# 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 ...

105 views

### Calculating this derivative

I'm currently having trouble to verify this fact on page 90 in Robert L. Devaneys "An Introduction to Chaotical Dynamical Systems": Let $F(x,\lambda) = f_\lambda (x)$. Assume $f_\lambda (x) = 0$ for ...
46 views

The two following theorems appear to be contradictory to me. I'm sure I must have overlooked something significant here. The Poincaré-Bendixson II(a) says that if $R$ is a Type I invariant region, ...
78 views

### For what value of $v_0$ is the solution periodic?

A solution of the second-order differential equation $$x''+x-x^3=0$$ satisfies the initial condition $x(0)=0$ and $x'(0)=v_0$. For what value of $v_0$ is the solution periodic? I have tried ...
42 views

### The behavior of the trajectory of the phase portrait

For the plane autonomous system $$x' = ax+by$$ $$y' = cx+dy$$ If the solution to this system is, say, $\binom{x}{y}= c_{1}\binom{1}{1}e^{-5t} + c_{2}\binom{1}{2}e^{-t}$, then it is ...
59 views

29 views

### How to find periodic solutions for dynamical system?

I have the Hamiltonian system given by $$H=\frac{1}{2}x^2-\frac{1}{3}x^3+xy^2+\frac{1}{2}y^2$$ Using computer software I managed to plot the dynamical system in the phase plane. I am aware that the ...
38 views

41 views

### Differential equations question: Follow-up on Dynamical systems?

Yesterday I asked a question on here. Unfortunately I closed off the page without fully signing up for my account so I could not comment on the answer I received, whilst the answer was very good there ...
33 views

### Time-$t$ map of a Hamiltonian flow: how to check twist property?

I would like to obtain a general formula to verify if a certain time-$t$ map of a Hamiltonian flow is twist. I have a Hamiltonian $1$ degree of freedom system $H=H(q(t),p(t))$, such that all orbits ...
46 views

### How to determine which initial conditions will make the solution of a Hamiltonian system periodic?

I have a Hamiltonian system given by: $$\dot{x}=x+y-x^2\\\dot{y}=2xy-y$$ I have found that the Hamiltonian function for the system is given by $$H(x,y)=xy+\frac{1}{2}y^2-x^2y$$ and I have managed ...
17 views

### Intermittent Coupled Oscillations

A forced harmonic oscillator has equation of motion x''+ 4x = F(t) where F(t) = pi^2 -t^2 for tpi I know how to find the solution i.e. we use initial conditions given (x=0,t=0,x'=0) and solve the ...
27 views

### How to rearrange this doubly infinite sum for a diffeomorphism using the existence of a first integral?

Let's take two diffeomorphisms $F,G: \mathbb{R}^{n} \mapsto \mathbb{R}^{n}$. Let $x \in \mathbb{R}^{n}$, and $x^{n} = F^{n}(x)$, where $n \in \mathbb{Z}$. Suppose that $F$ has a first integral, i.e. ...
20 views

15 views

### Reduced crossed product $C(X)\rtimes_r G$ in terms of orbits and groupoids

Let $G$ be a discrete group, and let $X$ be a compact Hausdorff $G$-space. It can be shown that the reduced crossed product $C^*$-algebra $C(X)\rtimes_r G$ is isomorphic to the reduced $C^*$-algebra ...
36 views

### Finding bifurcation of trigonometric system

I'm really struggling to find the bifurcation(s) of the system $x'=x^2 + \cos(x+ \mu)$, $\mu \in [0,2\pi)$. I've tried substituting $y=\mu+x$, taylor expanding, and just about everything else I ...
29 views

### Applied nonlinear dynamics: the onset of chaos in biological cycles (reference request)

I have seen some applied research in the onset of chaos in the study of current regulation in the human heart and the transition into cardiac arrest. I would like to review any literature that exists ...
39 views

### Every fiber bundle with Cantor set fiber is the suspension of a homeomorphism of the Cantor set.

I've heard that every fiber bundle (over $\mathbb S^1$?) with Cantor set fiber is the suspension of a homeomorphism of the Cantor set. Can someone explain the intuition behind the fact? Is there a ...