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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Notation for orbits in Ergodic Theory
Consider a map $T:X\to X$ from a set to itself. In ergodic theory, we say that the orbit of a point $x\in X$ is the set \begin{equation} \{x,Tx,T^2x,T^3x,\cdots\}.\end{equation}
Is there any standard ...
- 193
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Locally asymptotic stable equilibrium of a non-linear difference equations system
Theorem 4.8 in the book "discrete dynamical system" gives a sufficient condition for an equilibrium to be locally stable in a nonlinear difference system.
It basically says the following: a ...
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14
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Jacobian of a trajectory given by a matrix exponential
I need to get the jacobian of the function $x(t) = e^{At} x_0 $.
so,
I was thought about applying the vectorization and Kronecker product:
$ d \, vec \, x = (x_0^T \otimes I) \, d \, vec (e^At)$.
But ...
- 59
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1
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Schwarzian derívate of a polynomial with different real roots
I’m working on the exercise 7.5.2 of the book Brin and Stuck introduction to dynamical systems.
It says: Show that any polynomial with distinct real roots has negative schwarzian derívative.
The ...
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14
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Extension of pseudo-orbit for continuous action
Let $G$ be a finitely generating set and $H\leq G$ be a subgroup with finite index, this means that there is finite set $\{g_i\}_{i=1}^n\subseteq G$ such that $G= \bigcup_{i=1}^n g_iH$. Let $A$ be a ...
- 1,275
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What are the rules for solving differential inequalities using Laplace Transforms?
I am looking to solve a non-autonomous differential inequality of the form
$$\frac{dV}{d\tau}-\epsilon V-\frac{E^2}{2\epsilon}\leq\frac{A^2}{2\epsilon}\tau^2+\frac{2AE\tau}{\epsilon},$$where $V:\...
- 43
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1
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First digits of the iterated powers of 2
I wanted to show that the first digits of $(2^{2^j})_{j=1}^\infty$ are not periodic. By the standard Dirichlet trick I can show that any array of digits forms the initial digits of some number of the ...
- 159
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Identify the dominant degree of freedom in eigenvectors
I have a physical system with n number of 6 degree of freedom bodies coupled by springs. So the system has 6n degrees of freedom (3 translational degrees of freedom + 3 rotational degrees of freedom ...
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43
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Identity all bifurcation that occur in the system for $\mu$ and bifurcation diagram
Consider the system
$$
x'=-x^4+5\mu x-4\mu^2, \, y'=-y
$$
I try to find bifurcation points, and identity all bifurcation that occur in the system for $\mu$ and bifurcation diagram.
My work: note that ...
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Solution for closed form of Brockett integrator
$\textbf{Preamble}$: I solve a statement about Lie transport $\mathcal{L}_a \nabla = 0$ i.e. 1-form $\mathcal{L}_a \omega = 0$. I found Cartan's formula of a 1-form equal to $\iota_a(d\omega) + d\...
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There exists a unique stable limit cycle around $(0,0)$
Consider the system
$$
x'=\mu x-y-x\sqrt{x^2+y^2}, \, y'=x+\mu y-y\sqrt{x^2+y^2}
$$
I am working on that show that (1) as $\mu>0$, there exists a unique stable limit cycle around $(0,0)$ (2) as $\...
- 1,420
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1
answer
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Application of the Poincaré-Bendixson theorem
I am trying to solve the following exercise (19) from this magnificent notes but I am encountering some problems:
Prove that the system
$$\begin{cases}
\dot{x} = 2x-x^5-xy^4 \\
\dot{y} = y-y^3-yx^2
\...
- 81
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Example of a buried Julia component of a transcendental meromorphic function.
We know examples of buried Julia components for rational functions. In 1998, McMullen gave the following example of a family of rational maps having buried Julia components: $f_{c,\lambda}(z)=z^m+c+\...
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How to show that is unique asymptotic stable
Based on this question:Poincaré-Bendixon show periodic solutions.
Show that the system $x^{'}=x-y-x^{3}$,$y^{'}=x+y-y^{3}$ has a unique periodic orbit on annulus $A:=\{(x,y): 1\le x^2+y^2\le 2\}$...
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There exists a function that satisfies $\sum_{n=1}^\infty |f^{[n]}(x) - f^{[n]}(y)| < \infty$ but is not a contraction?
There exists a function $f: \mathbb{R} \to \mathbb{R}$ that satisfies
$$
\sum_{n=1}^{\infty} |f^{[n]}(x) - f^{[n]}(y)| < \infty \quad \forall x,y \in \mathbb{R}
$$
where $f^{[n]}(x) = f(f(f...(f(x))...
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Transform a differential equation into Hamiltonian form
I am currently doing the following exercise in the book Modern Geometry - Methods and Applications Part I by Dubrovin, Fomenko, and Novikov.
Exercise 33.4.1: Consider the differential equation
\begin{...
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Extension of a continuous function $f:X\rightarrow X$ to a function $g: E\rightarrow E$ such that $X$ is embedded in $E$, a Banach space.
