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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8 views

smoothness of invariant manifolds

Suppose M be compact manifold and f be a diffeomorphism on M.and A be hyperbolic set respect to f.How can we proof that the global stable and unstable manifolds of A are embedded manifolds?
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7 views

Suspension flow and topological equivalence(s)

Let $M$ be a compact smooth manifold, $\tau:M\to\mathbb{R}_{\geq 0}$. Let $f:M\to M$ is a surjective piecewise-smooth map. There is a standard construction of suspension allowing to extend $f$ to a ...
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1answer
36 views

What are the fixed points of $f_ c = c · \sin$ for $c > 1$?

I’m doing an exercise for a lecture on dynamical systems. We are asked to classify all bifurcations of the dynamical system $f_c = c·\sin$ for real $c > 0$. We are given that bifurcations of ...
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0answers
67 views

Help in understanding the prove whether a map is one-to-one from the book

I found the same Question here Prove that $S: I \rightarrow \Sigma_2'$ is one-to-one. But it is unsolved. Moreover, I do not understand the answer in the Question. Definition 1 : $\Sigma_2 = ...
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1answer
26 views

How can this function be considered to have a saddle node bifurcation?

Say I have the function $f(x,\mu) = (1 + \mu)x − x^2 − 0.1$. By definition a Saddle Node bifurcation occurs if: $f_{\mu_0}(0) = 0$ $f'_{\mu_0}(0) = 1$ $f''_{\mu_0}(0) \neq 0$ $\frac{\delta ...
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1answer
26 views

How to describe behavior of population system, given by system of ODEs?

So I have a system of equations:$$x'(t)=x(t)(4-2x(t)-y(t))\\y'(t)=y(t)(3-x(t)-y(t)) $$ What I understand so far is: if we have x being the population of prey and y is the population of predators. x ...
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12 views

Computing the index around a curve with respect to a field, invariance?

If I understood the course book Nonlinear Dynamics and Chaos right, The index can be found by $$\newcommand{\dd}{\mathrm{d}} \newcommand{\id}{\mathrm{d\,}} I_{C}=\frac{1}{2\pi}\oint_C ...
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1answer
12 views

Velocity field arrows along null clines as well as outside null clines

For Question 8 (as well as in general), I don't understand how to sketch velocity field arrows along the null clines as well as outside the null clines. For this question the f1 null cline would be ...
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0answers
35 views

Lotka-Volterra Problem From Arnold's Ordinary Differential Equations

Problem 1 of section 2.7 of Arnold's Ordinary Differential Equations book asks to prove that the period of the oscillations in the Lotka-Volterra model tends to infinity as the initial condition ...
9
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1answer
118 views

Bifurcations in the Duffing oscillator

I'm trying to describe all the bifurcations in the two parameter Duffing oscillator: $$\ddot{x} + ax + bx^3 = 0$$ In phase space with $y = \dot{x}$ I've found the origin to be a centre for $a>0$ ...
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1answer
23 views

Locally evaluate nonlinear dynamic system's stability using eigenvalues

I don't have a large mathematical background, but I'm working with Computational Neuroscience. I have a large Synaptic Matrix (x axis: presynaptic NeuronID, y axis: postsynaptic NeuronID) in a Modular ...
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45 views

Prove that $T$ has an orbit of period 3

Suppose that $T$ is continuous map from an interval $I$ to itself. Moreover, suppose that there exists $x_1 < x_2 < x_3 < x_4 $ such that $$T(x_1) = x_2, T(x_2) = x_3, T(x_3) = x_4\ \ ...
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0answers
33 views

Idea behind Lyapunov functions? [on hold]

Despite my best efforts I am having some trouble understanding the concept of Lyapunov functions. Suppose we have a 2D dynamical system $$\dot{x} = f(x,y) \\ \dot{y}=g(x,y)$$ with a fixed point at ...
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179 views
+100

Equality by iteratively applying $(a,b)\rightarrow [(a+1,2b)\text{ or }(2a,b+1)]$?

I play a game starting with $2$ positive integers $a$ and $b$. At each step of the game I can double one of the integers and add $1$ to the other integer. Is there always a procedure for any ...
5
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1answer
553 views

Why is unique ergodicity important or interesting?

I have a very simple motivational question: why do we care if a measure-preserving transformation is uniquely ergodic or not? I can appreciate that being ergodic means that a system can't really be ...
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1answer
44 views

Show that f: $\mathbb{R}$/$\mathbb{Z}$ $\to$ $\mathbb{R}$/$\mathbb{Z}$ orientation reversing. Then f(x) = x has exactly 2 solutions.

