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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Prove that if $f$ and $g$ are topologically conjugate functions and $f$ has chaos, then $g$ has chaos.

Question: Prove that if $f$ and $g$ are topologically conjugate functions and $f$ has chaos, then $g$ has chaos. I can see a bit of the reason behind of the claim but I can't prove it. To prove ...
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78 views

Periodic Solution of Damped Pendulum with Constant Torque

I have a system of ordinary differential equations $ \theta' = v$ $ v' = -bv - \sin \theta + k$ These are the equations for a pendulum with $\theta$ being angular position, and $v$ being angular ...
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22 views

Lyapunov function or functional

I'm wondering when we call it Lyapunov function, and when Lyapunov functional? Does it differ from whether the system is a finite or infinite dimensional one? Thanks. Best,
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17 views

Does uniform hyperbolicity requires both or any of the stable and unstable spaces?

Consider the bernoulli shift map, From the definition in this article in scholarpedia, We say that f is uniformly hyperbolic or an Anosov diffeomorphism if for every x∈M there is a splitting of ...
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20 views

How to formalize the subjective perception of perspective?

Suppose you're moving in a car or a train in the direction of the $x$ axis with some velocity with respect to the ground. If you look through the window (that is, at the $y$ or $z$ axis), what you ...
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45 views

dynamical systems applied to economics

I'm ending my undergraduate economics course and I'd like to extend my MA research program to dynamical economic systems. Knowing that my mathematical basis is calculus of 1 and 2 variables, linear ...
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19 views

What does omega limit sets have with invariant sets?

What does omega limit sets have with invariant sets? I was thinking of omega limit set as the limit of a sequence inside the invariant set. But... if I look at the definition of Invariant set, it's ...
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37 views

Geometry / dynamics analogues

In 3-space geometry we have curvatures when a point is proceeding along a curved arc. Similarly when particle motion occurs with respect to time we have accelerations. Is there a one to one ...
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25 views

For a general nonlinear ODE, does continuous dependence on a parameter imply continuous dependence on initial conditions

If the solution of the differential equation $$\dfrac{dy}{dx}=f(x,y,\lambda)$$ under initial conditions $x_{0},y_{0}$, is continuously dependent on the parameter $\lambda$, does it imply that it will ...
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34 views

Determine which sets are local attractors and determine global attractors

Consider the discrete-time dynamical system on $X=\mathbb R_0^+$ given by iteration of the map $f(x)=x^{1/2}$ I want to determine which of the sets $I_1=\{ 0 \}$, $I_2=\{ 1 \}$, $I_3=[0,1]$ are local ...
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28 views

Can an unstable limit cycle be contained directly within a stable one?

Can both the alpha and omega point sets of a trajectory be part of two different limit cycles? I.e. can trajectories being 'repelled' from one limit cycle be pulled into an 'attracting' (stable) limit ...
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Elliptic and Fredholm partial differential operators

As I learn from the comments to this question a non elliptic operator on a compact manifold can not be a fredholm operator. On the other hand it seems that the following operator is fredholm(At ...
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34 views

Detection of Cycles without a Center in an ODE

In my classes in dynamical systems theory, we were taught how to detect cycles or cyclic behavior in an ODE (be it dampened, sustained or growing) around a fixed point by looking at the eigenvalues ...
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1answer
38 views

Discrete Dynamical Systems & Credit Card Debt: How to solve for payment

I have the following problem, taken out of Giordano, Fox, and Horton's A First Course in Mathematical Modeling: Your current credit card balance is $\$12,000$ with a current rate of $19.9\%$ per ...
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27 views

Logistic and Quadratic map

I am trying to understand the relation between a logistic map and a quadratic map. For example, how can you modify a logistic map for the quadratic map, i.e., modifying the logistic map ...
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63 views

Are those Locally Lipschitz definitions equivalent?

Let $f:\mathbb{R}^{+}\times \mathbb{R}^{n}\rightarrow \mathbb{R}^{n}$ be locally Lipschitz in the sence that there exists a positive $C^{0}$ function $\ell :\mathbb{R}^{+}\times \mathbb{R}^{+} ...
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106 views

(Still open) Convergence of a certain matrix product representing a class of piecewise linear dynamical systems.

Consider a discrete-time dynamical system of the following kind. Assume $x(t) \in \mathbb{R}^n$. $x(t+1) = A_{\Omega(x(t))} x(t)\quad$ where $\quad A_i = \left [ \begin{array}{c|c} 1 & 0\\ \hline ...
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41 views

How to find a superstable period-$2$ orbit of the logistic map.