We consider a compact metric space $X$ and a continuous function $f:X\rightarrow X$. I know it's possible to embed $X$ into a Banach or separable Hilbert space $E$ using the weak star topology. ...
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Confusion regarding the definition of the state of a physical system
I'm currently covering Jan de Vries' Elements of Topological Dynamics and in it he gives a brief introduction to the field through the lens of classical mechanics. He defines the state of a mechanical ...
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Recurrent neural networks stability
I'm reading some papers on stability of neural networks mainly a dynamical system point of view.
RNN can be thought of as $h_t=f(h_{t-1},x_t,\theta)$ where $\theta$ represent some parameters that are ...
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Flow of dynamical system evolves towards its interior guaranteed by imposing strict inequality in Nagumo's theorem?
To prove a closed set is positive invariant (i.e. the flow of ode either tangent to the boundary of the set or point inwards the interior of the set), we could use Nagumo's theorem. Consider the ...
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Reducing a system of differential equations to canonical form
I have the system (where w is a constant):
$\dot x = wx+wy$
$\dot y = -2wx-wy$.
I want to reduce it to canonical form (which I'm not entirely sure what that means...)
From what I've read I need to ...
- 1,901
2
votes
1
answer
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Positive/Negative Definite Functions Confusion
I have the Lyapunov function $V(x_1,x_2) = x_1^2+x_2^2$ for the nonlinear system:
$$\dot{x_1} = x_2$$
$$\dot{x_2} = -x_1 - x_2-x_2^3$$
Now obviously $V$ is a positive definite function but in order to ...
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0
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Linear Differential Equation as $t \rightarrow -\infty$
Consider the linear diffeq $$\dot{x} = Ax$$
The general solution has the form $$x(t) = c_{1}e^{\lambda_{1}t}v_{1}+c_{2}e^{\lambda_{2}t}v_{2}$$ if $A = V \Lambda V^{-1}$ and so is diagonalizable.
An ...
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1
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Why do we need a $T$-invariant probability measure to define ergodicity?
I am studying the concepts of ergicity in a dynamic system. Let $(M, \mathcal A)$ be a measurable space and $T: M \to M$ a measurable map. Given a probability measure $P$, we can say that $P$ or $T$ ...
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Topological conjugacy of the logistic map at different parameter values
I am wondering whether the dynamical systems generated by the discrete 1 dimensional map $g(x;p) = px(1-x)$ (the logistic map) at different values of $p$ are topologically conjugate.
Of course, this ...
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Conjugacy of expansive flows
I'm reading "Expansive one-parameter flows" by Bowen-Walters.
Let $(X,d)$ be a compact metric space and $\Phi:X\times \mathbb{R}\to X$ be a continuous flow on $X$.
They consider the ...
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Intuition of the concepts behind Ergodic Theory
As a graduate math and physics student, I am introducing myself to the study of Ergodic Theory, reading Introduction to the Modern Theory of Dynamical Systems, by Katok and Hasselblatt. I understand ...
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Expanding endomorphism is mixing - proof details
When I read Introduction to Dynamical Systems by Brin and Stuck, I didn't understand one detail. This is in the proof showing that expanding endomorphism is mixing. I do not understand why when $n>...
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Is there a Julia fractal that contains uncountable many copies of itself?
We know that the Mandelbrot fractal contains a countable number of copies of itself.
See :
Does the Mandelbrot fractal contain countably or uncountably many copies of itself?
Where that is explained.
...
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Seeking Help with a Proof Step in Measure-Preserving Systems and Kronecker Factors
I need your help to finalise my proof of the following theorem:Let $ (X, B, \mu, T) $ be a measure preserving system, where $(X,B ,\mu)$ is a
standard probability space. $ (X, B, \mu, T) $ is a ...
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Embedded subshift in $[0,1]$ has dimension $0$?
I was wondering whether most embeddings of one-dimensional subshifts have zero Hausdorff dimension? Given a finite alphabet $\mathcal{A}= \{ 0,...,d-1 \}$ and $\Omega\subseteq \mathcal{A}^\mathbb{N}$, ...
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Closed form experssion for gradient dynamics on energy $E = \frac{1}{1 + x^2} (s - xy)^2$
With the following energy $E = \frac{1}{1 + x^2} (s - xy)^2$, where s is a constant and x, y are two variables. The dynamics of gradient descent on this energy are
$\dot{y} = -\frac{\partial E}{\...
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Best references for a more intuitive introduction to Dynamical Systems, Ergodic Theory and Quantum Chaos
As a graduate math and physics student, I am introducing myself to the study of Smooth Dynamical Systems and Ergodic Theory, with the aim of applying it to Quantum Chaos and Quantum Ergodic Theory. I ...
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Calculating the expected hitting time of a specific birth and death chain
I am working with a specific birth and death chain, defined as follows.
Consider a set of states $X = \{0,1,2,...,n\}$, where $x^* \in (0,n)$ is a recurrent state. Transition probabilities are defined ...
