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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102 views

Equivalence of $x,y\in G$ given that $xzy^{-1}z^{-1}$ is a commutator for some $z$

Let $G = \langle a,b,c\:|\: a^2, b^2, c^2\rangle$. Let $\tilde{}$ by the equivalence relation on $G$ generated by conjugation and inversion (i.e., $x\tilde{} y$ if there is a finite sequence of ...
3
votes
3answers
1k views

Poles and Zeros of Linear Systems

This period I follow a course in System and Control Theory. This is all about linear systems $$\frac{dx}{dt}= Ax + Bu $$ $$y = Cx + Du $$ where A,B,C,D are matrices, and x, u and y are vectors. ...
2
votes
0answers
70 views

Pencil of conics and periodic orbits

Let $\dot{x}=P(x,y)$ and $\dot{y}=Q(x,y)$ be a quadratic polynomial differential equation. Prove that if the pencil of conics $P+\lambda Q$ contains an imaginary conic, a real conic reduced to a ...
1
vote
2answers
96 views

Question about dynamical behavior near point

Let $x' = f(x)$ be autonomous first–order equation differential with an equiliburiium point $x_0$. Suppose $f'(x_0) = 0$ what can I say about the behavior of soluton near $x_0$? If $f'(x_0) ≠ 0$ and ...
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0answers
142 views

Prove the following saddle-node bifurcation theorem

Prove the following saddle-node bifurcation theorem: Suppose that $f_λ$ depends smoothly on the parameter $λ$ and satisfies: (a) $f_{λ_0}(x_0)=x_0$ (b) $f_{λ_0}'(x_0)=1$ (c) $f_{λ_0}''(x_0) \ne 0$ ...
7
votes
1answer
180 views

Vectors fields structurally stable

(This was a question on my doctoral qualifying exam.) Let be $X$ a vector field defined in $\mathbb{R}^2$ such that $X$ is structurally stable in every compact set of $\mathbb{R}^2$. Is $X$ ...
2
votes
1answer
54 views

Calculating the equilibrium of a time series system

Hi there math experts. I would like to calculate the equilibrium of two linear equations. However, they're part of a time series, where $a_{-1}$ defines the lagged value of $a$. I don't know how to ...
4
votes
1answer
625 views

Why are hyperbolic toral automorphisms (e.g. Arnold's cat map) ergodic?

Let $\varphi : \mathbb{T}^2 \to \mathbb{T}^2$ be a hyperbolic automorphism of the torus, induced by a linear map $A : \mathbb{R}^2 \to \mathbb{R}^2$ of determinant $\pm 1$ with no eigenvalues of ...
15
votes
1answer
499 views

Kakutani skyscraper is infinite

Karl E. Petersen's book "Ergodic Theory", chapter 2, exercise 9, on page 56 Prove that for any ergodic measure preserving transformation $T:X\rightarrow X$ on a non-atomic probability space $(X, ...
0
votes
1answer
104 views

Meaning of index in matrices

Question is, what does "index" mean? For systems of order greater than the number of characteristic roots of $C$ of index one Also, can anyone explain why is $u_1 + u_2 + n -1 =0$ and what ...
3
votes
1answer
352 views

Convergence of fixed-point iteration for convex function

Let $f:[0,1]\to[0,1]$ be a smooth, convex (downward) function satisfying $$ f(0)=f(1)=1,\quad \lim_{x\to 0}f'(x)=-\infty,\quad \lim_{x\to 1}f'(x)=+\infty. $$ I am confident to be able to argue that ...
4
votes
1answer
208 views

Solution of non linear ODE system is always positive if its initial valus is positive

Given a system of nonlinear differential equation \begin{eqnarray}\frac{dx}{dt}=2x(3-y) \\ \frac{dy}{dt}=3y(4-x)\end{eqnarray} If $r(t)=$($x(t)$,$y(t)$) is a solution of the system with initial value ...
2
votes
0answers
74 views

