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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5
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1answer
151 views

two fixed points, same fractional iteration

Suppose a function f(z) has two fixed points, one repelling, and the other attracting. Call the repelling fixed point f(-1)=-1, and the attracting fixed point f(+1)=+1. I'm interested in functions ...
0
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2answers
77 views
+50

approximate vanishing in Pontryagin dual

Let $\{n_k\}\subseteq \mathbb{Z}$ to be any given sequence of integers, and suppose it satisfies the following property: (*) For any $\lambda\in A\subseteq \mathbb{T}$(the unit circle), ...
0
votes
0answers
9 views

Generalized Rational Difference Equation

The difference equation is $$y_{n+1}=\frac{\beta y_{n-l}+\gamma y_{n-k}}{A+y_{n-k}}$$ for $n=0,1,2,\dots$ where $l$, $k$ are two positive integers such that $l\neq k$. The parameters are positive ...
1
vote
1answer
16 views

Closed Form Solutions To Simple Iterated Polynomial Building Blocks

I've been doing some work on fractals and simple iterated polynomials lately. I admit, I've only taken classes up through Calc 2, although I've done a decent bit of reading on many topics over the ...
15
votes
2answers
310 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 ...
0
votes
0answers
24 views

Normal form calculation

I am working on a problem involves 4 dimensional dynamical system. Is there any ready package (for maple ,matlab...) which calculate the normal form of nonlinear continuous dynamical systems? The ...
0
votes
0answers
43 views

Points where limit cycle intersects set: $S=\big\{(x_1,x_2)|x_2=0\big\}$

I really need some help figuring this out! The system $\dot{x}=f(x)$ with $x(t) \in\mathbb{R^2}$ has a limit cycle that passes trough the set $S=\big\{(x_1,x_2)|x_2=0\big\}$. The ...
0
votes
0answers
12 views

how reducing the number of parameters work? can anyone help me, please.

The number of parameters of differential equations in the first image had been reduced and the solutions are as in the second image. Can anyone explain to me how does the "reducing number of ...
4
votes
1answer
56 views

Dynamical system $x_{n+1} = \frac{1}{2}(x_n - \frac{1}{x_n}) \ \ , \ \ n = 0, 1 , 2,…$

Consider the dynamical system $$ x_{n+1} = \frac{1}{2}(x_n - \frac{1}{x_n}) \ \ , \ \ n = 0, 1 , 2,... $$ So by using the substitution $x_n = \cot(y_n)$, I have found: $$ x_n = \cot(\cot^{-1} ...
0
votes
2answers
37 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 ...
0
votes
1answer
22 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 ...
0
votes
0answers
32 views

For which $r$ is this system dissipative $\ddot{q}+rq^2\dot{q}+\sin(q)\cos(q)=u$

Could someone please help me to understand the following: Having the differential equation: $\ddot{q}+rq^2\dot{q}+\sin(q)\cos(q)=u$ that models the electrical charge $q$ of a particle in an ...
0
votes
2answers
32 views

How to modify this bump function so that the “bump” is at $y=1$?

$$f(x) = \begin{cases} e^{-1/(1 - x^2)} & -1 < x < 1\\ 0 & \text{otherwise} \end{cases} $$ I noticed that when I multiply the denominator of the fractional part of this ...
3
votes
2answers
31 views

Lotka-Volterra model with two predators

In this, Lotka-Volterra model, we have two predators: $$\frac{dp}{dt} = ap\left(1-\frac{p}{K}\right) - (b_1q_1+b_2q_2)p$$ $$\frac{dq_1}{dt}=e_1b_1pq_1-m_1q_1$$ ...
0
votes
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\{ ...
1
vote
1answer
25 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 ...
0
votes
0answers
16 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 ...
-6
votes
0answers
46 views

Determine Stability of nonlinear ODE in Biological Mathematics

Please help me find the stability of the critical points given in problem 3 below. I was able to determine using matlab, but he is telling me to use partial derivatives to do this. Do I need to make a ...
0
votes
2answers
400 views

Closed form solution of this second order linear difference equation?

$$ y(k + 2) - 3y(k + 1) +2y(k) = 2^k + k $$ Transform into a system of $n$ first order equations (Step 1) $$\begin{align} x_1(k) &= y(k)\\ x_2(k) &= y(k + 1) \end{align}$$ It follows that ...
9
votes
1answer
548 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 ...
0
votes
0answers
19 views

Calculating Average Velocity of Population

I'm trying to find a formula to calculate the average velocity given the following information: growth rate, shrinkage rate, transition from growing to shrinking rate and transition from shrinking to ...
2
votes
2answers
31 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 ...
0
votes
0answers
39 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 ...
0
votes
0answers
11 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 $f$?
0
votes
1answer
30 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 ...
0
votes
1answer
39 views

Are fixed points and equilibria the same thing?

Are fixed points and equilibria the same thing, in terms of a logistic map?
0
votes
0answers
14 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 $v-w$ ...
1
vote
0answers
463 views

How to numerically find Floquet multipliers (e.g., characteristic multipliers or Lyapunov exponents for periodic orbits from chaotic systems)?

Anyone have any suggestions for the following situation/question? (help wanted, please!) I understand the theory (c.f., Perko or Nayfeh and Balachandran, Ch.3), but I do not understand how this is ...
2
votes
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 ...
0
votes
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 ...
0
votes
0answers
49 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 $n$" , ...
2
votes
1answer
36 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 ...
1
vote
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. ...
1
vote
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) ...
1
vote
0answers
20 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 ...
-2
votes
0answers
79 views

Write a function such that its global minimum is the closest local minimum of another function

Suppose I have a function $f: \mathbb{R}^n \to \mathbb{R}$ that is continuous and differentiable and bounded below with many local minima. I want to write a function $g(x, y)$ such that the global ...
0
votes
0answers
25 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 $x_{t+1} = \alpha A x_t + b$ ...
2
votes
1answer
34 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, ...
1
vote
0answers
24 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 ...
1
vote
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 ...
0
votes
2answers
39 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 ...
0
votes
1answer
48 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 ...
0
votes
0answers
34 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 ...
0
votes
0answers
34 views

Dynamical Systems and translated Dynamical Systems

Consider a function $f:\mathbb{R}^ + \times \mathbb{R}^+ \to \mathbb{R}^+ $. Define a dynamical system, $x_{n+1}=f(x_n, x_{n-1})$. Suppose the dynamics (# of periodic points, fixed points, chaotic ...
4
votes
1answer
537 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
vote
1answer
46 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: $$P'(t) + A(t)^TP(t) + P(t)A(t) + Q(t) = 0,$$ ...
1
vote
0answers
24 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 ...
2
votes
1answer
54 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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votes
0answers
74 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 ...
1
vote
0answers
17 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 ...