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 ...
4
votes
2answers
63 views
Numerical search for periodic orbits
Suppose that $\phi$ is a map defined in all of $R^n$, with image also in $R^n$. Is there a known good algorithm to find the periodic points of $\phi$ of order $m$, (or fixed points of $\phi^m$) ...
2
votes
1answer
36 views
Why are log-probabilities of global states of Boltzmann machines linear in their energies?
While going through the Wikipedia article about Boltzmann machines, I read the following on the probabilities of global states of Boltzmann machines in the "Equilibrium state" section ...
1
vote
0answers
31 views
Monotonicity of Poincaré's Map
thanks for reading.
Consider a one-dimensional dynamical system $\dot{x} = f(t,x)$. Let's call $\phi(t,t_0,x_0)$ the solution passing through $x_0$ at time $t_0$ (where $t$ is the time argument of ...
1
vote
0answers
22 views
Lyapunov exponents for PDEs
How can one define and calculate (analytically or numerically) Lyapunov exponents for partial differential equations? Do there exist examples of nonlinear PDEs for which Lyapunov exponents can be ...
0
votes
2answers
53 views
why is it not possible to have a period-2 orbit(or greater) for a 2D autonomous system?
Why is it not possible to have a period-2 orbit(or greater) for a 2D autonomous system whereas the same is possible for a non-autonomous system?
I can explain the existence of period-2 orbits in ...
2
votes
0answers
69 views
Floquet's Theory, Hills Equation
Let us examine Hill's equation $\ddot x+Q(t)x=0$, where $Q$ is piecewise continuous and with a period $T$. Let $\mu_{1,2}$ be the multiplicators. Let $\lambda$ be the characteristic exponent. How can ...
0
votes
0answers
33 views
References that discuss systems of ODEs on the non-negative orthant of $\mathbb{R}^n$?
Does anyone know of any references discussing initial value problems on the non-negative orthant? More specifically, consider the initial value problem
$\frac{dx}{dt}=f(x),\quad\quad ...
0
votes
1answer
85 views
Continuous-time versus discrete-time stochastic models
When modeling a dynamic phenomenon, (from a general point of view) people can use two type of models: (1) continuous-time models, (2) discrete-time models.
To be more precise, assume that we try to ...
4
votes
0answers
111 views
Rewriting the advection-diffusion equation
This is mostly a reference request question, although I certainly appreciate any insights and/or comments.
Let us assume $p:R^n×(0,∞)\to \mathbb R$ is a scalar concentration, $u\in R^n$ is the ...
1
vote
0answers
192 views
Question about $f_n=f_{n-1}+\ln f_{n-1}$ with $f_0=2$ [closed]
Let $n,m$ be strictly positive integers.
Let $f_0 = 2$. Let $f_n=f_{n-1}+\ln f_{n-1}$.
Let $h_{n,1}=\sinh^{-1}\left(\dfrac{n}{2}\right)$ and $h_{n,m}=\sinh^{-1}\left(\dfrac{h_{n,m-1}}{2}\right)$.
...
1
vote
0answers
75 views
Dynamical system and Riemann Hypothesis [closed]
In this article
http://jp4.journaldephysique.org/index.php?option=com_article&access=standard&Itemid=129&url=/articles/jp4/abs/1998/06/jp4199808PR625/jp4199808PR625.html
they claim ...
2
votes
1answer
54 views
Tent map: show that $x$ is a periodic point IFF it is a rational number of the form $\frac {m}{p}$ where $m$ is even and $p$ is odd
Let us consider the tent map: $f [0,1] \rightarrow [0,1]$ where
$f(x) = 2x$ if $0\leq x \leq \frac{1}{2}$ and $f(x) = 2(1-x)$ if $\frac{1}{2}\leq x \leq 1$.
Show that $x$ is a periodic point IFF it ...
0
votes
0answers
56 views
Show that the set of all periodic points of the tent map are dense in $[0,1]$ [duplicate]
Let us consider the tent map:
$f: [0,1] \rightarrow [0,1]$ where
$f(x) = 2x$ if $0\leq x \leq \frac{1}{2}$ and $f(x) = 2(1-x)$ if $\frac{1}{2}\leq x \leq 1$.
I am facing an issue with this problem:
...
2
votes
3answers
86 views
Dynamical Systems problem — a function of ONLY period 3
I am trying to construct an example of a function $f: [0,1] \rightarrow [0,1]$ such that it has a periodic point of period 3 and NO other periodic points.
Any ideas? how can I even start envisioning ...
0
votes
1answer
64 views
An isometry does not have the pseudo-orbit tracing property.
Let $X$ be a metric space with metric $d$ and let $f:X\rightarrow X$ be dynamical system.
We say a sequence $\{y_i\} \subset X$ is a $\delta$-pseudo-orbit of $f$ if $d(f(y_k),y_{k+1}) < \delta,$ ...
0
votes
1answer
25 views
Can there be a backward orbit for a Discrete Dynamical System?
