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 ...

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Bifurcations of dynamical systems, different parameters [closed]

I've found the following excercise and I've broken my head about it, but I don't know how to answer it. So we have a system $$\frac{dx}{dt} = f(x,a_1,a_2,...,a_n)$$ And say we first take $a_1$ as ...
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23 views

Periodic Shifts of finite type

I am learning about periodic shifts and want to look at finite types of periodic shifts. So I have this theorem: A shift space $X$ is an $M$-step shift of finite type iff whenever $uv, \; vw\; \in ...
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41 views

Existence of a recurrent point [duplicate]

I need to show that if $f:S^{1} \rightarrow S^{1}$ is a preserving-order diffeomorphism and $f$ has irrational rotation number, then $f$ has at least one recurrent point. How can I prove that? ...
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1answer
27 views

Continuous Time Dynamical Systems with 2-cycles

I know that discrete time dynamical systems such as $x_{n+1} = rx_n(1-x_n)$ exhibit 2-cycles for some parameter values of r. I'm curious if there exist continuous time dynamical systems that exhibit ...
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3answers
88 views

How to interpret complex eigenvectors of the Jacobian matrix of a (linear) dynamical system?

Consider a linear ODE system of the following form: $$ \frac {dx} {dt} = Ax $$ In case $A$ has real eigenvectors, I can interpret them as the directions in which the system will move, if the initial ...
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18 views

colvolution function

I am trying to understand the equaliti denoted in the attache picture. Any help? Thank you!
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1answer
31 views

writing a state of a dynamical system

Is it un/common to write the state of a dynamical system in the following manner: $$ \begin{pmatrix}x_{t+1} \\ v_{t+1} \end{pmatrix} = \begin{pmatrix}A & B \\ C & D ...
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1answer
33 views

modelling the behavior of a particle

I want to study the evolution of a particle as a function of time, then in dynamical systems, the usual thing to do would be to define the state of the particle. Usually we are able to do this by ...
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1answer
47 views

Proving conjugacy to the Logistic Map

I have a map which I have to show is a conjugate to the Logistic Map ( $x_{n+1} = rx_n(1-x_n)$ ). The map in question is as follows. $x_n = \sin^2(\pi\theta_n)$ $\theta_{n+1} = N^n\theta_0$ mod $1$ ...
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47 views

What is corresponding Lie group for Lie algebra of vector fields in dynamical systems?

According to Ado's theorem, for every finite dimensional abstract Lie algebra there is a Lie group. Finite dimensional analytic (or meromorphic) Vector fields (in dynamical systems) over the filed of ...
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1answer
37 views

A generalization of Poincare-Birkhoff theorem

What could be the statment of a possible generalization of Poincare Birkhoff theorem for $M\times [0,\; 1]$ where $M$ is a compact orientable manifold?
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74 views

Solution for a differential equation

I am stuck in getting the solution for the following non-linear differential equation: \begin{equation*} x^2 + B\frac{dx}{dt} = A\sin(wt) \end{equation*} Is there any method to solve this kind of ...
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1answer
100 views

Does every homeomorphism of a compact metric space lift to the Cantor set?

This is a follow-up to this question. It is well-known that any compact metrizable space can be expressed as a quotient of the Cantor set. But can every homeomorphism of such a space be lifted to a ...
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28 views

What type of Hopf bifurcation takes place here?

Consider the system: $\dot{x} = \mu x-y-xy^2-x^3$ $\dot{y} = x+\mu y - x^2y-y^3$ I have shown that a Hopf bifurcaiton takes place at the origin $(0,0)$ as a stable spiral becomes an unstable spiral ...
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56 views

How I can make a proof to this conjecture if it is possible?

Is there someone who can show me the way helping me to proof this conjecture. If it's not open, at a least show me links or papers which to be helpful. Conjecture: Assume $c > 0$ and that an ...
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1answer
95 views

Understanding stability of fixed points in 2D maps.

I'm trying to understand the stability analysis for a map of the form $$(x_{n+1}, y_{n+1}) = A(x_n,y_n)$$ Where A is a 2x2 matrix - assumed to be diagonalisable and with distinct eigenvalues. I ...
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1answer
42 views

Necc. and suff. conditions for a canonical transformation.

