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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Undamped Forces

I want to make sure I am doing this problem correctly, especially when it comes to drawing the potential function V(x). Consider the system of differential equations: $$\dot {x}=y$$ $$\dot {y}=-x+x^...
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2answers
153 views

Closed orbits of vector fields under perturbation

Consider a vector field $V$ on an annulus $U$, say. Also, assume that the vector field $V$ has a closed orbit. I am looking for a reference that gives stability results of the following type: If the ...
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103 views

Stable and unstable manifolds of fixed points

I want to make sure I understand the definition of these terms. If someone could correct me or let me know if I am right I would appreciate it. The stable manifold of a fixed point is the set of ...
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168 views

The “muscle” behind the fact that ergodic measures are mutually singular

This is really motivated by the soft question at the end, but let me begin with something more circumscribed: Let $(X,\mathcal{B})$ be a measurable space and let $T:X\circlearrowleft$ be a self-map ...
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77 views

Hausdorff dimension of a ball

Let $\{f_1,\dots,f_m\}$ be an IFs and $E_n$ be the associated self similar set. It's known that $E_n$ is a union of disjoint balls $B(x_i,R\cdot r^n)$ (balls with same radius but not the same center). ...
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77 views

Question about having periodic solution.

Assume $a>0$ and $b>0$ and $g(x)=0$ when $|x|>1$ , $g(x)=k$ when $|x|\le1$ . Now show that in system of differential equation $$x'=y $$ $$y'=-[2b-g(x)]ay-ay^2$$ if $k<2b$ ...
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51 views

Rational dynamical system with nonnegative paramaters and with nonnegative initial conditions

let $A$ be a rational system of the form :\begin{cases} x_{n+1}=\frac{\alpha_{1}}{y_{n}} \\ y_{n+1}=\frac{\alpha_{2}}{z_{n}} \\ z_{n+1}=\frac{\alpha_{3+}+\beta_{3}x_{n}+\sigma_{3}y_{n}+\lambda_{3}z_{...
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1answer
44 views

Proving that the Bernoulli self similar measure is doubling

Let $\mu_p$ a measure which is the push forward of the bernouli product measure $(p,1-p)^\mathbb N$. Let S=$\{f_1,\dots f_m\}$ an IFS, a system of functions with attractor $K$, means $$K=\bigcup_{i=1}^...
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1answer
45 views

Omega limit set of omega limit set $\omega(\omega(a))$

Consider a dynamical system with a flow $\phi(t;a)$, and let $A\subset \mathbb{R}^n$. The omega limit set of $A$ is defined as the union of all $\omega(a)$ over all $a\in A$. Since for a given $a$, $\...
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1answer
97 views

Closed orbits of dynamical systems

Consider the system $$\dot{x}=x-rx-ry+xy, \qquad \dot{y}=y-ry+rx-x^2,\qquad r=\sqrt{x^2+y^2},$$ which can be written in polar coordinates $(r,\theta)$ as $$\dot{r}=r(1-r), \qquad \dot{\theta}=r(1-\cos ...
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136 views

Chaos in Newtons Method

Im trying to prove that Newtons method applied to ${\rm f}\left(\, x\,\right) =x^{2} + c$, is chaotic for $c > 0$. I know I need to prove: (a) The periodic points of ${\rm f}$ are dense in $X$, ...
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0answers
61 views

Dichotomy for global existence or blow up for solutions of evolution problems.

Consider the problem (Nonlinear Schrödinger equation) \begin{equation} \left\{ \begin{array}{rl} iu_t + \Delta u\mp u|u|^{\alpha}=0\\ u(0) =\varphi\in H^{1}(\mathbb{R}^N), \\ \end{array}\...
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1answer
84 views

Particle under central force (Dynamics)

So I have this question: There is a particle moving in response to a central force per unit mass of $$F(r) = {\alpha\over r^2} + {\beta\over r^3}$$ where $\alpha$ and $\beta$ are constants. Initially ...
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1answer
61 views

Hysteresis Model with two real parameters

I would like to ask the following: I am trying to make a throughout analysis of a Hysteresis model in one dimension, with two real parameters: $\frac{dx}{dt}=f(x,ν,μ)=νx-x^3+μ$, where $ν,μ\in{\...
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0answers
137 views

Find all sinks/sources/saddles for a certain diffeomorphism

I'm trying to do the following exercise from Devaney's Introduction to Chaotic Dynamical Systems, exercise 2.6.1. The problem is this: Consider the diffeomorphism $Q_\lambda$ of the plane given by ...
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1answer
135 views

Bifurcation Example Using Newton's Method

I am studying dynamical systems as part of a research project. I have been using Newton's Method and studying the dynamic properties. Does anyone know where I could find a relatively simple example ...
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43 views

Show there is only one trajectory passing through each point

I have to show the following: Let $\varphi$ be a flow on the manifold M and suppose that that the orbits {$\varphi_t (x_0)$} & {$\varphi_t (x_1)$} intersect. Prove that the orbits coincide.
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229 views

