views ### Superposition of solution of second linear differential equation defined by an integral Consider $$\ddot q_k=-\omega_0^2q_k-a(2q_k-q_{k+1}-q_{k-1})\qquad\qquad\qquad (1)$$ with$k\in \mathbb{Z}$. To find a solution one can consider the function$q_k(t)= e^{i\omega t}e^{isk}$with$s \in ...
Suppose $(X,f)$ is a dynamical system which consists of a discrete space $X$ with $|X| \geq 2$ and consisting of a single periodic orbit. I need to show that $(X,f)$ is topologically transitive. Is ...
I am studying invariants for systems of ODEs. For example, I have proved that if $x'(t)=f(x(t))$ such that $x\, (t_0) > c$, and if $f(k)>0$ for all $k \geq c$, then the derivative will keep ...