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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36 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: ...
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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, ...
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1answer
15 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 ...
1
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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?
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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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0answers
30 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 ...
2
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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
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 ...
1
vote
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 ...
2
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0answers
27 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
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 ...
2
votes
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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0answers
33 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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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: ...
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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 ...
1
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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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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
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 ...
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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)& ...
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 ...
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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 ...
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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 ...
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. ...
3
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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 ...
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 ...
5
votes
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
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 ...
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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 ...
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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 ...
4
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0answers
175 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 ...
3
votes
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 ...
0
votes
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 ...
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?
0
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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 ...
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1answer
29 views

Geodesic flow on a compact manifold is defined for all time

How can I prove that on a compact manifold, the geodesic flow is defined for all time? Is this as simple as citing the Hopf-Rinow theorem?
2
votes
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 ...
0
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1answer
25 views

Can a countably infinite compact topological space have isolated point? Can it admit a minimal subsystem?

Examples I could think of are all sequences with their limit. But is every countably infinite compact space admit atleast one isolated point?
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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} ...
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0answers
29 views

Effect of weight perturbations on eigen values

I have a dynamical system of coupled differential equations: $\dot{X}$ = -X + S( AX + BI) where X = 7x1 vector, A = 7x7 and B = 7x1 vector. S(x) is a nonlinear sigmoid function. I want to find the ...
2
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0answers
36 views

Finding conditions of non existence of Periodic orbit

$$ x'=y \mbox{ and } y'=ax-by-x^2y-x^3 $$ I need non-existence of periodic orbits. Which conditions $a$ and $b$ in $\mathbb{R}$ must satisfy? First, one can see that if $a\leq 0$, then the system has ...
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0answers
17 views

“Damping factor” for a set of non-linear ODEs

I am not sure if this question is on-topic here I have a set of four non-linear ODEs representing a negative feedback. I have done parameter variation by random sampling to study the sensitivity of ...
3
votes
2answers
44 views

Meaning of the expression “orientation preserving” homeomorphism

The only time that I've heard the term "orientation-preserving map" was in Linear Algebra, but today I read the term orientation-preserving homeomorphism of the circle in the following context: If a ...
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0answers
27 views

Are these definition equivalent about a periodic point?

Given a dynamical system $(X,G)$, def1. A point $x\in X$ is called periodic, if there exist a syndetic set $S\subseteq G$, such that $Sx=\{x\}$. def2. A point $x\in X$ is called periodic, if $Gx$ is ...
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0answers
17 views

A question on recurrent points of group acitons

Given a dynamical system $(X,G)$, A point $x\in X$ is called recurrent, if for any neibourhood $U$ of $x$, there exist a $g\in G$, $g\neq e$ such that $gx\in U$. If $G$ is a topological group and ...
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0answers
17 views

Linear affine random dynamical systems - positive Lyapunov index proof check?

Consider the affine random dynamical system $$ X_n = \mathbf A_n X_{n-1} + R_n, $$ starting from an initial non-zero position $X_0$, where $\mathbf A_n\in\mathbb C^{d\times d}$ and $R_n\in\mathbb ...
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
16 views

Trajectories in orthogonal systems

Please forgive any awkward phrasing or misuse of terminology. My education isn't entirely formal. Question Am I right in guessing these orbits trace lissajous-ish figures on hyperspheres? ...