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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97 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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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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20 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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38 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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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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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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24 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
70 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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29 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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37 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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121 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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126 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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53 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
35 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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84 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
14 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
32 views

First Order Difference Equations

I'm trying to solve this first order difference equation. I usually go through the usual avenue (eigenvalues of homogeneous function then eigenvectors then solution) but here I don't know what I can ...
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1answer
37 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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8 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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62 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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25 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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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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59 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
29 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
29 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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83 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: ...
2
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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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62 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
83 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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22 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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456 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
62 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 ...
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1answer
61 views

Difficulty in understanding the Dyadic map and its application

The Dyadic map also called as the Bernoulli Shift map is expressed as $$x(k+1) = 2x(k) \bmod 1$$. Consider a discrete map $F : X \rightarrow X$ in the interval. Let this map be the Tent Map. In Link1: ...
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47 views

Could all iterates of $s(n)=2n+1$ be composite for some starting $n$?

Let $s(n)=2n+1$ and $\sigma(n)=\{n,s(n),s^2(n),s^3(n),\ldots\}$, where $s^3$ denotes functions composition, $s^3(n)=s(s(s(n)))$. For example $\sigma(11)=\{11,23,47,95,\ldots\}$. As another example ...
3
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1answer
51 views

Gentlest ascent in dynamical system

I have a question about the following excerpt from the paper("An Iterative Minimization Formulation for Saddle-Point Search") by Gao,Leng, Zhou on gentlest ascent in dynamical systems. I am having ...
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1answer
66 views

Slow fast systems

I have some questions concerning fast slow system like the van der pol equation say we have $\epsilon x′_1=-\frac13 x_1^3+x_1 − x_2$ and $x′_2= x_1$ Does $\epsilon x'_1$ means that $x_1$ is faster ...
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65 views

Topological Conjugacy between tent and skewed tent map

Consider the family of skew tent maps $\mathcal{S}$ on $[0,1]$, such that: $S(0)=S(1)=0$; The peak (maximum) of the tent occurs at $S(a)=b$; $\max(a,1-a)<1$ which implies the map to be locally ...
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38 views

Connection between possibility of non-monotonic solutions to first-order delay differential equations and 1-d discrete dynamical systems?

Is there a connection between the possibility of non-monotonic solutions, including periodic or other oscillatory solutions, arising in first-order autonomous delay differential equations such as the ...
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1answer
23 views

Approximating monotonically increasing differential equation

I am trying to make sense of the Appendix of the paper (Cooper, 1986). The following model is presented: $$\dot{(BX)}=\gamma_1BX \\ \dot{(BXB)}=\gamma_2(BX)B \\ \dot{B}=\gamma_3(BXB)$$ Without ...
2
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1answer
69 views

Is the topological entropy of the “caterpillar waves” 0?

Please let me first describe the general background. The state of the system at time $t$ will be described by a scalar or phase $u=u^t$. Both $t$ and $u$ are discrete. $u$ take values from ...
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43 views

Function not equal a.e. to continous function on real line and on circle

I am looking for a proof of the following fact: Suppose that $H: \mathbb{R} \rightarrow \mathbb{R}$ is a periodic function with period $1$. Suppose further that there is no continuous function ...
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1answer
34 views

Why is T continuous?

Let $T\colon X\to X$, with $X=\left\{0,1,2\right\}^{\mathbb{Z}}$, desribe the following dynamics: 1 becomes 2 2 becomes 0 0 becomes 1 if at least one of its two neighbours is 1, otherwise it remains ...
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1answer
33 views

controlling position of pendulum with motor

There is a pendulum with a motor mounted at its point of rotation. The motor can generate a rotational force at any time, thus changing the dynamics to $θ" = −a*sinθ − b*θ" + u$ where $u$ is the ...
3
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1answer
31 views

What does a 3D periodic solution of a differential equation look like?

The Pointcare-Bendixson Theorem implies that if a solution stays in a bounded region with no equilibrium points then it is either a periodic solution or it approaches a periodic orbit as t goes to ...
0
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1answer
57 views

Lyapunov function

My lecturer gave us the definition of a Strong Lyapunov function. She then said that if V is positive definite but $dV/dt$ is also positive definite (instead of negative definite) in a region ...
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1answer
69 views

Lyapunov equation for stability analysis - what's the point?

Straight from Wikipedia: In the following theorem $A, P, Q \in \mathbb{R}^{n \times n}$, and $P$ and $Q$ are symmetric. The notation $P>0$ means that the matrix $P$ is positive definite. ...