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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41 views

How to construct a diffeomorphic function using another function with certain properties

A $C^{\infty}$ function $f(x)$ on the interval $[a, b]$ satisfies the following 3 properties: 1) $f(x) = 1$ for $a \leq x \leq b$ 2) $f(x) = 0$ for $x < \alpha$ and $x > \beta$ where $\alpha ...
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1answer
20 views

How to modify a function to meet certain properties?

I want to modify $$B(x) = \left\{ \begin{array}{lr} e^{-\frac{1}{x^2}} & : x > 0\\ 0 & : x \leq 0 \end{array} \right.$$ so that the new function $$C(x) = \left\{ ...
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1answer
27 views

prove $H(x)=x^\top x$ is constant along solutions of the sytem if $A(x)^\top + A(x)=0$

Could someone please help me to understand the following: Having the differential equation: $\dot{x} = A(x)x$ where $A(x)$ is a real -valued matrix of dimension $n\times n$ How can I prove ...
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1answer
27 views

Phase portrait in 2 dimensions

I am trying to plot the phase portrait for the system: $\dot{x} = 1+y -e^{-x}$ $\dot{y} = x^3-y$ Now I worked out my eigenvalues to be $\lambda_1 = 2, \lambda_2 = -1$ and these correspond to 2 ...
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49 views

How can we construct Markov partitions for Smale Horseshoe and Solenoid?

I was reading the book Ergodic Theory, Hyperbolic Dynamics and Dimension Theory by Luis Barreira. In Chapter 7, it is asked to construct Markov partitions for the Smale Horseshoe and the solenoid. In ...
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14 views

If $f$ is a diffeomorphism is it true that $NW(f|_{NW(f)})=NW(f)$?

If $f$ is a diffeomorphism is it true that $NW(f|_{NW(f)})=NW(f)$ where $NW(f)$ is the nonwandering set of $f$?
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1answer
33 views

looking for a standard theorem for comparison principle for ode

Consider $$ y'_1(t)=f_1(t),\qquad y_1(0)=y_{10}$$ and $$ y'_2(t)=f_2(t),\qquad y_2(0)=y_{20}. $$ If $f_1>f_2,\quad y_{10}>y_{20}$, then $$y_1>y_2.$$ The above is what I was told by my ...
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1answer
41 views

Are fixed points and equilibria the same thing?

Are fixed points and equilibria the same thing, in terms of a logistic map?
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18 views

How can I find the nullclines for this system?

Consider the following system of differential equations: $$ u' = a_1u(1-u) -a_2u(v+w)$$ $$ v' = a_3uv - v - Rvw $$ $$ w' = Rvw - w$$ I am interested in the nullclines projected onto the $v-w$ ...
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2answers
43 views

Tent map invariant density

Is there a formula for the invariant density for the tent map $f_t$ (for $\sqrt{2}\leq t\leq 2$)? $$ f_t = \begin{cases} tx, & 0\leq x<1/2 \\ t-tx, & 1/2 \leq x\leq 1. \end{cases} $$ By ...
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2answers
34 views

Unique Fixed Point

Let $G:\mathbb{R}^n \to\mathbb{R}^n$ be transformation such that $G(x):=Ax+b$ where $A\in\mathcal{M}_{nxn}(\mathbb{R})$ and $b\in\mathbb{R}^n$ such that $det(A-I)\neq0$ . How would you prove G has ...
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1answer
25 views

When does two curves do not intersect in the phase space

When can I say , that two curves in the phase space of the following equation never intersect: $x'=F(x,t)$ Where $x'= \frac{dx}{dt}$ and $F : \mathbb{R}^{3} \to \mathbb{R}^{2}$. The reason I am ...
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0answers
18 views

Periodic Points of $h:=f\times g: [0,1]^2\to[0,1]^2$ for Continuous $f,g$

Question: Let $f,g:[0,1]\to[0,1]$ be continuous, $h:=f\times g:[0,1]^2\to[0,1]^2,$ $(a,b)\mapsto(f(a),g(b))$. Then "Period Three Implies Chaos" applies to $h$, while Sharkovskii's Theorem does not. ...
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0answers
20 views

Denoting bijection (conjugation) in a commutative diagram

I have a simple notation question: I there a standard way how to denote in a commutative diagram that a map is a conjugation? I thought of the following three, but: The left one (simple arrow) ...
2
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1answer
37 views

Periodic Points of a Continuous $f:S^1\to S^1$

Question: Let $f : S^1 \to S^1$ be continuous. Suppose $f$ has a fixed point and a periodic point of prime period $3$. Then does it have to have a periodic point of prime period $2$? Motivation: I am ...
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0answers
23 views

Counterexamples for the Converse of “Topological Conjugacy Implies Equal Topological Entropy”

Question: I would like to find two topological dynamical systems that are not topologically conjugate but nevertheless have the same topological entropy. Two topological dynamical systems $f:X\to ...
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0answers
26 views

Does the iteration $e_i^\top x_{t+1} = \max_j e_i^{\top} (\alpha A^j x_{t} + b^j)$ converge?