Approximation by a Morse-Smale diffeomorphism on the cicle

Let $f$ be a diffeomorphism on the circle $\mathbb{T}$, where $f$ is in the $C^1$ topology and has a fixed point. It is asked to prove that $f$ can be approximated by a Morse-Smale diffeomorphism by ...
0
votes
0answers
137 views

Poincaré Maps of a planar system

Consider a planar system \begin{equation} \begin{array}{c} \dot{x}=x-y-x\left(x^2+y^2\right) \\ \dot{y}=x+y-y\left(x^2+y^2\right) \end{array} \end{equation} This is the cross section: ...
0
votes
2answers
121 views

Existence of invariant set in dynamical system generated by ODE

Is there any nonempty, compact and invariant set in dynamical system generated by this system of equations? $x'=x+\sin{(xy+2)}-7$ $y'=-y+\arctan{(x^2+y^3-6)}$ My idea is to use this fact: Not ...
2
votes
1answer
114 views

Simplifying a polynomial by a nice recursive formula

Let define a function $g(x)= (1+x^2 )/2 $ and then define again $G_i$ where $ G_1(x) = g(x) $ and $G_{n+1}(x) = g(G_{n}(x))$ . How can we approximate $G_{2n} $ and $G_{3n} $ with respect to $G_n$ ...
2
votes
1answer
210 views

Not empty omega limit set

Dynamical system is generated by: $x'=-x+f(x,y)$ $y'=-y+g(x,y)$ $f,g \in C^1$ and $f,g$ are bounded. Prove that the omega limit set of p: $\omega(p) \neq \emptyset$ for all $p \in \mathbb{R}^2$. ...
2
votes
0answers
93 views

Decreasing function with a fixed point and 2 cycle?

Can you give me an example of a decreasing function with a fixed point and 2-cycle?
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2answers
128 views

The system of $x(t+1) = Ax$ growing and retaining stability possible?

This is about general equilibrium: Suppose that $x(t)$ represents outputs of all sectors and parts of the whole economy - represented as matrix. How outputs evolve to $x(t+1)$ ...
2
votes
1answer
155 views

Stability of a system that has (Jacobian-like) matrix with eigenvalue of less than 1 that has $x$ as non-eigenvector

This is about general equilibrium: Suppose that $x(t)$ represents outputs of all sectors and parts of the whole economy - represented as matrix. How outputs evolve to $x(t+1)$ is ...
1
vote
1answer
71 views

Orbit of $\frac{1}{2}$ of the dynamical system defined by $f:[0,1] \to [0,\frac{3}{4}]$ where $f(x)=3x(1-x)$ converges.

I view the orbit of $\frac{1}{2}$ of the dynamical system defined by $f:[0,1] \to [0,\frac{3}{4}]$ where $f(x)=3x(1-x)$ as the sequence $(x_{n})$. So $$x_n = \frac{1}{2}, f\bigg(\frac{1}{2}\bigg), ...
2
votes
1answer
259 views

Periodic solution of nonlinear differential equation

Let $C$ be a positive constant. Consider the following system of differential equation with inial value \begin{eqnarray} z(t)+\frac{\sqrt{2}}{2} u'(t)-1=0 \\ 2u''(t)+C \sin(2u(t))=0 \end{eqnarray} ...
1
vote
1answer
90 views

(system of) nonlinear equations and instability

I heard that a system of nonlinear equations is unstable. I am curious of how "instability" is defined, and why do nonlinear equations show instability? Edit: OK, so what about contexts in matrices ...
1
vote
1answer
187 views

Maximal Positive Invariant Set — Some fine print

I would like to share something I noticed on the definition of Maximal Positively Invariant Sets. Definition 1. For a discrete-time system of the form $x_{k+1}=f(x_{k})$ (and $x_{k}\in ...
0
votes
1answer
60 views

