Tagged Questions

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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1answer
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questions about transversal surfaces (curves) to a vector field

The following is an excerpt from Dynamical Systems by Shlomo Sternberg: By a transversal, $L$, to the vector field $V$ we mean a surface of codimension one which is nowhere tangent to $V$ . In ...
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23 views

α and ω possible limit sets of points

What are all the $α$ and $ω$ possible limit sets of points for: $$A=\begin{pmatrix}-4&-2\\3&-11 \end{pmatrix}.$$ I am not really sure what to do.. ...
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1answer
29 views

Solutions of nonlinear equations

Show that the differential equation $x^.=x^1/5$ with initial condition $x(0)=0$ has non unique solutions. Why does the theorem on uniqueness of solutions not apply? ...
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18 views

Lubrication Theory: Quick Question!

Basically, I'm modelling the flow of a "coating" process -- a fluid flow between a flat moving plane and a stationary cylinder, 2D, cartesian coordinates. Subscript 0 is the at the minimum height b/w ...
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1answer
21 views

If a linear ODE system has a solution that tends to zero, it also has an unbounded solution

$a:[0,\infty)\to \mathbb{R}$ is a continous and bounded and $$x'(t)\ =\left(\begin{matrix}0&1\\-a(t)&0\end{matrix}\right) \ x(t)$$ has a non-zero solution like $y(t)$ such that $\lim_{t ...
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1answer
39 views

show that all other solutions are bounded

Suppose $G(x)$ is a solution of the differential equation $$x'(t)\ =\left(\begin{matrix}-5&2\\-4&1\end{matrix}\right) \ x(t)+ \ f(t)$$ where $f(t)$ is a continous function and ...
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1answer
7 views

Unique equilibrium that is not an attractor

Fix an ODE system $\dot{x} = f(x)$ where $f : \mathbb{R}^n \to \mathbb{R}^n$ is locally Lipschitz. In the case where $x^\ast$ is a global attractor of $f$, it obviously holds true that $x^\ast$ is the ...
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0answers
14 views

system dynamics: overshoot

My HW problem asks to look at the system dynamics in MATLAB of a feedback system using state-space models in Simulink. The system has no input other than its own output times a feedback, K: $\dot{x} ...
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1answer
24 views

Local asymptotical stability for an ODE

Consider a system: \begin{align*} \frac{d}{dt} \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} = \begin{pmatrix} x_2 \\ a x_1 + b x_1^2 \end{pmatrix} \end{align*} with $a < 0$ and $b\ne0$. My question ...
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25 views

Complex Dynamics of the map

Consider a dynamical systems $$ Z_{n+1}=f(Z_n, Z_{n-1}), $$ where $f$ is a mapping from $\mathbf{C}^2$ to $\mathbf{C}$, defined as $f(z,w)=\dfrac{\alpha}{z}+\dfrac{\beta}{w}$. $\alpha$ and $\beta$ ...
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13 views

Solving system with non-symmetric, indefinite coefficient matrix

So I have a non-linear system that I have linearized using the small angle approximation. When I linearize, I noticed that the coefficient matrix is not symmetric and it has positive real, negative ...
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0answers
18 views

Show that the action variable is $J = \sqrt{A^2 + 2E} - A$

I have the hamiltonian: $$H = \frac{1}{2}p^2 + \frac{1}{2}A^2 \tan^2(q)$$ And I would like to show that the action variable is $J = \sqrt{A^2 + 2E} - A$, where $E$ is the energy. I'm having a ...
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1answer
36 views

Hopf bifurcation and limit cycle

I am studying bifurcation and had a system like this: $$dx/dt=ux-y-x(x^2+y^2),$$ $$dy/dt=x+uy-y(x^2+y^2).$$ I want to determine whether a Hopf bifurcation would occur. I wrote the system into polar ...
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1answer
37 views

Nonlinear Dynamics and Chemical Reactions (Ivanova Reaction System)

I have a homework problem in which I'm given an Ivanova reaction system $X+Y \longrightarrow 2Y$, $Y+Z \longrightarrow 2Z$, $Z+X \longrightarrow 2X$, and I'm asked to write the mass-action ODEs ...
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1answer
23 views

Showing that the linear twist map is sensitive dependent

Choose $\Delta=\frac{1}{2}$ (I believe this value should work). let $\delta > 0$ and let $\textbf{x}_1=(x_1,y_1) \in X$. I assuming that $d$ is the Euclidean distance. Somehow I think we ...
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1answer
19 views

State space system gives different bode plot then transfer function matrix

I have a discrete state space system with matrices $A$,$B$,$C$ and $D$ with sampling period $T_s$. I can either create a state space system, sys1 = ss(A,B,C,D,Ts), ...
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25 views

problem on dynamical system [on hold]

