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.

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1answer
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 ...
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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 ...
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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 ...
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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 ...
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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/......
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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 ...
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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 ...
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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, ...
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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: $$ \dot x = x - y + kx(x^2+y^2), \\ \dot y = x - y + ky(x^2+y^2), $$ ...
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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 $x$ ...
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Interpretation of a Parabolic Partial Differential Equation

Let $\Omega\subset \mathbb{R}^{d}$ ($d\geq 1$) be a bounded domain with a smooth boundary $\partial\Omega$. Let $S, I$ be dependent variables and $x, t$ their independent variables. Additionally, $q:\...
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SIR Model Specifics

I read on Wikipedia (under "Compartmental models in epidemiology") that the differential equations for the SIR Model was the following, $$S'(t)=-\frac{\beta}{N}I(t)S(t)$$ $$I'(t)=\frac{\beta}{N}I(t)S(...
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Literature request — uniqueness and existence of a specific type of ODE

I am looking for a proof of the existence and uniquenes of ODE's of the type: \begin{equation} \dot{f}(t,x,y) = F(h(t,x), f(t,x,y)), \end{equation} where $f : T \times X ...
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42 views

State equations for a linear mechanical system

The problem: The equations of motion of a lumped mechanical system undergoing small motions can be expressed as $$ Mq'' + Dq' + Kq = f ~~~~ (1) $$ (using primes instead of dots on top of the $q$s) ...
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Stabilizability, what does it mean to steer to zero

The definition of stabilizability for linear systems is: Stabilizability is the ability to steer a system to zero, with a control input that can be defined over an infinite amount of time. This is ...
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23 views

Using Lipschitz to Prove Solutions are Continuable on R

$x ^ { \prime } = \cos \left( x ^ { 2 } \right)$ Given the above equation, I need to show that it determines a dynamical system. So, since this cannot be directly solved, I tried using Lipschitz to ...
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Finding stable and unstable manifolds

Consider the system $x ^ { \prime } = 4 x + 2y ^ { 3 } \\ y ^ { \prime } = - 3 x$ Question: Find the stable and unstable manifolds around the fixed point $(0,0)$ and and sketch the phase portrait ...
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Continuous function preserves the immediate basin of attraction?

Suppose that $f$ is a differentiable function and $p$ is a fixed point of $f$ such that $|f′(p)|<1$. Let $K$ be the maximal interval about $p$ in which all points tend asymptotically to $p$ under $...
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Showing that system has a periodic orbit and has a limit cycle.

$x ^ { \prime } = y + \frac { x } { \sqrt { x ^ { 2 } + y ^ { 2 } } } \left( 16 - (x ^ { 2 } + y ^ { 2 }) \right)$ $y ^ { \prime } = - x + \frac { y } { \sqrt { x ^ { 2 } + y ^ { 2 } } } \left( 16 - (...
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25 views

Plotting Phase Portrait for Nonlinear Damped Pendulum for larger damping

I'm asked to sketch the phase diagram near the equilibrium points of the nonlinear damped equation: $x ^ { \prime \prime } + k x ^ { \prime } + \sin x = 0$. I've found that for any integer $n$, $( n \...
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On steering the state of an LTI system to a desired state and keeping it there

For sake of simplicity, consider the super classic LTI system \begin{equation*} \begin{cases} & \dot{{x}}(t) =A {x}(t)+B {u}(t) , \quad \quad \text{for } 0\le t \le t_f \\ & x(0)=x_i \end{...
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1answer
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Denseness of a sequence in 2-Torus

I want to show that if $\alpha$ and $\beta$ are rationally independent irrational numbers i.e. $\forall m,n \in\mathbb{Z}$ , $m\alpha + n\beta \not\in\mathbb{Z}$ , then the sequence $\{ (n\alpha$ (...
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Show that Ergodic Theorem is a special case of Kingman's Subadditive Ergodic Theorem.

A version of Birkhoff's Ergodic Theorem is the following: Theorem 1: Take $\xi\in L^{1}(\Omega,\mathcal{F},\mathbb{P})$. If $\theta$ preserves $\mathbb{P}$, then $$\dfrac{\xi(\omega)+\xi(\theta\...
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Find periodic orbit of the system given in polar coordinates

Consider the following system given in polar coordinates $r'=r(r^2-5r\cos\theta - 6)$ and $\theta ' = 1$. Prove that there is a periodic orbit for the system. I know that Poincaré-Bendixson ...
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Is this $a_n=(1-\frac{1}{2!})^{(\frac{1}{2!}-\frac{1}{3!})^{…^{(\frac{1}{n!}-\frac{1}{(n+1)!})}}}$ have a finit limit?

My question here is related to telescopic sum using factorial and it is related to my question here, I have computed some values of $a_n=(1-\frac{1}{2!})^{(\frac{1}{2!}-\frac{1}{3!})^{...^{(\frac{1}{n!...
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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} &\...
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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 $\pi/2$ ...
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1answer
54 views

Find Lyapunov function & determine the stability of the singularity origin for the vector field $f(x,y)=(y-x^2+3y^2-2xy,-x-3x^2+y^2+2xy)$

Let's first define the stability of a singularity of a vector field- Let, $f\in C^1(E)$, $E\subset \Bbb{R}^n$ be open and $x_0\in E$ is a singularity of $f$ i.e. $f(x_0)=\mathbf{0}$. Let, $\phi_t:...
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1answer
53 views

Derive the power series expansion for he local stable and unstable manifolds of the ODE $x'=(-x_1+x_2^2,2x_2+x_1x_2)$ for the singularity $(0,0)$

We are asked to find the power series expansion of the local stable and unstable manifolds of $x'=f(x)$ for the singularity $(0,0)$ where $f(x_1,x_2)=(-x_1+x_2^2,2x_2+x_1x_2)$ I have tried to solve ...
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14 views

One-Parameter Bifurcations of Fixed Points in Discrete-Time Dynamical Systems [closed]

I want to ask about One-Parameter Bifurcations of Fixed Points in Discrete-Time Dynamical Systems. Prove that in a small neighborhood of x = 0 the number and stability of fixed points enter image ...
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1answer
30 views

How to find the equation of the bifurcation curve of a cusp catastrophe?

The Cusp catastrophe corresponds to the equation $$F(x,a,b)=x^4+ax^2+bx$$ where $a,b$ are the control parameters. The following diagram of cusp catastrophe shows the curves that satisfy $\frac{dF}{dx}=...
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1answer
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 ...
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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 ...
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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 $R_\alpha$ ?
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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 ...
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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 ...
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2answers
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 ...
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1answer
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}(...
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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 ...
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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}^...
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1answer
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 ...
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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 ...
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1answer
44 views

Driving the state of a discrete system to zero in one step

I have the following system of difference equations: $\textbf{x}(k+1) = A \textbf{x}(k) + \textbf{b} u(k)$ where: $A = \begin{bmatrix} 1 & 2 \\ 3 & \alpha \end{bmatrix} $ and, $\mathbf{b} =...
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6 views

weak formulate for heat equation is this comparison lemma tru?

When I proved some lemma I don't know whether it is true or not because i want to use it in other theorem. Consider the following IBVP with Dirichlet BCs on a bounded open set $Ω ⊂ R^n$ for $u : Ω × [...
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1answer
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 ...
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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? ...
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1answer
30 views

Continuity of a topological conjugacy $h$

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^...
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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 ...
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1answer
22 views

showing system is topologically transitive

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 ...
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1answer
37 views

Show without solving the ODE that an equality is an invariant after initial condition

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 ...

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