Differential Equation and Stability

I have an equation: $V_{t+1}=V_t+r(S(V_t))$. r is a constant when$(r=?)$ is $V$ asymptotically stable and when otherwise? What I tried is, finding equilibrium points, I got: $S(V_t)=0 $ and $r=0$. ...
7
votes
2answers
225 views

Prove that Anosov Automorphisms are chaotic

Let me add some detail first. An Anosov automorphism on $R^2$ is a mapping from the unit square $S$ onto $S$ of the form $\begin{bmatrix}x \\y\end{bmatrix}\rightarrow\begin{bmatrix}a && b \\c ...
2
votes
1answer
220 views

Resources for learning non-linear Math

I have a basic math background (ODE's, calculus), thermodynamics and mechanics. I have looked through books by Ilya Prigogine and most of the math went way past me. What books or other resources ...
3
votes
2answers
378 views

Classification of points in the Mandelbrot set

I am trying to understand the classification of points in the Mandelbrot set. There are an infinite number of baby Mandelbrots, each associated with a defined set of landing rays. There are the pre ...
4
votes
3answers
566 views

Differential equation, Stability , Lyapunov function

Given a system of differential equations \begin{eqnarray} x'&=&2y(z-1)\\ y'&=&-x(z-1) \quad (1)\\ z'&=&xy \end{eqnarray} Note that $u_0$=(0,0,0) is an equilibrium point of the ...
2
votes
0answers
48 views

Question regarding continuous time systems

If $S_u$ is the set of all solutions to the continuous time problem x(dot) = Ax + Bu and $\phi_h :(R^n)^R to (R^n)^N$ be an operator which maps every state x into the sequence (x(0), ...
0
votes
1answer
62 views

$Re[f(x + n i)] = 0$ and $f(z)$ is not periodic

Let $z$ be a complex number and $f(z)$ an entire function such that For $x$ real and $n$ any integer. $Re[f(x + n i)] = 0$ and $f(z)$ is not periodic. What are typical examples of such $f(z)$ ? Is ...
1
vote
1answer
85 views

Derivation of a formula for discrete time case

Given is the following discrete system $$\begin{align*} &x(k + 1) = Ax(k) + Bu(k)\\ &x(0) = x_0\;. \end{align*}$$ How do we prove that the explicit solution formula for $x(k)$ (analogously ...
1
vote
1answer
201 views

Is the initial value problem of an ODE considered as a dynamic system?

Is the initial value problem of an ODE considered as a dynamic system? A dynamic system is defined as In the most general sense, a dynamical system is a tuple (T, M, Φ) where T is a monoid, ...
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0answers
70 views

Is there a “natural” embedding of the Henon map into a continuous flow?

I was wondering about this. Now, I'm a big fan of "continuous iteration", and I was curious about this problem. What I was wondering was whether or not there exists a continuous dynamical system which ...
0
votes
1answer
68 views

A Jacobian involving probabilities

I am working with the following system of OD equations: $$\frac{dE(I_1)}{dt}=-\mu E(I_1)+\lambda E(I_2)-\lambda E(I_1I_2)=f_1(E(I_1),E(I_2))$$ $$\frac{dE(I_2)}{dt}=-\lambda E(I_1I_2)+\lambda ...
1
vote
3answers
76 views

Dynamical equations involving step functions

This is not a specific problem, but a question about the application of a theory. We have these equations of the form $dx/dt=f(x_t)$ where a given point $x$ moves along some path over time. I am ...
10
votes
2answers
315 views

An equivalent condition for strong-mixing

For a measure-preserving (finite) system $(X,\mathcal{B},\mu,T)$, is it correct that the following are equivalent? For every $A,B\in\mathcal{B}$ , $\displaystyle\lim_{n\rightarrow\infty}\mu(A\cap ...
0
votes
1answer
124 views

Question about iterations : $g(c,z) = a$ always has a solution where $z$ is strictly real?

