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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22 views

Counterexample for a weakened definition of contraction mapping

I was trying to solve this exercise form a book on dynamical systems. There, after introducing the notion of contraction mapping, it is shown that, given a complete metric space $(X,d)$ and a ...
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23 views

Finding polynomial satisfying a certain rational equation

I'm stuck on this embarrassingly easy sounding problem, both in understanding and trying to recreate some code. Given two rational functions $\phi (x,y)$ and $C(x,y)$ we are looking for a polynomial $...
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Is it possible that a non-autonomous non-linear differential equation can be transferred into an autonomous? [closed]

Is it possible that a non-autonomous non-linear differential equation can be transferred into an autonomous? a) Give an example. b) what implications would it be to use one or the other scheme.
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Example of a dynamical system

Consider a dynamical system defined by a non-autonomous ordinary differential equation $$ \dot z = X(z,t)$$ where $X$ is periodic with respect to time of period $T > 0$. Consider the flow evaluated ...
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1answer
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Manifolds, Vector Fields and Foliations [closed]

Let $M$ be a Manifold, $X$ be a Vector Field on the manifold $M$ and $F$ be a Foliation of the manifold $M$. When is the vector field $X$ tangent to the foliation $F$?
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The Horizontal Flow, The Time-One Map, $C^r$ conjugation and Suspension of a Diffeomorphism

Let $M$ be a $C^r$ Manifold and $f:M \rightarrow M$ be a $C^r$ Diffeomorphism, where $r\in\mathbb{N}$. Consider then the Product Manifold $$M_1:=\mathbb{R} \times M$$ and the Horizontal Flow on $M_1$ $...
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36 views

Stochastic product rule for SDE

For a nonlinear SDE: $$dX_t = b(t, X_t) dt + \sigma(t)X_tdB_t, ~~~ X_0 = x,$$ the following stochastic integrating factor is obtained $$F_t(\omega) = \exp\left(\frac{1}{2} \int_0^t \sigma^2(s) ds - \...
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$f: [0,1] \to [0,1],$ such that $| (f^n)'(x_0)| \leq e^{-2n}$ $\Rightarrow $ $|f^n(x_0) - f^n(x_0 + h) | \to 0,$ for $h$ small

Let $f:[0,1] \to [0,1]$ be a smooth function and let $x_0\in[0,1]$ such that $$\left|\frac{\mathrm{d}f^n}{\mathrm{d}x}(x_0)\right| <e^{-2n}, \ \forall \ n\in\mathbb N,$$ where $$f^n := f \circ \...
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Why large linear systems of saddle point type are indefiniteness and often poor spectral properties?

I'm reading the paper "Numerical solution of saddle point problems" by Michele Benzi. In the abstract, he states that these types of large linear systems of saddle point are challenging due ...
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Suspension of Diffeomorphisms

There is no doubt that Suspension of Diffeomorphisms is a special case of Suspension of a Representation (that is also known as Suspension of Diffeomorphism Group). Therefore, I would be very grateful ...
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37 views

About Riesz's proof of the Birkhoff Ergodic Theorem

I'm trying to write a fine-grained version of Riesz's proof of the Pointwise Ergodic Theorem (sometimes called Birkhoff Ergodic Theorem) and I'm struggling with a couple of details. You can read that ...
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How to increase firs order system dynamic to second order system [closed]

So, I have system dynamic equations, and I don't know how to increase system order 1 to order second
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52 views

Show that $T : [0,\sqrt{3} + \sqrt{5} + \sqrt{7}] \to [0,\sqrt{3} + \sqrt{5} + \sqrt{7}] $ is mixing.

Is this shuffling map on the unit interval mixing ? $$ f(x) = \left\{ \begin{array}{ccl} x + (\sqrt{5} + \sqrt{7}) & \text{ if } & 0 < x < \sqrt{3} \\ x + (\sqrt{7} - \sqrt{3}) & \...
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16 views

Logistic model feasability implies stability [closed]

I have a model with logistic grown, as the lotka-volterra with limiting resources by carrying capacity. Does any one knows a theorem which says that in the logistic model, the feasability of the non ...
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1answer
35 views

How to calculate the height of centre of mass above an inclined plane

I am attaching the problem here: Since I want to use the energy approach I must require the height of the centre of mass of the rod at the initial and required (angle $\phi$) position. For the ...
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20 views

Resources for long term behavior of sequences

I have the problem where I am trying to minimize the long-term behavior of a sequence of the form $$\frac{h_{j + 1}/h_j}{f(h_j)},$$ are there any non-elementary techniques or resources for analyzing ...
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1answer
24 views

Definition of the tipping point

I have seen this mentioned in papers (e.g. here). I have a vague idea that a tipping point of a dynamical system is the state where small fluctuations can cause major changes in behavior. Is there a ...
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What is the fastest trajectory for constant acceleration rendezvous?