Suppose that $G\colon R \to R$ such that $G(x)=rx(1-x)$. I need to find the value for $r$ at which the super stable period-$1$ and period-$2$ points exist. I think I know how to get the super stable ...
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332 views

Concentric Equilateral Triangles

I'm currently researching a particular dynamical system that is very geometric in nature. As part of this, I need to prove the following results (the second obviously implies the first). They are ...
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56 views

Determine the stability of a fixed point

Consider $x'=f(x)$, where $f(0)=0$ and $f(x)=-x^3\sin\left(\frac{1}{x}\right)$ for every $x\neq 0$. How to determine the stability of the fixed point $x^*=0$?
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21 views

Proving that the function $\rho$ which sends a lifting of a circle map to its rotation number is continuous.

Let $\mathcal{L}$ denote all circle maps of degree one with nondecreasing liftings (a map $f \in \mathcal{L}$ is of degree one if its lifting $F$ satisfies $F(x+1)=F(x)+1$) . I need to prove that if ...
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33 views

Differential equations - bounded dynamical system

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be $C^1$-function and let $I_{x_0}=(a,b)$. Assume that there exist $M>0$ such that $|\varphi(\cdot,x_0)|_{[0,b)}|\le M$, where $\varphi(t,x)$ is dynamical ...
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31 views

Suggestion or help to choose a book for theory of ODE

I have recently have a course in theory of ordinary differential equations , I have learned main ideas and now I want to study theory of ode from a book for filling gaps. I have choosed three book byt ...
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33 views

If $f$ is a homeomorphism then any periodic point have period less or equal 2

How can one prove the followiong statment? Let $f:[0,1]\to [0,1]$ be a homeomorphism. If $x\in\operatorname{Per}(f)$ then the period of $x$ can't be greater than $2$, i.e, $f(x)=x$ or $f^2(x)=x$.
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36 views

Construct the equation of a linear vector field if the phase portrait is given

I already know that since that $0$ is a repulsor singular point over $x$-axis and $-x$ also works as a repulsor point in opposite direction. Can anyone help me out by finding the equation of this ...
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16 views

Dynamics Central Force spiral

A particle P of mass m moves in a central force field of magnitude $mkr^-3$. Initially r=a and the particle has velocity V perpendicular to OP where $V^2<k/a^2$. Prove that P spirals in towards O. ...
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26 views

Endomorphisms of the Torus

Question: Show that tfae: The endomorphism of the torus $T_A:\mathbb{T}^n\to\mathbb{T}^n, [x]\mapsto[Ax]$, where $[x]:=\{x+y:y\in\mathbb{Z}^n\}$ and $A\in M_n(\mathbb{Z})$, is invertible. ...
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20 views

Number of Periodic Points of the Expanding Map $E_m:S^1\to S^1$

Question: Let $\forall m: E_m:S^1\to S^1$, $x\mapsto mx (\mod{1})$ be the expanding map of the circle. What is the number of periodic points of $E_m$ of (minimal) period $n$? Motivation: In Barreira ...
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41 views

Unique Ergodicity

Show that unique ergodicity is a topological invariant. Is arguing as follows an overkill (hopefully if the logic is correct --- I have a feeling that there has to be a way a T-invariant measure has ...
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37 views

Does every matrix have periodic orbits of even length?

Let $A$ be an $n \times n$ matrix (let's say hyperbolic, but this might be irrelevant). Consider the action of $A$ on $(\mathbb{R} \backslash \mathbb{Z})^n$. Does this action always have periodic ...
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23 views

Some websites for numerical values of orbit of $x$ under $f$ in a dynamical system

I don't know anything about programming. Is there some website to calculate the orbit of $x$ under $f$ in dynamical system? For example, if $x_{n+1}=2x_n(1-x_n)$, how to a have a list of first many ...
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19 views

Determine whether a set is Invariant, Positively invariant or negatively invariant

I have just started a dynamical systems course and I am a bit confused as to how to determine if something is positively or negatively invariant, or just invariant. I know the defintions for ...
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22 views

Is there a fiber bundle approach to nonlinear oscillations?

I've recently been learning about nonlinear oscillations, and I noticed a seemingly strong connection between how the equations of motion are solved/approximated, and fiber bundles (or vector bundles ...
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23 views

How to determine if these r.v are independently Gaussian?

