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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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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43 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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2answers
522 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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108 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 ...
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1answer
64 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
64 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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67 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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130 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
42 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 $$\frac{x'}{x(1-\frac{x}{K})}=\frac{x'}{x-\frac{x^2}{K}}$$ ...
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90 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
138 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 ...
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1answer
73 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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68 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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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 ...
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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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93 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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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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100 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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73 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: ...
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198 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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159 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 ...
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1answer
103 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 ...
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1answer
92 views

How to prove symmetry of the following Maxwell-Bloch equations?

I have the following Maxwell-Bloch equations: $\dot{E}=-\alpha_{1} E+ k_{1}P$ $\dot{P}=-\alpha_{2}P+ k_{2}ED$ $\dot{D}=-\alpha_{3}(D-\lambda) -k_{3}EP$ In this system ...
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1answer
23 views

Sampling a matrix of an AR model

Let us consider a dynamic system $x_t = A x_{t-1}+v_t$ where $v_t$ is multivariate normal noise with zero mean, i.e. $v_t\sim\mathcal{N}(0,\Sigma)$ and $A$ is a matrix. As far as I know, for some $A$, ...
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107 views

Topological entropy of isometric extension

L.s., This is a homework question some of my fellow students and I are having great difficulty with. Let $Y,Z$ be compact metric spaces, $X = Y \times Z$, and $\pi$ the projection to $Y$. Denote $h$ ...
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432 views

Alpha and Omega limit sets (dynamical systems)

What are all the possible $α$- and $ω$-limit sets of points for the linear system of equations $x'=Ax$ given by the following matrices $A$: I guess I don't really know how to get started, Could I ...
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59 views

Alpha and omega limit sets [duplicate]

What are all the possible $α$- and $ω$-limit sets of points for the linear system of equations $x'=Ax$ given by the following matrices $A$: I guess I don't really know how to get started, Could I ...
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2answers
175 views

System of equations, limit points

This is a worked out example in my book, but I am having a little trouble understanding it: Consider the system of equations: $$x'=y+x(1-x^2-y^2)$$ $$y'=-x+y(1-x^2-y^2)$$ The orbits and limit sets ...
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1answer
414 views

Jacobian Matrix in dynamical systems

Can someone explain what exactly the Jacobian matrix is (specifically in its application to dynamical systems) and maybe give an example of how to compute it? It really confuses me...and I haven't ...
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2answers
40 views

Preserving orbits by multiplication with a non-vanishing function

I'm reading through some notes from a past course of mine, where a system of ODE's of the form$$ \begin{array}{c} x'=h(x,y)f(x,y)\\ y'=h(x,y)g(x,y) \end{array} $$ appears, such that ...
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110 views

Translational invariance and zero eigenvalue

Page 2 (506), line 18 of http://www-personal.umich.edu/~orosz/articles/NonlinScipublished.pdf says that "The presence of translational symmetry in the nonlinear equations gives rise to a relevant ...
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68 views

dynamical systems classifying an equilibrium point

Consider the differential equation $$x''+(2a)x'-(b^2/2)x+x^3=0 \tag{1}$$ where $a,b >0$ are constants. (i) Write differential equation (1) as a first order dynamical system. (ii) Determine the ...
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1answer
55 views

Uniqueness of homeomorphism

In the theory of dynamical systems, the Hartman–Grobman theorem states that there is a homeomorphism of a neighborhood which conjugates the original system and its linearization. A problem bothers me: ...
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1answer
378 views

Limit cycle of dynamical system $x'=xy^2-x-y$, $y'=y^3+x-y$

Consider a planar ODE system $z'=f(z)$ with $z=(x,y)$, $$ f(x,y)=(xy^2-x-y,y^3+x-y). $$ Using polar coordinates, one can get $$ r'=r(r^2\sin^2\theta-1),\quad \theta'=1. $$ With Mathematica one can ...
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2answers
92 views

Stability based on phase portrait

I wonder how can I find the largest delta in which if initial condition is inside circle (How do I specify radius of that circle) bounded by delta then the solution is always within the dashed circle ...
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2answers
385 views

Stability of fixed points for a differential equation

Consider the differential equation $x'=x^2-9$ a. find the stability type of each fixed point To find the fixed points, I set this equal to $0$, right? Would someone mind explaining why I do ...
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1answer
108 views

Stability of Linear Systems

for the following matrices A, classify the stability of the linear systems x=Ax as asymptotically stable, L-stable (but not asymptotically stable) or unstable and indicate whether it is a stable node, ...
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1answer
802 views

Verifying if system has periodic solutions

Given the following system $\dot{x} = y$ $\dot{y} = y(9-x^2-2y^2) - x$ verify whether it has periodic solutions and if so are they attracting or repelling. I thought: The critical points or fixed ...
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1answer
79 views

Induced sytem ergodic implies normal sytem ergodic

Okay, we consider a measure preserving system $(X, \mathcal F, \mu, T)$ and let $A \in\mathcal F$ be such that $\mu(A) > 0$ and $\mu ( \cup ^{\infty}_{n=1} T^{-n}A) = 1 $. Now I want to show that ...
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1answer
44 views

ODE and recurrence relation

I am trying to understand the following claim (I came across it while reading a paper): Consider the map (Standard/Arnold map) $T_{k}:(x,y)\mapsto(x+y+kf(x), y+kf(x))$, with ...
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1answer
62 views

Showing a second-order ordinary differential equation has periodic orbits

Is it possible to show that $x''(t)-x'(t)²+x(t)²-x(t)=0$ has at least a periodic orbit? I've made it a system by setting $y=x'$ and get $x'=y; y'=y²-x²+x$. I'm asking the question because I find a ...
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1answer
59 views

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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1answer
45 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
70 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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1answer
59 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
114 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 ...