All I have studied is forward orbits. So I was wondering whether there can be backward orbits. If that were to be the case, then shouldn't the transformation, T be invertible? This is not guaranteed ...
-1
votes
2answers
26 views
Working out the interval in which the algae becomes extinct (how to get the interval)
I have a birth rate that is
$$b(p) = \frac{p^2}{p^2 + 3}$$
and a death rate that is
$$d(p) = \frac{p}{4}.$$
I therefore have a reproduction rate as $r = b - d$. In order for my algae to become ...
1
vote
1answer
79 views
Why can't gradient systems have closed orbits?
I've seen the proof provided in Strogatz where he compares the change in the gradient function $V$ after one period $T$; on one hand, $\Delta V=0$ because $V(x(T)=V(x(0))$, while on the other, $\int ...
0
votes
1answer
58 views
why are fixed points of period-1 orbits also fixed points of period-2 orbits?(in the logistic equation)
$$x_{n+1}=x_{n}\mu \left( 1-x_{n}\right)$$
this has a 'STABLE' period 1 orbit with 2 fixed points upto a certain parameter value,its a fact.
$$x_{n+2}=x_{n+1}\mu \left( 1-x_{n+1}\right)$$
For the ...
1
vote
1answer
44 views
odd non linear system (simple)
Consider the system
$x^{'} = x(x^{2} + y^{2})$, $y^{'} =y(x^{2} + y^{2})$
Questio part 1) find the equilibrium of the point system and describe its behavior.
when i do this i just get the zero matrix ...
1
vote
1answer
82 views
Nonlinear phase portrait and linearization
Consider the nonlinear system
$x^{'} = y$, $y^{'}= -8 \sin x - 2y$
where $-2\pi$ < or = x < or = $2\pi$
Find the equilibrium points of the system.
$(-2\pi,0)$$(-\pi,0)$$(0,0)$ ...
2
votes
1answer
86 views
Picard iteration (general)
This a general question about Picard iterations and is as follows.
Let A be a $n\times n$ matrix. show that the Picard method for solving $X^{'}=AX$, $X(0)=X_{0}$
gives the solution $e^{tA}X_{0}$
I ...
3
votes
1answer
42 views
Poicare-bendixon show periodic solutions.
Show that the system $x^{'}=x-y-x^{3}$,$y^{'}=x+y-y^{3}$ has a periodic solution.
went to polar.
r$r^{'}=x^{'}x+y^{'}y$
thus
r$r^{'}=x^{2}-x^{4}+y^{2}-y^{4}$
collecting plus squares.
...
3
votes
0answers
102 views
Root Locus Diagrams - “Breakaway Point”
Say that we have a root locus diagram with n poles and m zeroes. And we determine that the root locus on the real axis lies between two of these poles and breaks away from the real axis and tends to ...
2
votes
0answers
50 views
Bifurcation in 3 dimensions (simple)
I am Doing a project i have a toy system that describes a bifurcation in 3 dimensions i am posting this in part because i can no longer understand what i have written down ( its been awhile) i have ...
1
vote
1answer
71 views
Lorenz equations and find a minimal trapping region.
Consider Lorenz's equations
$x^{'}= \sigma (y-x)$
$y^{'}= (rx-y-xz)$
$z^{'}= (xy-bz)$
$\sigma, r, b>0$ are parameters of the system.
The question is as follows
Show that there is a certain ...
0
votes
1answer
63 views
What's the impulse response of this first order system?
I think my textbook may be wrong, I really can't get this. I'll ask to my teacher, but in the meanwhile I'd like to discuss this with you.
A first order system:
$$
y(k+1) = a(k)y(k) + g(k)
$$
Given ...
1
vote
3answers
76 views
conditions under which real-matrix exponential are equivalent
Consider $M_{1}$, $M_{2}\in\mathbb{R}^{2\times2}$, $k\in\mathbb{R}$, $M_{1}\neq M_{2}$. Under what conditions is $e^{M_{1}}=e^{kM_{2}}$?
Thanks!
2
votes
0answers
49 views
Linearization of a dynamic system, $\small 10 \frac{d^2 y}{dt^2} + \frac{d y}{dt} = u^2(t) $
I have a dynamic system, with in signal u and out signal y. The system is described with the following differential equation
$$
10 \frac{d^2 y}{dt^2} + \frac{d y}{dt} = u^2(t)
$$
The task is to ...
1
vote
1answer
45 views
Half-stable vs saddle node
What is the difference between a half-stable and a saddle node in two and three dimensions?
0
votes
1answer
73 views
Isolated Versus Non-Isolated Fixed Point, 2D Dynamics
I am trying to understand the classification of fixed points in a dynamical systems context (fixed points of a system of two linear differential equations are places where both $x_1' = x_2' = 0$).
...
0
votes
1answer
59 views
Solution to system of difference equations with repeated unit roots
Can anyone provide the forms of the solutions for the homogeneous part and particular solutions for a non-homogeneous system of two linear autonomous difference equations ...
0
votes
1answer
77 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 ...