I have a time series vector ${(X_i)}_{i=1}^n$ which is the output of a non-linear dynamical system in $R^d, d=1$ of unknown distribution. Using Takens delay emebedding theorum, if I embed the 1 ...
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28 views

Lotka-Volterra coordinates transformation

I would like to ask the following: Given a Lotka-Volterra predator-prey system, \begin{align} & \frac{dx}{dt}={\alpha}x-{\beta}xy \\ & \frac{dy}{dt}=-{\gamma}y+{\delta}xy \end{align} , with ...
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65 views

Lotka-Volterra First Integral and Fixed Point

I have the following problem that I am dealing with, quite a long time, I must say. Let us assume that we have a predator-prey, Lotka-Volterra system given to us by: \begin{align} & ...
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68 views

Lyapunov exponents of a linear upper block triangular system

I seem to be stuck at formally showing something that intuitively seems to be true. I have a linear non-autonomous system of the form $$ \dot{x} = A(t)x $$ where $A(t):\mathbb{R}\to ...
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49 views

Types of solutions of ODEs and periodic orbits.

I am currently studying specific types of solutions of ordinary differential equations. If a vector field can be autonomous or non-autonomous, in which of these cases are there periodic orbits if ...
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32 views

Homoclinic orbits

Consider an autonomous vector field on the plane having a hyperbolic fixed point with a homoclinic orbit connecting the hyperbolic fixed point. Can a trajectory starting at any point on the ...
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1answer
39 views

Row Reducing with Imaginary Numbers

So I have the system $$ X' =\begin{pmatrix} 0 & 1 \\ -k & -b \end{pmatrix} X $$ where I assume $$ 0 \le b < 2 \sqrt{k} $$ which results in one complex ...
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80 views

Logistic family and chaos

It is a well known fact that the map $f(x)=4x(1-x)$ is chaotic on $[0,1]$. By chaotic I mean the usual definition, i.e.: a) the periodic points of $f$ are dense in $[0,1]$, b) $f$ is topologically ...
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57 views

Writing ODE system with a complex variable

I'm looking at the system of ODEs: $$\begin{cases}\dot{x} = -y + kx + xy^2\\ \dot{y} = x + ky - x^2\end{cases}$$ I'm trying to introduce a complex variable $z = x+iy$ to write this as a single first ...
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100 views

Measure of the set of periodic points of a measure preserving map

Given a continuous (Lebesgue) measure preserving map $T$ from a compact convex region to itself that has an aperiodic point (i.e. a point $p$ such that $p \ne T^n(p)$ for any $n$), does the set of ...
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123 views

Bump Functions in Dynamical Systems.

Define $B(x) = e^{-1/x^2}$ for $x > 0, B(x) = 0 $ otherwise. sketch the graph of B(x); prove that B'(0) = 0. When $x = 0$, $B(x) = 0$. It follows that the rate of change of a constant ...
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23 views

Uniformly convergent iterates for a function analytic on the complex unit disk

I have asked this question to several leading mathematicians in dynamical systems, and they all told me to ask someone else, until I was directed back to asking my advisor, with whom I first posed the ...
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1answer
36 views

Help figuring out output signal of LTI system.

Would greatly appreciate any help in figuring out the output signal of my discrete time LTI system. My input signal is cos(ωn) and my frequency response is H(e^jω)=(1+e^−jω)/2.
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57 views

Strongly mixing uniquely ergodic dynamical system

I'm looking for an example of a dynamical system which is both (measure-theoretically) strongly mixing and uniquely ergodic. I've looked around and found lots of discussion of systems which are ...
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54 views

Stability of dynamical system described in polar coordinates

Near a fixed point, a dynamical system $\dot{\bf{x}}=\bf{F}(\bf{x})$ can be approximated by $\dot{\bf{x}}=A\bf{x}$, where $A$ is the Jacobian matrix. From the trace and determinant of the Jacobian ...
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130 views

Limit points of the differential system $\dot {x}=y-x+x^3$, $\dot{y}=-x$

Consider the following system of differential equations: $$\dot {x}=y-x+x^3,\qquad \dot{y}=-x.$$ By linearization, it's easy to see that $(0,0)$ is a (nonlinear) sink. Show that there exists an ...
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26 views

Substitution in a system of ordinary differential equations when terms of the same order derivative for different variables occur in the same equation

Let's say I have a differential equation such as: y'' - 2ty' + y = 0, y(0) = 2.1, y'(0) = 1.0 I can solve this (among other ways) by substitution and conversion ...