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Is there a general solution to $f(x)=f^{\circ n}(x)$?
This question has crossed my mind, and I tried finding some solutions to that functional equation, then to find a pattern.
It's surprisingly hard to find real functional equation calculators online, ...
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Why a linear map with a negative double eigenvalue arising from the solution of a linear differential equation is proportional to the identity?
I'm trying to solve problem 3.2.5 from the book A First Course in Dynamics of B. Hasselblatt and A. Katok that says:
Suppose a linear map with a double negative eigenvalue arises from the
solution of ...
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How to find the summation of the above trigonometric series without using desmos?
How to find the summation of
$$
\sin(2 + \sin(2 + \sin(2 + \cdots \infty)))?
$$
I am trying this question by denoting the above summation as $S$. Therefore,
$$
S = \sin(2 + \sin(2 + \sin(2 + \cdots \...
- 1,159
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votes
1
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Existence of center manifold
I've been working on the following exercise:
Prove that the system
$$
\begin{cases}
\dot{x} = -x^3,\\
\dot{y} = -y + x^2
\end{cases}
$$
has no analytic center manifold (supposed in the following way $...
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0
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Soft question - Index theory in nonlinear dynamics vs Complex analysis
The video https://www.youtube.com/watch?v=wZvFKcQ_3Rc&t=8s mentioned something called the Index Theory. I can't find it on wikipedia. Where could I find more about the theory?
Here index is just ...
- 369
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Unipotent closure in classical groups
Let $G=\mathrm{SL}_n(\mathbb{R}),\mathrm{Sp}_{2n}(\mathbb{R}),\mathrm{Spin}_n(\mathbb{R})$ be a semi-simple simply connected classical group, $\Gamma\subset G$ a discrete and cocompact subgroup. Then ...
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Question on the proof of $\mathrm{scl}(a) = |\mathrm{rot}(a)|/2$ in $\mathrm{Homeo}^+(\Bbb R)^{\Bbb Z}$
I'm currently reading a part of Calegari's scl and encounter a proof I can't understand well.
Theorem 2.43. Let $\mathrm{Homeo}^+(\Bbb R)^{\Bbb Z}$ denote the full preimage of $\mathrm{Homeo}^+(S^1)$ ...
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Is it possible to derive an explicit expression for the inverse branches of $f(x)=x + x^{1+p}\mod 1, p > 0, 0\leq x\leq 1$?
My dynamics course has talked a bit about the Pomeau-Mannville map, defined as $f(x)=x + x^{1+p}\mod 1,p > 0, 0\leq x\leq 1$. I was wondering whether there is any known trick or a special function ...
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nonlinear odes: stabilizing terms in a subcritical pitchfork bifurcation
I am reading through Strogatz's book on nonlinear odes and dynamical systems. One thing that is a little confusing is his description of stabilizing higher order terms to control the dynamics of a ...
- 2,159
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0
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Differences between Poincare map and Poincare section
I am self-studying dynamical systems, and wanted to double-check my understanding of these concepts.
In Strogatz's "Nonlinear Dynamics," the author plots a periodic solution to the forced ...
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0
answers
17
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Reference for introductory topology for dynamical systems
I'm going to take an introductory class on dynamical systems at the level of Guckenheimer & Holmes, or Arnold's ODEs book, but I don't have any topology background. I know analysis at the level of ...
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0
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49
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Do other resources exist on Rate-Independent Systems?
Are Rate Independent System new in the world of mathematics? I can't tell because I certainly am new to mathematics.
I couldn't find any more resources than this book:
Rate-Independent Systems: Theory ...
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0
answers
22
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Inconsistency Between Approaches Solving System of Differential Equations
This requires a bit of setup, so bear with me. I'm working through a physics problem. The physics itself is irrelevant, but the setup gives you a system of differential equations. The details in how I ...
0
votes
0
answers
12
views
Modulational instability: Structures beyond linear stability analysis
Modulational instabilities occur in waves which occur in nonlinear system. A linear stability analysis is done to examine the dynamics in the short-time regime to observe the structures which ...
- 361
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0
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43
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$\ln(\exp(x) + \exp(y) + \exp(z)) = x + y + z$ , iterates of $x - \ln(\exp(x)-1)$ and generalizations
Let $x,y > 0$
Consider the equation
$$\ln(\exp(x) + \exp(y)) = x + y$$
Now it is clear (by symmetry) that if we express $y$ as function of $x$ or $x$ as a function of $y$ that is the same function.
...
- 15.3k
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0
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Behaviour and limits of $f(n+1) = \frac{f^5(n)}{2} - f(n-1)$
Let $f(0) = 0,f(1) = \frac{1}{2}$
and
$$f(n+1) = \frac{f^5(n)}{2} - f(n-1)$$
where $*^5$ is a power.
Then it seems
$$ \sup f(n) = \lim \sup f(n) = \frac{1}{2}$$
and
$$ \inf f(n) = \lim \inf f(n) = \...
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