Im having some problem with the following question. Show that if $f: \mathbb{R}/\mathbb{Z} \to \mathbb{R}/\mathbb{Z}$ orientation reversing, then $f(x) = x$ has exactly $2$ solutions. ($f$ has $2$ ...
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0answers
28 views

Phase Diagram Bifurcations and much more. [on hold]

this is a problem I came across while solving a paper. A temporary guide would be appreciated, because I just need a kick-start,I'm not familiar how I should be tackling the question, so a guide on ...
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1answer
18 views

Lagrangian of bead on a rotating hoop

I'm trying to find the Lagrangian for a bead on a rotating circular loop (constant angular velocity $\omega$, radius $a$) in two different ways and I'm unsure why these are giving different answers. ...
6
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1answer
89 views
+150

Properties of the space of $ T $-invariant probability measures over a compact topological space.

Let $T: X \to X$ be a continuos map defined on the compact space (Maybe Haussdorff) $X$. Denote by $M_T$ the space of $T$-invariant probability measures defined over the Sigma Algebra of Borel sets. ...
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1answer
23 views

Periodic cycles of the Poincare map

For a dynamical system $\dot{x} = f(x)$, I understand the Poincare map is defined by successive intersections of an (n-1) dimensional surface $\Sigma$ with trajectories in n dimensional phase space. ...
9
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1answer
562 views

Complicated exercise on ODE

I have this exercise extracted from a examination of qualitative theory of ODE (in which we study the existence and uniqueness of solutions, and stability using the function of Lyapunov) I don't know ...
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1answer
26 views

Numerical phase plane?

In my Differential Dynamical Systems text book, I came across the following question: Sketch the local behavior you obtained in the phase plane and compare with a numerical phase plane plotter that ...
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30 views

3-cycles of the iterated quadratic maps and the Galois Theory of $x^3 = a(x-1)$

Taken from these notes [1] on Galois Theory, I would like to show that iterating the map $$p: x \mapsto x^2 - x - 2 $$ has a cycle of order 3 when you start with the root of $x^3 - 3x - 1 = 0$. ...
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2answers
379 views

Reference book for “Dynamical Systems”

I want to do my thesis about oscillations. I am a math student so I enjoy rigorous texts and hate sketchy ones. I am looking for a textbook or a good source that could help me with dynamical systems. ...
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3answers
3k views

Recommendation for a book and other material on dynamical systems

I currently have the book Dynamical Systems with Applications Using Mathematica by Stephen Lynch. I used it in an undergrad introductory course for dynamical systems, but it's extremely terse. As an ...
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3answers
295 views

Textbook Recommendation: Topological Dynamics

I need to take credits satisfying a topology requirement, and can structure it myself. My field of study is dynamical systems, can someone recommend a textbook that handles differential ...
3
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1answer
33 views

Recommended second textbook for dynamical systems?

I recently finished a course on dynamical systems supplemented by Strogatz's textbook. There are a few parts of the book that we didn't cover (in particular, the material on fractals), but the ...
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1answer
32 views

Angle-doubling map is mixing

Let $$ T:\mathbb{S}^1\to\mathbb{S}^1\\ x\mapsto 2x $$ be the angle-doubling map on the circle. We know that this transformation is ergodic. We want to prove that is mixing. I have to show that $$ ...
3
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1answer
54 views

$A\in \Bbb R^{n\times n}$ and $B \in \Bbb R ^{n\times m}$. Show that $\exists p\in\Bbb R ^m$ s.t. $(A,Bp)$ is controllable iff $(A,B)$ is controllable

Let $A\in \Bbb R^{n\times n}$ and $B \in \Bbb R ^{n\times m}$. Show that there exists a vector $p\in \Bbb R ^m$ such that $(A,Bp)$ is controllable iff $(A,B)$ is controllable. Here when I say $(A,B)$ ...
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1answer
20 views

What is the difference between a trajectory and an orbit in dynamical systems?

In dynamical systems, the integral curves for a differential equation that governs a system are referred to as trajectories or orbits, then what is the difference bewteen they?
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47 views

What is the slow manifolds? and how to calculate?

I'm a newbie in slow manifolds and dynamical system. I cannot understand the concept of slow manifolds and how to calculate that. Please explain the concept of slow manifolds intuitively and ...
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0answers
52 views

What's the difference between $Df$ and $Tf$?

