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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86 views

Henon Map Parameter

In case of Hennon map two parameters $a$ and $b$ to be set.The Hénon map takes a point $(x_n, y_n)$ in the plane and maps it to a new point $x_{n+1} = 1-a x_n^2 + y_n$, $y_{n+1} = b x_n$. The map ...
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0answers
40 views

Considering bank-interest and inflation rates to calculate remaining money in the account

Peter has A [35,000₤] in bank and banks gives B [350₤] per month as interest; he immediately puts C [100₤] back to the to account and spend the rest of it R [250₤] till next months. Every month, ...
2
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2answers
34 views

stability of equilibria for $n$-dimensional nonlinear systems of differential equations: examples

I'm currently self-studying dynamical systems. I'm trying to summarize what can be said about the stability of equilibrium points for an $n$-dimensional non-linear system of differential equations: ...
1
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1answer
14 views

System of separable diff. eqns, explicit solution and curves, Lotka-Volterra model

In the book on p.68 is a system of differential equations for a Predator-Prey model (Lotka-Volterra) given as: $$ \dot x=x(\alpha-c\gamma) \\ \dot y=y(\gamma x -\delta) $$ On the next page, it is ...
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1answer
23 views

Infinite closed shift-invariant set of binary sequences with zero entropy?

Does there exist an infinite closed shift-invariant $X \subset \{0,1\}^\Bbb Z$ with zero topological entropy? How to think of an example? Will periodic points of shift$|_X$ have zero entropy?
4
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0answers
57 views

How do i find formula for the recurrence relation :$x_{n+1}= x^2_{n}-x^2_{n-1}$ with :$x_{-1}=0,x_{0}=\frac{3}{4}$?

How do i find formula for the recurrence relation according to the below initial conditions: $x_{n+1}= x^2_{n}-x^2_{n-1}$ with :=$x_{-1}=0,x_{0}=\frac{3}{4}$ ? Note:$n=0,1,2\cdots$ Thank you for ...
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1answer
593 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 ...
2
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0answers
25 views

convergence of systems of the form $V(t+T) = e^AV(t)$

Let assume I have a generic positive function $V:[0,\infty)\rightarrow \mathbb{R}^n$ which satisfy $$V(t+T) = e^{A} V(t),\qquad t\in[0,\infty)$$ where $A$ is a non diagonalizable real valued $n\times ...
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0answers
29 views

A continuous function with a dense set of periodic points but without sensitive dependence on initial conditions

The Question: Give an example of a continuous function, $f$, on the interval, $I$, such that the set of periodic points of $f$ is dense in $I$, but f does not have sensitive dependence on initial ...
0
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1answer
18 views

Difficulty understanding the concept of writing an L-System?

I've recently tried my hand at L-Systems, but I'm having some difficulty wrapping my head around it. I watched this video on the subject which is pretty good, but I had a question around the 1:43 ...
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0answers
17 views

does an exponential bound on a Lyapunov candidate implies asymptotic stability?

if I have a Lyapunov candidate $V:[0,\infty)\rightarrow \mathbb{R}$ and I'm able to show that $$ V(t)\le k e^{-\eta t} V(0),\qquad \forall t\in[0,\infty) $$ can I conclude something about ...
2
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0answers
26 views

Stability for a infinite dimensional dynamical system

Suppose I have a infinite dimensional dynamical system as $\dot{x_n}=Ax$ where $A$ is an infinite-dimensional matrix and $\{x_n\}$ is a sequence of states. I was wondering if you can help me find ...
1
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1answer
16 views

Local center manifold theorem.

Local center manifold theorem, under certain assumptions, state that for the \begin{cases} \dot x = Cx+F(x,y) \\ \dot y = Py+G(x,y)\\ \end{cases} there exist a function $h(x)$ such that ...
1
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1answer
33 views

About expansive homeomorphim

We say $(X,f)$ is expansive if there is $c(f)>0$ such that if $d(f^{n}(x), f^{n}(y))< c(f)$ for every $n\in Z$ then $y=x$. Let $(X,f)$ is expansive with constant $c(f)$ and for infinite set ...
4
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3answers
4k views

Linear Algebra: Finding a steady state matrix

Here is the problem: And here is what I tried to do: I tried doing it in my calculator and got that the answer is 40 and 160 as you keep multiplying PX. Can anyone point out what I'm doing wrong ...
2
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0answers
23 views

Non-Conservative System

I'm having a bit of trouble understanding the concept of a conservative system mathematically. A problem in a textbook (Arnold's Mathematical Methods for Classical Mechanics) is asking me to give an ...
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1answer
2k views

find equilibrium points in matlab

I need help with this material: The dynamics growth of two populations is expressed by the system of equations: ($x=$ prey, $y=$predator, $0 \leq t \leq 30$) $$\dot ...
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0answers
32 views