Given a constant $0 < \alpha < 1$, a matrix $A \in R^{n \times n}$ and a vector $b \in \mathbb{R}^n$, it is well-known that a sufficient condition for the iteration $x_{t+1} = \alpha A x_t + b$ ...
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2answers
318 views

Do any of these sequences have infinitely-many distinct iterates under run-length substitution?

Let $$S = \{x \in \{1,2\}^\mathbb{N}: \ \text{every run in }x\text{ has finite length}\}$$ and define $$T: S\to \mathbb{N}^\mathbb{N} $$ such that for any $x\in S$, ${T}x$ is the sequence of ...
2
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1answer
37 views

Reference request : Ergodic theory and Number theory

I would like to work on relation between Ergodic theory(Or Dynamical system ) and Number theory but I am looking for a good reference book, Lecture note and Also I like to get familiar with Articles, ...
2
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1answer
31 views

Perron-Frobenius theorem applied to continuous-time dynamical systems

I'm publishing a series of papers in which I make use of a fairly basic result that allows me to apply the Perron-Frobenius theorem in a case where the matrix is not non-negative but has negative ...
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0answers
25 views

What do we know about the orbits of $f^n(x)$ if $f$ is strictly decreasing?

If $f$ is strictly decreasing, then we first have either: 1) $x < f(x)$ 2) $x > f(x)$ 1) If $x < f(x)$, we know $f^2$ is strictly increasing, so $x < f(x) < f^2$. But $f^3$ is ...
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1answer
49 views

Understanding proof that a homeomorphism cannot have eventually periodic points

Prove that if $f: \mathbb{R} \rightarrow \mathbb{R}$ is a homeomorphism, then $f$ cannot have periodic points of primitive period $3$. The proof was given as follows: Suppose that we have a ...
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0answers
46 views

trapezoidal trapping region for the Van der Pol equation?

For demonstration purposes, I have been constructing analogue computers using op-amps. These circuits provide insight into dynamical systems without the use of numerics. One of the analog devices ...
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1answer
32 views

Time derivative of an invariant probability measure

Consider a dynamical system defined through a vector field $F$ in $M \subset \mathbb{R^n}$ that generates a flow $\Phi^t$ of the form $$\bf{\Phi^t X_0 = X} \ , $$ being $X_0 \in \mathbb{R}^n$ the ...
2
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1answer
52 views

Solution of differential lyapunov equation

How would I solve for following, else any implemented algorithms or solvers in matlab (even ways to solve it) for Lyapunov differential equation of form: $$P'(t) + A(t)^TP(t) + P(t)A(t) + Q(t) = 0,$$ ...
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0answers
26 views

Lyapunov exponents of dual / adjoint / transpose random dynamical system (RDS)

Consider the the state of a system at time $n$, $X_n$, as the action of a product of i.i.d. $d\times d$ random matrices acting on a $d$ dimensional vector $X_0$, so we have $$X_n = A_n \cdots ...
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0answers
55 views

Beyond taylor series?

Consider functions $f(z)$ that are analytic for $Re(z) > 0$ and are also analytic for $(Im(z))^2 > 0$. Let $n$ be a nonnegative integer. Now I define some series expansion of "order $n$" , ...
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19 views

Alpha representation of probability distribution

Probability vector is an $n$-dimensional vector $p=(p_1,...,\ p_n)$ that the sum of whose components equals one, i.e. $p_1+...+p_n=1$. If we take the square root of each component of probability, we ...
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95 views

The Curvy Rebound: One of the most interesting (Geometric) Probability problems.

An infinitely small ball is placed at a random point on the red line shown on the right. The line is 10 metres long. The semi-circle that stems from this line is its resulting size too. Also ...
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1answer
57 views

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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0answers
75 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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0answers
21 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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1answer
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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0answers
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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2answers
44 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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0answers
17 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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2answers
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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1answer
24 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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1answer
33 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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1answer
26 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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51 views

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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1answer
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
35 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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1answer
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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1answer
62 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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0answers
105 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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2answers
39 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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4answers
330 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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0answers
55 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 ...