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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212 views

Unfolding a Billiard Trajectory

The following image is from page 1019 of http://www-fourier.ujf-grenoble.fr/~lanneau/references/masur_tabachnikov_chap13.pdf From what I understand about unfolding billiards we are representing the ...
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3answers
110 views

Repeated nested roots

Quite some years ago, I remember being asked the following question: Suppose $\alpha = \sqrt{2+\sqrt{2+\sqrt{2+\ldots}}}$, what is $\alpha$. The solution was given by squaring $\alpha$ and solving ...
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43 views

Are there solutions when the boundary conditions are particle positions at 2 different times instead of positions and speeds at an initial time?

Is it possible to find solutions for a dynamic system when the boundary conditions are particle positions at 2 different times instead of positions and speeds at an initial time? The question is ...
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85 views

The Starry Rebound

An (infinitely small) ball starting out in the middle of a 5 pointed star table (outer 5 points 10m radius, inner 5 points 5m radius) has a starting angle of a random value from 0 to 360 degrees. The ...
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63 views

Results for $y^{\prime\prime}(x) = a(x)y(x)$, where $a(x) > 0$.

I'm looking for references to any known results regarding solutions to the following 2nd order ODE $y''(x)=a(x)y(x)$, where $a(x)>0$ and $x \in \mathbb{R}$. Any help would be appreciated.
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63 views

Doubling Map and Measure

First off! This is a homework question, so I DO NOT want an answer to the question I'm writing, I really just want an explanation of the final bit (which I'll make clear). So if we have $T:[0,1)\to ...
2
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1answer
306 views

How to classify equilibrium points

I have the two differential equations: $$\frac{dN_1}{dt} = N_1(2 - N_1 - 2N_2)$$ $$\frac{dN_2}{dt} = N_2(3 - N_2 - 3N_1).$$ I worked out the equilibrium points to be at $N_1 = 0, \frac{4}{5}$ and ...
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308 views

What is the shape of parabolic critical orbit?

The parabolic critical orbits of discrete dynamical system form n-th arm stars : which shapes are conjugated with "regular" n-th arm stars Here are 2 images of parabolic critical orbits for 2 ...
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144 views

Bernoulli shift on $S^\mathbb{Z}$

Why is the Bernoulli shift ergodic? I know the proof for $S^\mathbb{N}$. The best would be if you could argue from this fact to the integer case, so that way I can see how one usually goes from ...
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399 views

Proof Strategy for a Dynamical System of Points on the Plane

I have a rather simple-looking system which exhibits a particular behaviour in simulation, and I would now like to attempt to prove this formally. The problem is, I don't really know where to start, ...
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68 views

Is $g(z)=\frac{1}{z}+\frac{1}{z^2+1}+\frac{1}{(z^2+1)^2 +1}+…$ analytic for $|z|>2$?

Let $z$ be a complex number. Let |.| denote be the absolute value. Let $n$ be a positive integer. Let $f_1(z)=z^2+1$. Let $f_n(z)=f_1(f_{n-1}(z)).$ Is ...
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216 views

Trapping region of Van der Pol system

Van der Pol's system of differential equations is given by $$ \dot{x}=y \\ \dot{y}=-(\mu+x)y $$ where $\mu\geq0$ is a constant. I want to show that any disk centred at $(0,0)$ is a trapping region, ...
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2answers
378 views

What's the point of a Horseshoe map?

At the moment I'm doing a project about the smale horseshoe map. This is a function which maps a square $D= \{(x,y)\in \mathbb{R}^2: 0\le x\le 1,\text{ } 0\le y \le 1 \} $ to a 'horseshoe'. It ...
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1answer
334 views

Ergodic Theory (Weak Mixing)

If $T$ is weak mixing then we know that $T\times T \times \ldots \times T$ is also weak mixing. Does anyone know if this is true for $T\times T \times \ldots$?
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158 views

How can the proof of a local theorem on a manifold involving a map with a fixed point and a differential be reduced to the case of $\mathbb{R}^n$?

Case in point: the Hartman-Grobman theorem (for maps). In the book "Geometric Theory of Dynamical Systems: An Introduction" by Palis and De Melo, the theorem is stated as follows (on page 60). ...
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161 views

Finding periodic points of every period, Sharkovskii's Theorem

Consider the map: $C_c(x)=c\cos(x)$ (a) Find a value of the parameter c for which this map has prime periodic points of every period, and provide an explanation with graphs supporting your argument. ...
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78 views

Boltzmann machines - motivation for the energy function

I've been studying Boltzmann machines lately and was wondering if anyone could give me a "high-level" explanation or motivation for the energy function used: $$E = -\sum_{i<j} w_{ij} \, s_i \, s_j ...
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176 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 $\phi^m$) ...
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63 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 ...
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56 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 ...
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48 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 ...
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2answers
116 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 ...
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264 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 ...
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1k 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 ...
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199 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 ...
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1answer
442 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 ...
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3answers
159 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 ...
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101 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 $d(f(y_k),y_{k+1}) < \delta,$ ...
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61 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 ...
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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 ...
2
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1answer
314 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 ...
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275 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 ...
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1answer
60 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 ...
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1answer
664 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)$ ...
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255 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 ...
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1answer
517 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. ...
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1answer
2k 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 ...
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123 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 ...
2
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1answer
596 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 ...
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1answer
188 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 ...
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135 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!
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54 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 ...
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1answer
160 views

Half-stable vs saddle node

What is the difference between a half-stable and a saddle node in two and three dimensions?
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1k 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 $x_1' = x_2' = 0$). ...
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1answer
88 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 ...
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2answers
410 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 ...
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1answer
92 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 ...
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1answer
53 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 ...
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5answers
320 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 ...
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234 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 ...