Let $c$ be any positive real number. Let $z$ be a complex number. Let $g(c,z)$ be some locally analytic function that is the $c$ th iteration of the entire function $f(z)$ with $g(0,z)=f(z)$. Let ...
2
votes
2answers
196 views

Periodic orbits

Let $x\in\mathbb R$ be a periodic point with lenght 2 of the recursion $x_{n+1}=f(x_n)$ My book about Dynamic systems says that this recursion has a fixed point now, because a periodic point with ...
0
votes
1answer
92 views

Logistic function

Let $$f=\lambda x(1-x)$$ be a logistic map with $\lambda>4$. How to show that set of all periodic points of $f$ in $\Lambda$ is countable and the set of point in $\Lambda$ with dense orbits is ...
3
votes
1answer
68 views

Inverse of itinerary function

Let $S^{-1}:\Sigma \to \Lambda$ be inverse of itinerary function. I showed that $S$ is continuous and bijective. How to show that $S^{-1}$ is continuous?
2
votes
1answer
127 views

Dynamical changing of an eigenvector

Consider a matrix $A\in\mathbb{R}^{n\times n}$. One of the eigenvalues of $A$ is zero and all the others are positive. Suppose $w\in\mathbb{R}^n$ is an eigenvector with the zero eigenvalue, i.e, ...
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votes
2answers
174 views

What kinds of maths to learn for understanding dynamical systems in cognitive science?

A current trend in cognitive science is to view the mind as a dynamical system (e.g., Continuity of Mind by Spivey, in which cognition is understood as a "continuous and often recurrent trajectory ...
3
votes
2answers
71 views

Defining an f-invariant measure

Suppose I have a compact oriented manifold $M$ with an orientation preserving self-diffeomorphism $f$. I wish to define a volume form on $M$ which is invariant under $f$. Certainly, it is necessary ...
0
votes
1answer
106 views

find the critical value of r.

For the following equations sketch the bifurcation diagram, determine type of bifurcation, and find the critical value of $r$. $$\dot{x} = rx + \cosh x$$ I seem to understand how to do the first ...
1
vote
1answer
142 views

Use Euler's method with step size 10^-n to estimate x(1), where f(x) is the solution of the initial-value problem below. f(x)=-x x(0)=1

Use Euler's method with step size $10^{-n}$ for $n=1,2,3,4.$ to estimate $x(1)$, where $f(x)$ is the solution of the initial-value problem below. $x'=f(x)=-x$ $x(0)=1$ EDIT / UPDATE: x_n+1=x_n + ...
1
vote
1answer
535 views

Find the values of r at which bifurcation occus and classify those as saddle node, transcritical, or pitchfork bifurcation

$$f(x)= rx-\frac{x}{1+x}$$ Find the values of $r$ at which bifurcation occus and classify those as saddle node, transcritical, or pitchfork bifurcation. I found the fixed points as ...
1
vote
2answers
577 views

Use linear stability analysis to classify the fixed points of the following system.$ f(x)=ax-x^3$ for $a>0$, $a=0$ and $a<0$.

Use linear stability analysis to classify the fixed points of the following system. $f(x)=ax-x^3 $ where a can be positive, zero or negative. I have found that for $a>0$ we have $2$ fixed points ...
0
votes
1answer
83 views

Prove that $f(x)=x(1-x)$ on $I$ is conjugate to $g(x)=x^2-\frac{3}{4}$ on a certain interval in $\mathbb{R}$. Determine this interval.

Prove that $f(x)=x(1-x)$ on $I$ is conjugate to $g(x)=x^2-\frac{3}{4}$ on a certain interval in $\mathbb{R}$. Determine this interval. Suppose $I$ and $J$ are intervals and function $f$ from $I$ ...
0
votes
1answer
99 views

Related questions on the Hausdorff dimension and local dimension of a Cantor set

Suppose $I$ is an interval on a Cantor tree with $m$ children $I'$, each of length $\varepsilon$. I have that $\sum_{\hat I\text{ is a child of }I}|\hat I |^s=|I|^s \Rightarrow ...