Trajectories of predator prey equation

I am studying the predator prey equation recently, and here is an example: Let $x'=x(1-0.5y)$ and $y'=y(-0.75+0.25x)$. This is a predator prey equations. Then ...
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1answer
26 views

Books on Catastrophe Theory

I'm looking for a technical introduction to catastrophe theory, preferably something short. I have a good background so graduate level texts are welcome. Thanks in advance.
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2answers
85 views

Fundamental Matrix (Floquet theory)

Let $\begin{pmatrix} \dot{x}_1 \\\dot{x}_2\end{pmatrix}=A(t)\begin{pmatrix}x_1\\x_2 \end{pmatrix}$ where $$A(t)=\begin{pmatrix}\alpha(t)+\cos(t)&\sin(t)\\ -\sin(t)& ...
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0answers
8 views

Transform stiff systems to non-stiff

Many dynamic systems tend to be stiff, so an explicit integrator is unstable. The solution is to use an implicit integration scheme. I am curious if there is some way to change the dynamics of the ...
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0answers
22 views

Linear Operator on real (positive definite) symmetric matrix; Generalization of Lyapunov theorem

I am wondering if there is any results on a somewhat "generalization of Lyapunov theorem". By which I mean, as we know from Lyapunov theorem, for a Lyapunov operator on real symmetric matrix, $L_A: ...
4
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0answers
174 views

Has this difference equation :$x_{n+1}=Ax^2_{n}-Bx^2_{n-1}$ been studied before?

I posted this problem before in MO but only I would like to know if this difference equation :$$x_{n+1}=Ax^2_{n}-Bx^2_{n-1}$$ where $A(\theta)= \cos(\theta)$ and $B(\theta) =\sin(\theta)$ are ...
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0answers
14 views

Incidence matrix of invertible substitution

A statement I am reading says: An invertible substitution $\sigma$ over $\{1,2\}$ is non-primitive iff $M_{\sigma}$ (it's incidence matrix) has one of the following forms:$$\left( \begin{array}{cc} 1 ...
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1answer
37 views

Convergence of a particular fixed point iteration scheme

Setup I have the following non-linear system of equations: $$ \mathbf{x} P(\mathbf{x}) = 0 $$ where $\mathbf{x} \in \mathbb{R}_{>0}^n$ is a probability distribution, i.e., $\sum_i x_i = 1$, and ...
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0answers
102 views

Periodic orbits of “even” perturbations of the differential system $x'=-y$, $y'=x$

Fix some even functions $f$ and $g$, differentiable, such that $f(0)=g(0)=0$ and $f'(0)=g'(0)=0$, and consider the associated autonomous differential system $$x'=-y+f(x)\qquad y'=x+g(y)$$ Is every ...
1
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1answer
30 views

If $f(x) \le f(Tx)$ then $f(x)=f(Tx)$ almost everywhere ( $T$ is $\mu$-invariant )

Let $X$ be a probability space with probability $\mu$. Let $T:X\to X$ be a measurable and $\mu$-invariant transformation, i.e $\mu \left(T^{-1}A \right) =\mu A. $ for each measurable subset $A\subset ...
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0answers
13 views

Two vector fields are cojugate but not take orbits

Let X and Y be C1 vector feilds on R^m. Suppose that 0 is an attracting hyperbolic singularity for X and Y. Show that there exists a homemorphism h of a neighborhood of origin which conjugate the ...
1
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1answer
20 views

Show that if f in Diff^r(M), r >=1, is structurally stable then all the fixed points off are hyperbolic.

i think since f is structurally stable so there exists an open nbd u containig of g then f and f are topoligy equivalent.i think since hyperbolic fixed pints dence and open there exists neighberhood v ...
2
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0answers
16 views

Question about periodic points in shift spaces

Let $A$ be a finite set endowed with the discrete topology. Then, the pair $(A^{\mathbb{Z}}, \sigma)$ is said to be the full shift over the alphabet $A$ where $A^{\mathbb{Z}}$ is endowed with the ...
0
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1answer
59 views

Trapping region for $\ddot x + (x^2-2) \dot x + x + \sin(x) =0$

I need to show that the system $$\ddot x + (x^2-2) \dot x + x + \sin(x) =0 $$ Have a periodic orbit. I always use polar coordinates to find a trapping regio, but with the sine term, I am kinda lost. ...
0
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1answer
39 views

From multivariable system transfer function matrix to state space representation

I have the transfer function matrix $H(s) = \begin{bmatrix} {1\over s+1} & {2\over s+2} \\ {-2\over s^2+3s+2} & {2s\over s+1} \\ \end{bmatrix}$ And I want to ...
3
votes
1answer
53 views

How to solve $x^2+x+a=0$ with fixed point iteration?

