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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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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12 views

How do I prove that for $ 0<\lambda<1$ the transformation $ T= x^3 - \lambda x $ is strusturally stable [on hold]

I don't know how to take the sup of the distances between T and a function g topologically conjugated to f, and their derivatives
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12 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
24 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. ...
2
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1answer
14 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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15 views

In a topological dynamical system every minimal subsystem is contained in every backward-upward subsystem

Definition. A topological dynamical system is a couple $(X,T)$, where $X$ is a compact, regular topological space, and $T:X\to X$ is a continue map. Definition. A subsystem of $X$ is a closed subset ...
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20 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
38 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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18 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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0answers
14 views

constructing a homenmorphism [on hold]

How to construct a homeomorhpism between these two linear one dimensional systems?: $$x'=-x$$ $$x'=x$$ I've been struggling to find one for a while, I failed. And in addition, is there a general way ...
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1answer
27 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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17 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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38 views

Differential equations, Dynamical Systems, and an introduction to chaos [closed]

Maybe this question doesn't belong here but i'm starting on a new book and it would really help if i had a solution manual of it. The book is 'Differential Equations, Dynamical Systems, and an ...
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1answer
27 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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13 views

Find the eigenvalues/vectors for the $N\times N$ matrix C summarizing the system $-X_{t-1} +2X_{t} - X_{t+1} = \lambda X_{t}$ [closed]

Consider the system $-X_{t-1} +2X_{t} - X_{t+1} = \lambda X_{t}$ for $k = 1,...,N$ Write down the NxN matrix C such that the above equations are equivalent to $CX = \lambda X$. Find the ...
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1answer
80 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
113 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
49 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
46 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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0answers
10 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
33 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
62 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
10 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
21 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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0answers
33 views

Complex map of two variables: Julia, Mandelbrot and Fatou Set [on hold]

$f_\alpha(u,v)=\frac{\alpha u}{1+v}$ where $u$ and $v$ are complex numbers. $v \neq 1$. The dynamical system is defined by $z_{n+1}=f_\alpha(z_n,z_{n-1})$. How to get the Julia, Mandelbrot and Fatou ...
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1answer
31 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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0answers
15 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 $$U(\bar{V})=U(V_{i_0})\cdots U(V_{i_{m-1}})$$ $$U(\Gamma_m)=\{U(\bar{V}):V\in \Gamma_m\}$$ I double ...
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1answer
33 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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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
17 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
21 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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0answers
24 views

Phase paths of a node not at origin [on hold]

When you have a node in the origin, you draw on the two eigenvectors and then if its a stable node, then the paths come along the eigenvector corresponding to the dominant eigenvalue. If unstable, ...
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25 views

Boundedness of a dynamical system [on hold]

Consider $$Z_{n+1}=\frac{\alpha Z_n}{1+Z_{n-1}}$$ where $z_0$ and $z_{-1}$ are given complex initial values. Note, $\alpha$ is a complex number. Here, $$|Z_{n+1}|\leq|{\frac{\alpha ...
2
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67 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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0answers
25 views

Non-wandering set of L and R?

I outsource a question related to this: How to determine the non-wandering set $\Omega(T)$ (if possible at all)?. Hope this is okay. Let $X=\left\{0,1,2\right\}^{\mathbb{Z}}$ and let $T\colon X\to X$ ...
2
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1answer
61 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
38 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
52 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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16 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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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
324 views

How to determine the non-wandering set $\Omega(T)$ (if possible at all)?

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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16 views

Help in understanding the correspondence between numerical and symbolic orbits

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
34 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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1answer
45 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
44 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
40 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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32 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 ...
2
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0answers
29 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 ...
1
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1answer
18 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 ...