Let $\mathbf{P} = C^{−1}\mathbf{p} + B\mathbf{q}, \mathbf{Q} = C\mathbf{q}$, where $C$ is a symmetric nonsingular matrix. Determine necessary and sufficient conditions on $C$ for the transformation ...
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19 views

Transfer function from state variable expression

I have a 3x3 state variable system. I need to choose where to place my poles according to some criteria. For example: (a) Percent overshoot < 20% (b) SettlingTime < 1.5s, and (c) steady-state ...
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1answer
32 views

Sufficient and necessary condition for a local contraction

I have an iterative map $X^{n+1} = T(X)$ which maps a k-tuple of vectors in $\mathbb{R}^n$, that is, $X = (\bf {x_1,x_2,x_3,\dots,x_k})$, where $\bf x_i\in\mathbb{R}^n$, into another $k$-tuple, i.e. ...
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1answer
20 views

The two forms of Henon map

The widely-used form of Henon map, according to Wikipedia, is $$ \begin{cases}x_{n+1} = 1-a x_n^2 + y_n\\y_{n+1} = b x_n.\end{cases} $$ However, in some other places, for example in the manual of ...
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25 views

Finding invariant manifolds

$$x'=y$$ $$y'=-x+x^3$$ from above system, one gets hyperbolic equilibria $(1,0)$ and $(-1,0)$. and both equilibria have same eigenpairs $(\lambda,v)$, such as $(\sqrt{2},(1,\sqrt{2})^T)$ and ...
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1answer
44 views

Relatively simple system of nonlinear ODEs

There are a lot of questions like this on MSE as well as online resources on the subject, but a) the MSE questions are either unanswered or correspond to systems substantially different from this one, ...
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22 views

Topological conjugacy in a dynamical system

Given nonlinear dynamical system, if one is asked to show that this system is topologically conjugate, is it asking that the flow of nonlinear system and the flow of linearization of the nonlinear ...
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1answer
50 views

Graph Theory + Dynamical Systems

Suppose you had a dynamical system $\dot{\vec{x}} = \vec{f}(\vec{x})$. In theory, one could represent this as a directed graph where the vertices are fixed points of the dynamical system and the edges ...
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28 views

How is this an untable limit cycle?

I am investigating the Lorenz equations and in MATLAB I have plotted a case with $\sigma = 10, b = 8/3, r = 21$ and I have this phase portrait: However I am not exactly sure how this is an unstable ...
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1answer
35 views

What does it mean take the determinant of the Jacobian in: $ V_{k+1} = \int_{M_{k}} \Bigg\vert det(\frac{\partial y}{\partial x}) \Bigg\vert dx$

In this Lecture, in the subsection Evolution of Volumes tell us: Let $M \subset D$ be a compat subset of phase space. We can define its volume by a usual Riemann integral: $$ Vol(M) = ...
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1answer
114 views

Is there any relationship between the Riemann z function and strange attractors?

I have this question in mind since the first time I saw a graphical representation of the zeta function (like in the sample below). Just by looking to them I wondered if there is any relationship ...
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2answers
117 views

solution of $y^{\prime \prime} + y^n = 1$ [closed]

I am not able to figure out the solution for the differential solution $$y^{\prime \prime} + y^n = 1$$ I want to specifically find an answer for $$y^{\prime \prime} + y^2= 1$$and $$y^{\prime \prime} + ...
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1answer
54 views

Concluding from limiting behavior

I've recently seen the following question on the internet: If I wanted to randomly find someone in an amusement park, would my odds of finding them be greater if I stood still or roamed around? ...
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0answers
52 views

Is this a spontaneous symmetry-breaking?

I have a system of equations: \begin{cases} f\left(x_{1}\right)+f\left(x_{2}\right)+P=0\\ \\ g\left(x_{1}\right)+g\left(x_{2}\right)=0 \end{cases} where $f,g$ are some functions, ...
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1answer
14 views

Circular solution of Kepler Problem

How can I get a circular solution of the $2-$dimensional Kepler problem of the form $$q=\exp(Kt)a$$ being $$\exp(Kt)= \begin{pmatrix} \cos(t) & \sin(t)\\ -\sin(t) & \cos(t)\\ ...
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1answer
34 views

How to program a little code that shows me the evolution of the system when starting with initial values?

I have rather no programming skills, neither with Matlab nor with other languages. I need a little "program" that shows me the evolution of a dynamical system when giving it some initial values. The ...
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1answer
81 views

Using Poincaré-Bendixson to prove that there is a periodic solution

I want to use the Poincaré-Bendixson theorem to show that there exists a nontrivial (and periodic) solution to $$z'' + [\log (z^2 +4(z')^2)]z' + z = 0.$$ Therefore I substituted $u = z'$ to get $$u' ...
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1answer
13 views

Calculating the inverse of a continuous map for a certain interval in order to calculate the Perron-Frobenius operator.