So when the constant is negative, iteration of $f=\sqrt{-a-x}$ converges quite easily. Also the derivative is less than 1 as long as $-2 \lt a \lt {1 \over 4}$, I don't think that's relevant as the ...
5
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0answers
88 views

Properties of join of open covers

I am studying the basics on Ergodic Theory from Fundamentos da Teoria Ergodica by Oliveira & Viana (you can actually find the pdf on their website), particularly, the properties of topological ...
0
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1answer
40 views

Analyzing the singularity of ODE system

It is asked to analyze the singularities of the system $$\dot{x} = y e^y$$ $$\dot{y} = 1-x^2$$ I've found that the singularities are (1,0) and (-1,0) The linearization of the sysyem give the matrix ...
0
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1answer
368 views

Separatrix curve in dynamical systems

I have this question: What is a separatrix of a equilibrium point of a continuous dynamical system and why it is flow-invariant? Thanks Hello and thanks for the answer. I explain better. I'm ...
8
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0answers
315 views

Overview of nonlinear analysis, differential equations (ODE and PDE), dynamical systems, and mathematical physics, and their relationships [closed]

The fields of (i) nonlinear analysis, (ii) ODE and PDE, (iii) dynamical systems, and (iv) mathematical physics (e.g., electromagnetism, general relativity, gravitation, etc,...) are very huge, ...
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0answers
32 views

Time period of oscillations of a point about the function's minimum value?

How am I to go about the following problem? Please do not explicitly solve it. Let $E_0$ be the value of the potential function at the minimum point $\xi$. Find the time period $T_0=\lim_{E\to ...
5
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1answer
597 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 ...
1
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2answers
25 views

what if an orbit is contained in its $\omega$- limit set?

I guess it should be a periodic orbit, but I'm not sure whether there is an counterexample or not. can you give me a proof or an counterexample?
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0answers
18 views

Dynamics - (stable/unstable) focus - motion direction - CW/CCW?

How to determine the direction a stable focus (source) or unstable focus (sink) is rotating, given the eigenvalues $\lambda=\alpha\pm\beta i$ ? I know that if $\alpha > 0$ then it is source and if ...
0
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2answers
121 views

Stability of fixed points for a differential equation

Consider the differential equation $x'=x^2-9$ a. find the stability type of each fixed point To find the fixed points, I set this equal to $0$, right? Would someone mind explaining why I do ...
3
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0answers
24 views

What can be said about the space of vector fields for which a given, say $C^1$, function is a Lyapunov function?

I am learning Morse homology and I have been thinking about the following observation. One way of doing, say finite-dimensional, Morse theory is by fixing a Morse function $f\in C^{\infty}(M)$, where ...
7
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1answer
228 views

Checking the stability of an equilibrium point

I have the linearization of a non-linear system about an equilibrium point as follows $$ \dot x = (-A+M)x, $$ where $x\in\mathbb{R}^3$, $A$ is a positive definite matrix and $M$ has its eigenvalues ...
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2answers
37 views

Fixed points that are NOT convergent points

Are there any fixed points that are NOT converget (aka attractig fixed points) in the sequence $x_n = 5\ln x_{n-1}$? How do you determine this?
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24 views

writing a recurrence relation into a matrix

I have the following equations which I want to turn into matrices for 'simplicity' $$x_{t+1} = x_t + \beta v_t \exp(-\gamma v_t) \\ v_{t+1} = v_t - \beta v_t \exp(-\gamma v_t)$$. So I thought ...
2
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1answer
22 views

Geodesic Flow is an Anosov Flow

I am trying to understand why geodesic flow on a compact surface of constant negative curvature is an Anosov flow. Klingenberg's book, Riemannian Geometry, says that in this case, the proof is very ...
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0answers
51 views

How can I solve this variable-coefficient ODE system?

I originally have a linear, homogeneous, second-order variable coefficient ODE system of this form: $X''(x) = A(x)X(x)$, where $X(x) = $\begin{bmatrix} f(x) \\ g(x) \\ \end{bmatrix} ...
0
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1answer
31 views

Show that the first quadrant of a dynamical system is invariant.

I have the following dynamical system $${dp \over dt} = p(1-p-q)$$ $${dq \over dt} = q(p-{1 \over 2}-q)$$ and I have to show that the first quadrant ( $p, q \ge 0$ ) is an invariant set. I know what ...
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0answers
25 views

about Poincare map

I saw that the Poincare map is defined by the flow of the periodic system with the least period $T$. that is, $$P(x):=\phi_T(x)$$ is a Poincare map with flow $\phi$ of time $T$. but I think if we ...