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 ...

learn more… | top users | synonyms (1)

0
votes
1answer
46 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
36 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 $v-w$ ...
2
votes
2answers
63 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
2answers
35 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
1answer
30 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 ...
1
vote
0answers
19 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
22 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
votes
1answer
39 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
26 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 ...
0
votes
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 $x_{t+1} = \alpha A x_t + b$ ...
15
votes
2answers
319 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
votes
1answer
46 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
votes
1answer
40 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 ...
1
vote
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 ...
0
votes
1answer
58 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
61 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 ...
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 ...
2
votes
2answers
75 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
27 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 ...
0
votes
0answers
67 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 $n$" , ...
1
vote
0answers
22 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 ...
0
votes
0answers
108 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 ...
2
votes
1answer
70 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 ...
2
votes
0answers
87 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 ...
1
vote
0answers
23 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,
1
vote
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 ...
0
votes
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 ...
0
votes
2answers
52 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
0answers
22 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 ...
1
vote
2answers
39 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 ...
1
vote
1answer
27 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 ...
0
votes
1answer
41 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 ...
1
vote
1answer
30 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 ...
5
votes
0answers
58 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 ...
2
votes
1answer
35 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 ...
1
vote
1answer
41 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 ...
2
votes
1answer
30 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 ...
0
votes
1answer
66 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}^{+} ...
1
vote
0answers
107 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 ...
3
votes
2answers
49 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 ...
8
votes
4answers
349 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 ...
0
votes
0answers
59 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 $x^*=0$?
0
votes
0answers
23 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 ...
1
vote
1answer
35 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 ...
0
votes
1answer
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 ...
2
votes
1answer
41 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 $f^2(x)=x$.
0
votes
1answer
38 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 ...
0
votes
0answers
19 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. ...
1
vote
1answer
29 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. ...
0
votes
1answer
21 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 ...