What is the difference between disturbance and noise for dynamic systems

In most references from dynamic system theory, the following linear continuous dynamic system is considered. $$\frac{\text{d}x(t)}{\text{d}t}=Ax(t)+Bu(t)+Dd_{1}(t)\quad (1)$$ $$y(t)=Cx(t)+Ed_{2}(t) \...
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1answer
292 views

Resolvent matrix

Suppose $A$ is a triangular matrix. What is the most efficient known algorithm to compute the polynomial (in $x$) matrix $(xI-A)^{-1}$? Of course, $(xI-A)^{-1}= N(x)/p_A(x)$, where $p_A$ is the ...
3
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0answers
52 views

Kinematics of gravity in a non uniform field

I am a first year physics student. I am trying to figure out how to compute position in terms of time for an object falling through non uniform gravity towards the earth, and by extension towards any ...
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183 views

Why use the Lefschetz Zeta function?

Given a compact, triangulable space $X$ and a continuous function $f: X\rightarrow X$, then we define the Lefschetz number $\Lambda_{f}$ by $$\Lambda_{f} = \sum_{k\geq0}(-1)^{k}Tr(f_{\ast}\vert H_{k}(...
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0answers
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Question about solutions of $x''+(1+r(t))x=0$ when $\int_1^\infty |r(t)| dx <\infty$ .

Let $x''+(1+r(t))x=0$ where $r(t)$ is continous and $\int_1^\infty |r(t)| dx <\infty$ show that the equation has solutions $\phi_1$ and $\phi_2$ such that $$\lim_{t\to\infty} [(\phi_1(t)-e^{it})=0$...
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1answer
111 views

Dynamical Systems Periodic Orbits existing

Consider the nonlinear dynamical system $(1)$ : $x' = y(1 + x−y^2)$, $y' = x(1 + y−x^2)$, where $(x,y)\in\mathbb{R}^2$. (i) Determine the equilibrium points of $(1)$ (ii) Classify the ...
13
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4answers
488 views

Why solutions of $y''+(w^2+b(t))y=0$ behave like solutions of $y''+w^2y=0$ at infinity

Assume $w>0$ and $b(t)$ be continuous on $[0,+\infty)$ and $\int_0^\infty |b(t)| dt <\infty$ show that $y''+(w^2+b(t))y=0$ has solution $\phi(t)$ such that $$\lim_{t\to\infty} [(\phi(t)-\sin(wt))...
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0answers
90 views

Transitive Actions, Primitive Actions, and Ergodicity

A group action is transitive iff it has one orbit. Intuitively, this seems to say $G$ shuffles around all the elements of the $G$-set. A group action is primitive iff it has no nontrivial blocks, ...
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44 views

Existence of minimal sub-systems

A topological dynamical system is a topological space $X$ together with a continuous function $f : \ X \to X$. In the following, I will assume that $X$ is compact and Hausdorff (in other words, I work ...
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2answers
50 views

Do solutions of $\dot{x} = \frac{x}{t^2} + t$ exist satisfying $x(0) =0$

Suppose we have the 1-dimensional ODE \begin{equation} \dot{x} = \frac{x}{t^2} + t \end{equation} Do there exist solution curves with initial condition $x(0)=0$? If you proceed in a standard way ...
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546 views

Black's formula and feedback system stability

Consider a hypothetical system with open-loop transfer function $G(s)$. Place it in positive feedback with unit gain. (That is, take its output and directly add it to its input.) The closed-loop ...
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2answers
37 views

Eigenvector multiplication

I don't understand how multiplying eigenvetors by an expression like $e^{-2t}$ works, and results in this graph. Can someone explain this to me?
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113 views

Nullclines for differential equations

Consider the system of differential equations $$\dot {x}=y-x^2$$ $$\dot {y}=x-y$$ a. Determine the fixed points (1,1) (0,0) b. Determine the nullclines and the signs of $\dot {x}$ and $\dot {y}$ ...
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21 views

Alpha and Omega Limit Sets in Polar Coordinates [duplicate]

I guess here I am not sure how to get started, I know the definitions: The $ω$-limit sets of points are the set of points that the system of equation approach as time goes to infinity, and the $α$-...
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1answer
65 views

Expressing σ as a binary shift map.

Having shown that the only fixed point of $\sigma$ is $x=0$, I've now got the show that the fundamental period-$2$ points of σ are of the form $x=0 . ababab \ldots$ where $a,b \in \{0,1\}$ and $a\neq ...
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1answer
65 views

Differential equations, stability of fixed points

Consider the differential equations: $$\dot{x}=x^2-9$$ $$\dot{x}=x(x-1)(2-x)=-x^3+3x^2-2x$$ a. Find the stability type of each fixed point. (I am not sure about the stability of the points. Do I ...
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2answers
68 views

Polar coordinates, Differentiation

Can someone clarify this step for me please, "The polar coordinate r satisfies $r^2=x^2+y^2$, so by differentiating with respect to t we get $r\cdot\dot r=x\cdot\dot x+y\cdot\dot y$" I am totally ...
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1answer
140 views

