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 ...
5
votes
3answers
268 views
KAM theory and the Ergodic hypothesis
I have seen several authors mentioning that KAM theory contradicts the Ergodic hypothesis. Unfortunately, the authors do not elaborate on this. I have some background in KAM theory but very little in ...
1
vote
0answers
26 views
Normalize to scale
I have an 2-D array of data (C), where C(:,1) has values corresponding to C(:,2). C(:,2) varies from 0.0001:0.0001:1, i.e. 10,000 values. I need to calculate the d(log(C(i,1))) / d(log(C(i,2))), which ...
1
vote
2answers
199 views
Why do we want to know the poles and zeros of a linear system?
I know that I already asked this kind of question on the website, but meanwhile I have much more knowledge about the subject and ready to describe my real problem with enough background information I ...
2
votes
1answer
85 views
how to solve this nonlinear equation system by changing this system to linear system
consider following nonlinear equation system how solve it?
$$x'=|y|$$
$$y'=x$$
and whats the matrix that associated to this system
1
vote
0answers
35 views
How to model a system with multiple probability distributions, each for a part of the system?
I need to build a complex probability model to describe some "real world" scenarios. The system consists of several types of objects, and the contraints upon these objects and their interactions are ...
2
votes
3answers
135 views
How to solve simple systems of differential equations
Say we are given a system of differential equations
$$
\left[ \begin{array}{c} x' \\ y' \end{array} \right] = A\begin{bmatrix} x \\ y \end{bmatrix}
$$
Where $A$ is a $2\times 2$ matrix.
...
2
votes
0answers
46 views
1
vote
1answer
78 views
Kalman filter and data extrapolation
Context of the situation:
I have a system set up that can give me the position of a person in a room. I also have a light that shines on this position. However, the light are lagging behind by 0.300 ...
2
votes
3answers
110 views
Quadratic equation with matricial coefficients
If I have a equation in the form
$${\lambda ^2}{I_N} + \lambda {M_1} + {M_2} = {0_N}$$
where ${I_N}$ is the identity matrix of order $N$, $M_1$ and $M_2$ are matrices of ($N\times N$) order and ...
4
votes
2answers
378 views
How to go about studying chaos theory/dynamical systems/fluid dynamics in grad school with a physics background?
I'm turning to you as I find myself in need of help as to how to best go about studying something related to chaos theory/dynamical systems/fluid dynamics in postgraduate school.
I'm currently in my ...
4
votes
1answer
178 views
Method of isoclines
I have this exercise and I do not know how to solve it.
By using the method of isoclines represent the integrals of equation corbes nonautonomous $x'=x^2-t$.
There are some indications: Let $P = ...
1
vote
1answer
61 views
Lyapunov Exponent
Suppose $(X,A,\mu)$ a probability space, where $X$ is a compact Riemann manifold, $T:X\to X$ a diffeomorphism and $T$ is a measure-preserving transformation( over the borel $\sigma$ algebra).
Prove ...
0
votes
1answer
62 views
How to model a system for tracking a person using kalman filter?
I need to model a system for human motion. The following link shows for to build a system for a plane.
I am currently reading the documentation for a kalman filter library ...
1
vote
1answer
109 views
Explanation for a Z-transform solution of a difference equation
I am studying Feedback Control of Computing Systems. (specifically using Hellerstein's book, section 3.1.4, page 76)
To solve difference equation Z-Tranform can be applied. In the book there is an ...
1
vote
1answer
165 views
How to calculate Inverse Z-Transform by long division
I am studying Feedback Control of Computing Systems. (specifically using Hellerstein's book, section 3.1.4, page 74)
An inverse Z-Tranform also can be obtained by a long division. In the book there ...
0
votes
0answers
54 views
Studying a nonlinear system under constraints in linear fashion
Suppose that $a = aBc$ where $a$ and $c$ are vectors and $B$ is some matrix that changes as time "continuously" goes on - making this system dynamical system. But suppose that at any time, if $B$ is ...
4
votes
2answers
84 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 ...
2
votes
3answers
361 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
54 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
88 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 ...
0
votes
0answers
62 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
131 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
36 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
264 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 ...
14
votes
1answer
321 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
65 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
140 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 ...
2
votes
1answer
69 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
45 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
71 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
1answer
66 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
81 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
2answers
89 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
40 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?
1
vote
2answers
66 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
83 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 ...
0
votes
1answer
56 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), ...
1
vote
1answer
133 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
49 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
73 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
47 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$.
...
6
votes
2answers
135 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
114 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 ...
1
vote
1answer
87 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 ...
3
votes
3answers
260 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
41 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
54 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
42 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
110 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, ...
1
vote
0answers
33 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 ...