Suppose you were controlling a space probe floating around in a gravity-less environment, and you wanted to rendezvous with a moving target. That is, intercept the target while matching its velocity, ...
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2answers
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Periodic solutions are Lyapunov Stable

Consider the ODE $$ \left\{ \begin{align*} \dot{x}=&y\\ \dot{y}=&-x^2-bx-c. \end{align*}\right. $$ Under the assumption that $b^2-4c>0$, we find the equilibria $P_1=\left(\frac{-b+\sqrt{b^2-...
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Equilibrium points non-linear system of differential equations with parameters

I am studying a paper about populations dynamics (for the interested reader:https://link.springer.com/article/10.1007/s10955-014-0989-8). Eventually we have a system of differential equations where, ...
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1answer
35 views

Derivative of a state variable as an output of an affine system

Given simple system of ODE. \begin{cases} \dot{x_1}=-x_1+u \\ \dot{x_2}=-x_2-x_1 \end{cases} As an output, I want to use $y=\dot{x_1}$. I'm trying to convert to an affine state-space, but in the case ...
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How to conclude on the equilibrium point for the Jacobian matrix

I am trying to classify the equilibrium point of this system x' = $-2xy$ y' = $-3x^2 -y^2 + 4$ When I find the equilibrium points, I get $$(0,0) , (0,-2), (\frac{-2}{\sqrt3},0), (\frac{2}{\sqrt3},0)$$ ...
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Relationship between the inverse of the right eigenvector of a matrix $A$ and the transpose of the right eigenvector of $A^T$

I should start by stating that this question is related to the Koopman's theory in dynamical system, which gathers growing interests in the fluid dynamics/data science areas such as dynamic mode ...
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1answer
47 views

Foliations induced by Submersions

Let $T:M^m \rightarrow N^n$ be a submersion where $n \le m$. It is well-known, in this case, that a foliation (called a simple foliation) of dimension $m-n$ (or codimension $n$) is defined. Its leaves ...
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33 views

Counting functions and geodesic flows

Let $(M,g)$ be a riemannian manifold of dimension $n$ and $TN^{\perp}$ denote the normal bundle of a submanifold $N\subset M$. Now let's define $SN^{\perp}:=\{(x,v)\in TN^{\perp}: |v|=1\}, $ so this ...
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1answer
18 views

Limiting points and fixed points of a system of differential equations

Consider a system of differential equations $$ \frac{d}{dt}f(t) = F(t, f(t), g(t)), $$ $$ \frac{d}{dt}g(t) = G(t, f(t), g(t)). $$ Assume $F, G \in C^{\infty}$. What is the necessary and sufficient ...
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1answer
98 views

How to write an ODE with a Darboux polynomial?

Question: Given a polynomial ODE $\dot{x}=f(x)\in\mathbb{R}^n$ that possesses a Darboux polynomial$^*$ $p(x)$ satisfying $\dot{p}(x)=c(x)p(x)$ for some function $c(x)$ (called the cofactor) how can ...
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What does the non-existence of Lyapunov number mean? [closed]

For discrete dynamical system, $\mathbf{F}:\mathbb{R}^m\rightarrow\mathbb{R}^m$, $k$-th Lyapunov number of the orbit beginning from $\mathbf{x}_0\in \mathbb{R}^m$ is defined as follow. \begin{align} \...
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1answer
86 views

Finding an approximate value of $\sqrt 5$

I'm having some trouble understanding the following: The point of this exercise is to computationally approximate a value for $\sqrt 5$. My textbook does the following: $\sqrt 5$ is the solution of ...
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1answer
22 views

Linear autonomous dynamical systems and the definition of Linearly independent vector functions

I hope some of you will find some time to answer my following query, which has arisen in my math studies as an econ undergraduate: I know that, given a dynamical system of the type $\vec{x}'=A\vec{x}$,...
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26 views

Gradient flows (Gradient-like dynamic systems)

Finding the relationship between Gradient and Hessian to improve convergence rate Let's take the article as an example https://arxiv.org/pdf/1907.10536.pdf and look at the last formula on page 2. $\...
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39 views

Subset of the full shift is compact and transititve

Let Consider the set $\Sigma = \{0,1,2,,,n-1\}^{\mathbb{Z}}$ consisting of all two-sided sequence of elements in $\{0,1,2,...,n-1\}$. Define the full shift map $T:\Sigma \to \Sigma$ by $\sigma(...,x_{-...
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37 views

Existence of Solution to Newtons dynamics. [closed]

Hi I'm wondering how to obtain the existence of solutions to the following problem of particle dynamics : Let $k,d\in \mathbb{N}$, $k\leq d-1$, does there exist solutions, atleast on small time ...
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33 views

Properties of the window average of periodic function

I have been recently working on a periodically driven system and am interested in the window averaged quantities to study the long-time behavior. Define the window average as $$\bar{f}(t)=\frac{1}{T}\...
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1answer
25 views

Can I use $||J||_2$ as a “gradient descent” for the system $J = b - Ax$?