Suppose we are observing chaotic continuos maps, the Perron-Frobenius operator $P$ satisfies: $P\phi_{n}(t) = \frac{d}{dt} \int_{f^{-1}([a,t])} \phi(x)dx$ I don't understand how for the shift map, ...
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1answer
36 views

Poincaré lemma and conservative vector fields

Let $U$ be some contractible neighbourhood of $0\in\mathbb{R}^n$ and let $X=\sum_{i=1}^nX_i\frac{\partial}{\partial x_i}$ be a (smooth) vector field on $U$. This vector field can be thought as a ...
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7 views

controllable system, matrix exponential norm

suppose we have $(A,B)$ be a controllable pair. Can I find a feedback control gain $K$ such that $A_c=A+BK$ is Hurwitz, which also satisfies that $||e^{A_ct}||\leq a e^{(-\lambda t)}$ and ...
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1answer
54 views

Determining boundary of basins of attraction

Let's say that I have a dynamical system that displays multiple stable states with corresponding basins of attraction. The Lyapunov function for the system is not known. Is there an analytic or ...
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23 views

Notation from Bowen's Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms

On pdf page 39 (page 33 of the text) under Proposition 2.8, the following notation was used $$U(\underline{V})=U(V_{i_0})\cdots U(V_{i_{m-1}})$$ $$U(\Gamma_m)=\{U(\underline{V}):V\in \Gamma_m\}$$ I ...
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1answer
38 views

What are “relaxation constants” in ode systems?

In my reading, I came across the following ODE system: $$\lambda_1 \dot x = f(x,y)$$ $$\lambda_2 \dot y = g(x,y)$$, where $\lambda_1$ and $\lambda_2$ are positive constants. Then, I saw that the ...
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0answers
58 views

Baker's map is ergodic

Define Baker's map by \begin{align} f(x,y) = \begin{cases} (2x,y/2) & \mbox{ if } (x,y)\in[0,1/2]\times[0,1] \\ (2x-1,y/2+1/2) & \mbox{ if } (x,y)\in[1/2,1]\times[0,1] \\ \end{cases} ...
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1answer
26 views

Constructing a Poincare map for dynamical system

I'm trying to construct a Poincare map for the system: $$\dot{x} = y$$ $$\dot{y} = -a^2x + b\cos(\theta)$$ $$\dot{\theta} = a$$ I have always thought of the Poincare map as more of a theoretical ...
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1answer
28 views

The definition of C^r Structural Stability

I currently have a definition that states that given a flow $f$, $f$ is structurally stable if for any $g$ in some neighborhood of $f$, $f$ and $g$ are topologically conjugate. Would the definition ...
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82 views

Certain Lie algebra structure on $\chi^{\infty}(\mathbb{R}^{2})$ or $\chi^{\infty}(S^{2})$

Is there a lie algebra structure $ [ \;. ] $ on $\chi^{\infty}(\mathbb{R}^{2})$ or $\chi^{\infty}(\mathbb{S}^{2})$ which is not isomorphic to the standard structures but satisfies the following: ...
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1answer
79 views

Is it possible to give an explicit description of the set of recurrent points?

Consider $$ X=\left\{0,1,2\right\}^{\mathbb{Z}},~~~T\colon X\to X, $$ and let $T$ describe the following dynamics: $1$ becomes $2$ $2$ becomes $0$ $0$ becomes $1$ if at least one of its two ...
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1answer
53 views

Definition of Lebesgue measure on the Circle $X=\mathbb{R}/\mathbb{Z}$.

I was given the following definition: I feel this is not adequate and would only define Lebesgue on $[0,1]$? (Wikipedia uses a pushforard of the complex exponential) Also later I am given a proof ...
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1answer
75 views

Dynamical Systems- Plotting Phase Portrait

So, I understand when plotting the phase portrait of a dynamical system, one must find the equilibrium points, classify the equilibrium points, and straight line paths (if the equilibrium points are ...
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18 views

Multi time scales analysis on nonlinear system of ODEs

So I have this coupled set of nonlinear ODEs that I want to do a multi time scales perturbation analysis on. $ u'(t)+\frac{C \epsilon u(t)^2}{Cl}-\frac{2 \epsilon p(t)}{Cl}-\frac{2 q_1'(t)}{Cl}=0 ...
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15 views

Deriving the Particular Solution to a Linear Discrete Dynamical System

In my lecture notes it says that for a linear dynamical system of the form $ f(x) = Ax $ where A is diagonalisable d x d matrix, with $ \left \{ v_1 , v_2, \cdots , v_d \right \} $ a basis for $ ...
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1answer
454 views

How to determine $\Omega(T)$?

Let $X=\left\{0,1,2\right\}^{\mathbb{Z}}$ and let $T\colon X\to X$ describe the following dynamics: 1 becomes a 2, 2 becomes a 0 and 0 becomes a 1 if at least one of its two neighbours is 1, ...
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1answer
59 views

Help in understanding a conjugacy problem

I am studying the book Applied Symbolic Dynamics and Chaos By Bai-lin Hao, Wei-Mou Zheng The basic premise of the concept of Symbolic Dynamics is : "Symbolic ...