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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24 views

Introducing noise and time lag between two coupled Rössler systems

I have two Rössler systems mutually coupled by the second component. I want to introduce some small noise and a slight time lag of the coupling between the systems. I'm not sure 1. what the best ...
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26 views

Qualitative properties of eigenvalues that can be inferred from matrix structure?

I am doing a linear stability analysis of a 6-dimensional system, what I want to know is if the system is stable at numerically solved steady states by looking at the eigenvalues of the jacobian ...
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158 views

System of first order ODEs with coherent sinusoidal time varying coefficient

I have encountered equations of the form $$\frac{{d{\bf{y}}(t)}}{{dt}} = \left( {{A_0} + {A_1}\cos (\omega t)} \right){\bf{y}}(t)$$where ${\bf{y}}$ is a vector and ${{A_0}}$ and ${{A_1}}$ are square ...
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33 views

Topology equivalence in dynamical system

my name is Eric. I've got trouble when proofing that system $\dot{x}=\alpha+x^2+O(x^3)$ is topological equivalence with system $\dot{x}=\alpha+x^2$. I don't understand how to build the homeomorphism ...
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17 views

How to change the variables so $x_{n+3}=f(x_{n},x_{n+1},x_{n+2})$ becomes of the form $g(y_n)=y_{n+1}$?

How to change the variables so $x_{n+3}=f(x_{n},x_{n+1},x_{n+2})$ becomes of the form $g(\underline{y}_n)=\underline{y}_{n+1}$
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16 views

Flow Notation over Interval

Given a section of the flow $\Phi^t(x_0)$ (for finite $t$), I'd like to denote a subsection of this flow from times $\tau^{i-1}$ to time $\tau^{i}$ using similar notation. I was considering using ...
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45 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 ...
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1answer
36 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 ...
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69 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} ...
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36 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 ...
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2answers
135 views

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), ...
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2answers
36 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
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2answers
47 views

Lotka-Volterra model with two predators

In this, Lotka-Volterra model, we have two predators: $$\frac{dp}{dt} = ap\left(1-\frac{p}{K}\right) - (b_1q_1+b_2q_2)p$$ $$\frac{dq_1}{dt}=e_1b_1pq_1-m_1q_1$$ ...
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2answers
57 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
21 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
30 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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61 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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16 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
49 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
48 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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38 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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71 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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36 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
33 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
21 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
24 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) ...
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1answer
40 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
32 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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28 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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323 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 ...
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1answer
55 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, ...
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1answer
46 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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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
72 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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71 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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2answers
98 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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30 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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71 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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23 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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120 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
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1answer
81 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
117 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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24 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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2answers
64 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
25 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
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
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1answer
29 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 ...