Theorem on existence and uniqueness

Consider the differential equations and: $x'=x^2$ with initial condition $x(0)=x_0$≠0 $x'=x^2-1$ with initial condition $x(0)=x_0$ $x'=x^2+1$ with initial condition $x(0)=x_0$ a. Verify that the ...
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1answer
43 views

Partial Fractions to solve Logistic Equation

I am not really understanding how my book is getting $$\frac{x'}{x(1-\frac{x}{K})}=\frac{x'}{x}+\frac{x'}{K-x}$$ so $$\frac{x'}{x(1-\frac{x}{K})}=\frac{x'}{x-\frac{x^2}{K}}$$ $$=\frac{\frac{x'}{1}}{\...
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0answers
91 views

Properties of periodic solutions of nonlinear ODE system

Assume you have a complicated nonlinear ODE system with some parameter $p$. Numerical simulations of the system show, that for any initial conditions and $p$, the solution tends to a periodic function ...
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1answer
140 views

Wronskian of a fundamental set of solutions

(instead of the dot above, i used ' and ", am I correct in thinking that these are equivalent?) Consider the system of equations, $$x'_1=x_2$$ $$x'_2=-q(t)x_1-p(t)x_2$$ where $q(t)$ and $p(t)$ are ...
2
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1answer
74 views

Trace of a matrix

"the preceding 2 scalar solutions correspond to the vector solutions $x^1(t)=(t,1)^T$ and $x^2(t)=(t^2,2t)^T$ which have the Wronskian $$W(t)=\det\left[\begin{array}{lr} \mbox t & t^2\\ \mbox 1 &...
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1answer
35 views

Fixed Point Summary

I know someone has given me resources for this before but I can't seem to find them... Would someone please summarize stable vs unstable, attracting vs repelling, and node, saddle,etc fixed points? I ...
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70 views

Conjugacy map can be chosen Lipschitz

An exercise in Katok and Hasselblat's Introduction to the Modern Theory of Dynamical Systems (Section 2.1, exercise 2) goes as follows: Let $f$, $g$ be $C^1$ maps defined in a neighborhood of the ...
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1answer
35 views

Linear approximation to a system in the neighbourhood of the origin?

What would a linear approximation to the following system near the origin be? $${dx \over dt}=-y-x(x^2+y^2), {dy \over dt}=x-y(x²+y²)$$ I have no idea how to find this... I'm looking at this as an ...
2
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2answers
58 views

What does constant terms for complex roots represent when drawing a phase portrait?

For example if I have ODE where $x$ is a matrix of $x_1$ and $x_2$ $$x_1 ' = x_1 - 5 x_2$$ $$x_2 ' = x_1 - 3 x_2$$ and I found the solutions which are $$x_1 = c_1 e^{-t} (2 \cos(t)-\sin(t)) - ...
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94 views

Hamiltonian Dynamics and the canonical symplectic form

1- What kind of advantages does one have by having a canonical symplectic form on $T^*M$ apart from the form being exact? Would it for instance provide any advantage to studying Hamiltonian dynamics ...
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1answer
23 views

A sequence of numbers' question. (From Krengel's book on Ergodic Theorems).

On page 136 of Ergodic Theorem's by Ulrich Krengel, in the proof he sets: $c_n= \sup_j (n)^{-1} \sum_{i=0}^{n-1}x_{i+j}$ then he argues that for any $k,m$ positive integers one has: $$c_{km}\leq c_m$$ ...
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1answer
102 views

Hopf bifurcation and limit cycles

$$dV/dt=10(V-\frac{V^3}{3}-R+I_{input})$$ $$dR/dt=0.8(-R+1.25V+1.5)$$ Use $I_{input}$ as the relative parameter to prove that there these equations undergo 2 hopf bifurcations and indicate whether ...
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1answer
74 views

Fixed points and stability of them

Find the fixed points and classify them for the system of equations: $$x'=v$$ $$v'=-x+wx^3$$ $$w'=-w$$ $$0=v$$ $$0=-x+wx^3$$ $$1=wx^2$$ $$0=w$$ is the only fixed point (0,0,0)?? jacobian: \begin{...
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1answer
203 views

System of differential equations, phase portraits and stability of fixed points

Consider the system of differential equations: $$x'=-x-y+4$$ $$y'=3-xy$$ a. Find the fixed points. $x'=-x-y+4$ $x+y=4$ $x+3/x=4$ x=3,x=1 $y'=3-xy$ $y=3/x$ fixed points: (1,3), (3,1) b. ...
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1answer
168 views

Are they stable or unstable limit cycles?

I am using cl_matcont to perform a bifurcation analysis of a dynamical system of ten equations (equations are identical in two blocks, thus 8 of them and 2 of them are the same) During the ...
0
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1answer
107 views

System of differential equations, phase portrait

Consider the system of differential equations: $$x'=-2x+y-x^3$$ $$y'=-y+x^2$$ a. Determine the fixed points. (Am I correct in thinking that to determine the fixed points, I must set x' and y'=0? I'm ...