I'm reading Michael Shub's Global Stability of Dynamical Systems. In chapter 4, he defined hyperbolic set and said the splitting $E^s$ and $E^u$ are $Tf$ invariant. So I assume this $Tf$ is the ...
2
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1answer
59 views

Eigenvectors question

$x'=x-2y$ $y'=4x-x^3$ Equilibrium points are $(2,1),(-2,-1),(0,0)$ Consider equilibrium point $(2,1)$: Let $X=x-2$ and $Y=y-1$. Subbing this into the main and eliminating all the nonlinear terms ...
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0answers
25 views

slow manifold of Val Der Pol's equation

I'm newbie in dynamical system and confused the Direction of the point (1,-1) moving. $$\epsilon x_1'=-\frac{1}{3}x_1^3+x_1-x_2$$ $$x_2'=x_1$$ I think that point moves left and up. Because $x_1$ ...
2
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1answer
37 views

A differentiable manifold of class $\mathcal{C}^{r}$ tangent to $E^{\pm}$ and representable as graph

The stable manifold theorem tell us: A local stable manifold $W^{s}_{loc}(x^{*})$, which is a differentiable manifold of class $\mathcal{C}^{r}$ and dimension $n_{-}, $ tangent to the ...
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1answer
32 views

Stubborn dynamical system state

It is rather common to use matrices to represent the relationship of the states of dynamical systems. It is very natural to use matrices because of their ease in analysis. Stability, convergence ...
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58 views

How to find in a Stability of Linear Systems. BIBO Stable, but Lyapunov Unstable System [closed]

I need help in this question. Let the input u ≡ 0. Determine the states x1 and x2 and the output y. For what set of initial conditions [x10 x20]T will the output be zero? This is the reply of ...
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1answer
41 views

Conditions for Deriving $R_0$ for SIR Model Using Survival Function Method

I'm taking a look at the SIR model given by the system of differential equations \begin{align} \frac{dS}{dt} & = - \beta S I \\ \frac{dI}{dt} & = \beta S I - \gamma I \\ \frac{dR}{dt}& ...
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95 views

reworking attractor equations

The equations/algorithms of that define attractors such as these: http://www.chaoscope.org/doc/attractors.htm With given parameters and number of iterations output a set of positions, the size of ...
2
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0answers
39 views

Question related to the ballistic motion

A point mass will move in the gravitational field of the Earth according to the equation $$\ddot R =-\frac{GM_eR}{|R|^3},$$ where $R$ is the position vector of the point mass measured from the ...
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0answers
11 views

How to understand this verbal description of the dynamics?

Let $T$ map $\left\{0,r,l\right\}^{\mathbb{Z}}$ to itself by having the r's move right, the l's move left and an r and an l annhihilate each other when they meet or cross. How would you understand ...
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32 views

Terminology for functions such that $f(x)\ge x$ for all $x$

Is there a common terminology for a real function $f$ such that $$f(x)\ge x$$ for all $x$? same question for the conditions $\forall x,f(x)>x$; $\forall x,f(x)\le x$; $\forall x,f(x)<x$. (I'm ...
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1answer
35 views

Sketching phase portrait

$\dot{x}=-2x-2y$ $\dot{y}=-x-3y$ Equilibrium point is $(0,0)$. Eigenvalues are $\lambda_+=-1$ and $\lambda_-=-4$ which have corresponding eigenvectors $2\choose -1$ and $1 \choose 1$ respectively. ...
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2answers
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A (topological) transitive point must be a recurrent point?

Given a continuous maps f:X——>X (maybe X is a metric space, even X is a compact metric space), The point x is a transitive point, if x's orbit is dense in X. And x is a recurrent point if there exist ...
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1answer
81 views

Flow, dynamical systems

We have $f:\mathbb{R}^N \to \mathbb{R}^N$ Lipschitz and $x(\cdot; x_0 )$ is the unique solution of $$x'(t)=f(x(t))$$ $$x(0)=x_0$$ Every solution of this ODE is global. Show, that $\Phi_t ...
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0answers
18 views

Characterizing equicontinuity via ultrafilters

We have a compact metric space $(X,d)$ and a homeomorphism $T:X\to X$. For any ultrafilter $p\in\beta\mathbb{Z}$ we can define the map $T^p:X\to X$ given by $T^p(x):=\lim_{n\to p}T^n x$ (which can ...
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1answer
38 views

How to proof that bracket of two vector field can be computed by second derivation

Can some one give a hint how can I proof that where $\phi$ indicated the flow of vector fields.
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1answer
46 views

Show system of ODEs has a periodic solution by finding smallest annular trapping region

Find the smallest annular trapping region of the following: $r'=r(1-2r^2+sin(2\theta)r^2)$ $\theta ' = -1$ I really do not understand how to do this. I have been trying to figure it out from things ...
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17 views

A question regarding space-state representation

First of all I am not sure if this is the right place to ask this. Lets say we have a system in a form of a harmonic oscillator desribed by a second order DE. There will be 2 state variables - x ...
18
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467 views

This one weird thing that bugs me about summation and the like

Most of us know $$\sum_{n=a}^b c_n=c_a+c_{a+1}...+c_{b-1}+c_b$$ Some of us know $$\prod_{n=a}^b c_n=c_a \cdot c_{a+1}...c_{b-1} \cdot c_{b}$$ A few of us know ...