Sir plz tell me what is self organised criticality?in terms of statistical mechanics
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0answers
36 views

Existence of Periodic Solution

I'm working with the system of equations below that represents a Pendulum with constant forcing. \begin{align*} \theta'&=v\\ v'&=-bv-\sin(\theta)+k \end{align*} Where $\theta$ gives the ...
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1answer
57 views

How to derive the hamiltonian from a non-classical lagrangian

For the non-classical lagrangian of a hydrogen atom: $$L = -mc^2 \sqrt{1-\frac{v^2}{c^2}} + \frac{e^2}{4 \pi \epsilon r}$$ We get that two conserved quantities are: $J = \gamma mr^2 \dot{\phi}$ and ...
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1answer
34 views

Reference Request: Discrete Dynamical Systems for Undergraduates

I am looking for a primer text in discrete dynamical systems for an undergraduate level of understanding in mathematics. I have taken introductory courses in numerical analysis and computational math, ...
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0answers
46 views

Asymptotic stability of $\dot{x}=-x^3$

So for an assignment I have to show that $\bar{x}=0$ is an asymptotically stable solution of $\dot{x}=-x^3 (\in\mathbb{R})$, using the definition of asymptotic stability (an equilibrium point $x^*$ is ...
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2answers
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Index Theory: Can a closed curve around a single unstable fixed point have index $0$?

I know that a closed curve containing zero fixed points has index $0$. Is the converse also always true?
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1answer
42 views

Proving that $S=\bigcup_{j=0}^{2^k-1} S_{n-1+k}$ is a spanning set for the $2$-D Baker map

A set $S \subset X$ is a $(n,\epsilon)$-spanning set if $\forall x \in X$, $\exists y \in S $ such that $d_n(x,y)<\epsilon$. This is where we define $d_n(x,y)$ by $d_n(x,y)=\max_{0\leq k < ...
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0answers
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Polar coordinate system of DE's to be written in cartesian form.

Suppose we have a system in polar coordinates: $\dot r = -r$ and $\dot \theta = \frac{1}{\ln{r}}$, we are asked to solve for $r(t)$ and $\theta(t)$ explicitly, so I just integrated both equations so ...
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1answer
40 views

Classify the fixed point at the origin of a dynamical system.

If we have a system $\dot x = -y+ax^3$ and $\dot y = x+ay^3$ I need to classify the fixed point at the origin for all real values of a. So I know we have to make the change of variables $ x = ...
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2answers
51 views

How to find a conserved quantity in this differential equation.

Consider the system: $$\ddot x = x^3 -x$$ What is the method to follow to find a conserved quantity for this system? So far what I have is: $\dot x = y$ and $\dot y = x^3 - x$ and I can find the ...
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2answers
50 views

ergodic system has a.e. dense orbits

one more question: Let X be a metric space with probability measure $\mu$ and T. X $\to$ X ergodic. => f.a.e. x the orbit $O_x=\{T^n(x) : n \in Z\}$ is dense in X so I have to show that the set B ...
2
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0answers
20 views

Fluid Flow: lubrication, integration, ODE

Basically, I'm modelling the flow of a "coating" process -- a fluid flow between a flat moving plane and a stationary cylinder, 2D, cartesian coordinates. Subscript 0 is the at the minimum height b/w ...
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0answers
33 views

recurrence of a dynamical system on a compact space

I have a question to an exercise which was already posted (but I'm not allowed to comment it). ...
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1answer
26 views

Generalize discrete Lyapunov equation for n-th order linear dynamics system

My specific application is analysis of dynamic textures using linear dynamics systems of the form $$ I(t) = Cz(t) + w(t) \\ z(t + 1) = Az(t) + Bv(t), $$ where $I(t)$ is the original signal, $z(t)$ ...
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0answers
35 views

Why this two dynamical Systems are not isomorphic?

Given two dynamical Systems on [0,1) with the Borel $\sigma-Algebra$ and the lebegue measure l. $T_a (x) = x + a$ mod1 $T_2 (x) = 2x$ mod1. Show that this two systems are not isomorphic for any ...
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0answers
20 views

(2.1) 2.Wronskian of a Fundamental Set of Solutions

Consider the system of equations: \begin{align*} \dot x_1&=x_2\\ \dot x_2&=-q(t)x_1-p(t)x_2 \end{align*} (Sorry I don't know how to do subscript notation for the 1's and 2's, an edit would ...
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0answers
26 views

Linear systems, eigenvectors

For each of the following linear systems of differential equations, (i) find the general real solution (ii) show that the solutions are linearly independent (iii) draw the phase portrait a. $$\dot ...
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1answer
19 views

eigenvectors, linear systems

For each of the following linear systems of differential equations, (i) find the general real solution (ii) show that the solutions are linearly independent (iii) draw the phase portrait a. $$\dot ...
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0answers
29 views