One quick question. Let's assume that I want to solve $Ax = b$ but I want to do that in a special way. My idea is that I first find the difference between $b$ and $Ax$, we call it $J$. $$J = b-Ax$$ ...
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Persistence of homoclinic points - the noncompact case

It is well known that transverse homoclinic points of $C^1$-diffeomorphisms of compact manifolds $M$ persist under small $C^1$ perturbations. Does the same hold for non-compact manifolds (with the ...
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1answer
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Intersection between dynamical systems and stochastics?

Are there any subjects that deal with the intersection between dynamical systems and stochastics? (Specifically in pure mathematics.) I know about random dynamical systems, but there does not seem to ...
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Highly Recommended References on Harmonic and Stationary Measures

A harmonic or stationary measure is said to be Regular if it is Absolutely Continuous with respect to Lebesgue Measure and Singular if it is not. I need to understand the Regularity and Singularity of ...
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26 views

Particular class of time switching systems

I am interested in understanding the behaviour of a class of continuous time switching systems of the form $\dot{x}(t) = f_{\sigma(t)}(x(t))$ where $f_{\bar{\sigma}}(x)$ is continuous for any $\bar{\...
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16 views

Limit of an iterated function system

Consider the function $h(x) = 2x(1-x).$ My goal is to find $$\lim_{n \to \infty} h^n \left ( \frac{1}{4} \right ),$$ where $h^n (x_0)$ is the $n^{\text{th}}$ iterate of the function $h(x)$ at the ...
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How to adjust initial condition to nondimensionalization of coupled ODE?

I am dealing with system of ODEs in dimensional form: $\frac{dB}{dT}=rB(1-\frac{B}{K})-\beta BW-\gamma_B AB$ $\frac{dW}{dT}=\alpha BW-\delta W-\gamma_W AW$ To nondimensionalize the given ODEs, let $b=\...
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55 views

System of differential equation of the form $\dot x = (A x) \circ x$

I am interested in systems of first-order differential equations of the general form $\dot x=(A x) \circ x$, where $A$ is a square, constant, real matrix and $\circ$ is the elementwise product (...
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38 views

Canonical transformation and generating function

I am stuck on an exercise of which I do not understand the solution. The exercise is the following: Consider the Hamiltonian system $$H(\theta, p, t) = \dfrac{(p-\omega t)^2}{2} - k\cos(\theta)$$ ...
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1answer
52 views

If $F\in\mathcal D \implies \left\lvert \frac{F^n(0)}{n} - F(0)\right\rvert\le\frac{n-1}{n}\, ,\forall \,n\in\mathbb N$

Let $F\in\mathcal D$, $\phi=F-id$, show that $\left\lvert \cfrac{F^n(0)}{n} - F(0)\right\rvert\le\cfrac{n-1}{n} ,\forall\, n\in\mathbb N$ Note: We define $\mathcal D$ as the set of increasing ...
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1answer
64 views

Two questions on the existence of sets associated with LaSalle invariance principle

I am copying in verbatim a standard reference on Lasalle's theorem (which is Hassan Khalil's textbook, page 128) Theorem: Let $\Omega \subset D$ be a compact set that is positively invariant with ...
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24 views

Multidimensional obstacle avoidance in ODE. Part II

Multidimensional obstacle avoidance in ODE For some time, I studied this question more closely and came to the conclusion that it is best to assign its own constraint for each selected coordinate. ...
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1answer
52 views

Every hyperbolic matrix $A\in SL(3,\mathbb{Z})$ has 3 distinct eigenvalues

Let $A\in SL(3,\mathbb{Z})$ be a hyperbolic matrix (i.e., the absolute value of each eigenvalue is not $1$), then all its eigenvalues are real and different. I read this result here, Proposition 4.12. ...
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34 views

Trying to understand random dynamical systems [duplicate]

Let $\{\theta_t : \Omega \mapsto \Omega \ | \ t \in T \}$ be a family of measure preserving transformations on the probability space $(\Omega, \mathcal{F}, \mathbb{P}).$ A measurable random ...
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2answers
45 views

What is the best ODE solver for stability and speed?

I'm going to build a ODE solver in C-code and I'm looking for an ODE solver that are robust against stiff ODE:s and very fast. But still, it also need to have a very good accuracy. I would be happy if ...
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1answer
39 views

If $h$ is a diffeomorphism, $F,G$ vector fields, such that $dh(x)F(x)=G(h(x))$, so $h(\omega_F(x))=\omega_G(h(x))$

Let $F:U\rightarrow \Bbb{R}^d,G:V\rightarrow \Bbb{R}^d$ smooth vector fields, where $U,V$ are open sets in $\Bbb{R}^d$. Let $h:U\rightarrow V$ a diffeomorphism such that $$dh(x)F(x)=G(h(x))\quad\...

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