2
votes
1answer
49 views
Surfaces without conjugate points
I'm trying to understand some aspects of the geodesics of surfaces without conjugate points and the following question came out: consider the hyperbolic space and two geodesics wich are asymptotic to ...
1
vote
1answer
28 views
set of immersions on a manifold is an open set in the set of all mappings to $\mathbb{R}^{2m+1}$
I'm studying a proof of Takens' Delay Embedding Theorem. A key fact used is that the set of immersions is open in the set of all mappings (mappings from an $m$-dimensional manifold M to ...
7
votes
5answers
139 views
High-School Level Introduction to Dynamical Systems
In one month I'll be giving a talk to motivated high schools students on a topic of my choice from dynamical systems and/or ergodic theory.
I'm having trouble coming up with a topic compelling enough ...
5
votes
3answers
124 views
Mathematical Limitations of Computer Experiments
One problem that has always bothered me is the limitations of computers in studying math. With a chaotic dynamical system, for example, we know mathematically that they possess trajectories that never ...
3
votes
1answer
42 views
What is the relation between mixing (measure theory) and a map being topological mixing?
A map is said to be topogical mixing if given two sets $A$ and $B$ then there exists $N$ such that for all $n>N$
$f^n(A) \cap B$ is not empty
On the other hand, a measure \mu is said to be ...
2
votes
2answers
42 views
convergence of vectors
Let $x_i, y_i \in \mathbb{R}^n$, and $F,G: \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}^n$.
Consider sequences of vectors
$$x_{i+1} = F(x_i, y_i)$$
$$y_{i+1} = G(y_i, x_i)$$
I have two related ...
1
vote
2answers
81 views
Cat map like maps without period
Is there any area-preserving chaotic map other than Arnold cat map which can be applied on a rectangle as well as being reversible but not periodic?
0
votes
0answers
15 views
Upperbound of $y(t)$ in an application of the gradient dynamics with a time-varying and zero-crossing coefficient
Consider this system:
$$
\begin{align}
y &= ax+1,\\
\dot{x} &=-\gamma a y,\quad \gamma>0
\end{align}
$$
where $\dot{x}$ is the derivative of $x$ with respect to time $t$, and a is also a ...
1
vote
1answer
96 views
Deducing asymptotically stability if $|f(x)| <|x|$
let f be a continuous map such that $f(0)=0$ and $|f(x)|<|x|$ whenever $0<|x|<\epsilon$ for some $\epsilon$>0
a) show that the sequence $f^{n}(x)$ has a limit point $x_{*}$
b) show that ...
1
vote
0answers
43 views
Calculation Lyapunov exponents for infinite systems of differential equations
Can you give an example of a function $\varphi$ and sequences $\{b_{i}\}$ and $\{a_{ij}\}$
for which one can calculate Lyapunov exponents of such the infinite system of differential equations
and ...
3
votes
0answers
60 views
Equivalence of two definitions of Lyapunov exponents
Yesterday I asked a question on eigenvalues of $A^{T}A$.
Eigenvalues of $A^{T}A$
The reason why I asked this question was that I'd seen in articles two different definitions for Lyapunov exponents ...
1
vote
1answer
51 views
If one periodic orbit is attracting then $T$ is not topologically transitive
Any ideas, hints on the following would be great.
Suppose that $T\colon X \to X$ is continuous, and there exist at least two distinct periodic orbits.
Show that if one of the periodic orbits is ...
3
votes
2answers
68 views
Liapunov Stable question
Show that the system $x^{'}=y-x^{3}$ and $y^{'}=-x-y^{3}$ has no closed orbits by constructing a Liapunov Function $V= ax^{2}+by^{2}$ with suitable a and b.
all i really know is that this is an ...
2
votes
1answer
39 views
Poincare-Bendixson in a $2\times 2$ system
Show that the system
$x^{'} = x-y-x^{3}$ and $y^{'}=x+y-y^{3}$
Has a periodic solution using Poincare-Bendixson
I really have no idea how to apply this theorem i know that i need find an orbit and ...
0
votes
1answer
44 views
ODE with global inequality constraint on derivative
The problem is solving the ODE $f''(x)+a_1f'(x)+a_2f(x)=g(x)$ with boundary conditions $f(c)=h(c)$, $f(d)=h(d)$, where $c,d$ must be such that $f'(x)\geq b\;\forall x\in[c,d]\subseteq\mathbb{R}$. The ...
1
vote
1answer
54 views
Simplify the nonlinear system of dynamic equations
I am working with a set of nonlinear dynamic equations that Mathematica has problems with solving. It is of the form
$$
f_1(x_{t+1},y_{t+1},x_{t},y_t) = g_1(x_{t+1},y_{t+1},x_{t},y_t),\\
...
3
votes
0answers
54 views
Nice corollaries to Poincaré-Bendixson theorem
I am interested in the applications of Poincaré-Bendixson theorem not (explicitely) related to differential equations. For convenience, recall the theorem:
Let $X : \mathbb{R}^2 \to \mathbb{R}^2$ be ...