Looking for advice on a math project

I had an idea of modeling the evolution of population in the dining hall (just the section where people get their food, not where people eat their food) and discussing the properties of this Dynamical ...
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0answers
25 views

Wronskian of a fundamental set of solutions

Consider the system of equations: $$\dot x_1=x_2$$ $$\dot x_2=-q(t)x_1-p(t)x_2$$ (Sorry I don't know how to do subscript notation for the 1's and 2's, an edit would be appreciated. Also the $x_1$ ...
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0answers
25 views

Wronskian, Linear Independence

Show that the following functions are linearly independent: $$e^t\begin{bmatrix}1\\1\\0\end{bmatrix}$$ $$e^{2t}\begin{bmatrix}0\\2\\1\end{bmatrix}$$ $$e^{-t}\begin{bmatrix}1\\0\\1\end{bmatrix}$$ ...
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0answers
32 views

Quantitative almost sure recurrence

I'm struggling to prove the following result, which is a special case of a quantitative recurrence result which is due to Michael Boshernitzan: Let $(X,d)$ be a compact metric space with finite upper ...
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1answer
41 views

Formal solution needed to question that looks too easy to be true about the Gauss map

Using the itineraries of the Gauss map write the continued fraction expansion of the number $0 \leqslant \alpha \leqslant 1$ such that $$\displaystyle \alpha = \dfrac{1}{4+\dfrac{1}{3+\alpha}}$$ I ...
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1answer
20 views

Drawing toral automorphisms

How do we set about drawing a toral automorphism as in figure 5.1 in the picture above. How do we know where the points highlighted in yellow are? What happens if the eigenvalue (Im guessing some ...
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1answer
23 views

What does “tracktable trajectories” mean in this context?

In a dynamical system what does the following point to? "What range of trajectories seem to be trackable by XYZ method?"
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0answers
21 views

Is the conjugacy map between two distinct circle homeomorphisms unique?

Suppose $f,g,h_1,h_2$ are circle homeomorphisms with $f≠g$ and $fh_i = h_ig$ for $i=1,2$. Does it follow that $h_1 = h_2$? I restrict $f≠ g$ because I noticed that if $$f(x) := g(x) := R_\alpha(x) ...
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0answers
34 views

Convergence to a fixed point [closed]

When the following system is given: $x(k+1)=r-rx(k)$ where $r>=0 $ is a parameter Can someone explain why the fixed points are given by: $x(k+1)=x(k)=x^*$, so $x^*=\frac{r}{1+r}$? and how to ...
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0answers
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How I can explain this contradiction regarding the position of initial condition $x₀$? [closed]

The motivation to this question can be found in http://mathworld.wolfram.com/LogisticMapR=4.html. My question: In formula (4) the initial condition $x₀$ must be in the interval $(0,1)$ while in ...
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2answers
36 views

$\lim\limits_{t\to\infty}t-x(t)=0\ ?$

Let $\displaystyle\cases{ x'=\frac{t-x}{1+t^2+x^2} & \cr x(1)=1 }$ be the Initial value problem, prove or disprove $\lim\limits_{t\to\infty}t-x(t)=0$ We've already proved that: for $t>1, ...
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0answers
36 views

How would one justify the claim that this differential cannot be solved analytically?

The Wikipedia article on the subject of free fall claims that: when the air density cannot be assumed to be constant, such as for objects or skydivers falling from high altitude, the equation of ...
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0answers
33 views

Algebraic approach to topological equivalence of dynamical systems

For continuous dynamical systems there is a notion called topological conjugacy or (somewhat weaker) topological equivalence. I gather that equivalence sends fixed points to fixed points and limit ...
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0answers
12 views

Question on stable manifolds

If $x\in M$ is a hyperbolic fixed point of a diffeomorphism $\phi:M\to M$, then the stable manifold $$ W^s=\{y\mid \lim_{n\to\infty}\phi^n(y)=x\} $$ is the image of an injective immersion $$ ...
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1answer
26 views

Repairing solutions in ODE

Recently I encounter something interesting that I hope to hear from your opinions: Suppose we are given a ODE $\frac{dy}{dx}=y$, with no initial condition. Naively, we divide both sides by $y$ and ...
3
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0answers
65 views

How to adapt the discrete-time to continuous, $(A) \Rightarrow (B)$?

in class was proved oseledets theorem for discrete time, following guidelines Ricardo Mañe book. Theorem discrete Oseledets (A) : Let $ M ^ n $ be a Riemannian manifold, $ f: M \rightarrow M $ be ...