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 ...

32 views

Devaney's definition of chaos

I'm reading Banks, et. al. paper On Devaney's Definition of Chaos. In it, they say "It is not difficult to find transitive examples for which sensitivity is not preserved under conjugation." I'm ...
44 views

16 views

Reference request: monotone and strongly monotone with respect to derivatives

Recall, let $H$ be a real Hilbert space. A mapping $F:H \rightarrow H$ is said to be monotone if $$\langle F(u)-F(v), u-v\rangle\geq 0, \quad \forall u,v\in H;$$ strongly monotone if there exists ...
55 views

Dynamics of a three dimensional system

I have a dynamical system in three dimensions given by: $\dot x = (1-x^2-y^2-z^2)x+xz-y$ $\dot y = (1-x^2-y^2-z^2)y+yz+x$ $\dot z = (1-x^2-y^2-z^2)z-x^2-y^2$ I analyzed the system by first finding ...
46 views

Suppose the period of γn is λn. If there are points Xn ∈ γn such that Xn → X ∈ γ , prove that λn → λ.

I was wondering if someone could help me with an exercise from Hirsch, Smale, and Devaney, Differential Equations, Dynamical Systems, and an Introduction to Chaos. Let γ be a closed orbit of a ...
11 views

If $1$ is a simple eigenvalue of $T$, then $T$ is an ergodic measure-preserving transformation

Let $T : X \to X$ be a measure-preserving transformaton and $U_T : L^2 (X, \mu) \to (X, \mu)$, $$(U_T f) (x) = f(Tx).$$ I have to show that if $1$ is a simple eigenvalue of $U_T$, then $T$ is ...
18 views

Hamilton principle/dynamics teaching in earlier stages.

In finding dynamic motion of particles we use laws of conservation of energy and momentum. It is found the dynamics formulation using action integral $$\int (T-V)\, dt$$ builds ODEs for dynamic ...
15 views

16 views

the prerequisites for “an outline of ergodic theory” by Kallikow and Mccutcheon

I want to study an ergodic theory book, and the book "an outline of ergodic theory" seems to be smooth and introductory. But in its preface nothing has been mentioned about prerequisites. All I know ...
10 views

Let $\{ y_k \}$ that satisfies $y_k\le {2^k\over M}y_{k-1}^\beta$ , then $\lim_{k\to \infty}y_k=0$.

Let be a sequence $\{ y_k \}^\infty _{k=0} \subset (0,\infty)$ that satisfies $$y_k\le {2^k\over M}y_{k-1}^\beta ,$$ where $k=1,2,...$, and $\beta\gt 1$ , $M\gt0$. Prove that if ...
38 views

Topics in Geometry and Dynamical Systems

Dynamical Systems/Fractal Geometry and Differential Geometry/Topology are really interesting areas of study. My question is whether there is any direct connection between them? In other words, are ...
32 views

configuration spaces in mathematics and in physics

On the Wekipedia website Configuration space , there are two configuration spaces defined. One is Configuration spaces in physics, the other is Configuration spaces in mathematics. Question. Do ...
20 views

Different ergodic probability measures are mutually singular

Can someone, please, give me a hint on how to demonstrate that different ergodic probability measures are mutually singular? Thank you!
24 views

Topologically Conjugate under a homeomorphism

Let $f$ and $g$ be $C^{1}$ vector fields on open sets $E$ and $F$ on $\mathbb{R}^n$. Suppose their flows are topologically conjugate under a homeomorphism $h:E \rightarrow F$. Prove the following: ...
19 views

Dynamical system in a square

I am considering a problem that is asking me to explore a deceptively simple dynamical system and discover some of surprising properties. I want to consider the motion of four particles A,B,C and D in ...
40 views

47 views

A question about the “state-transition-matrix” of a physical system,

Here's a simple but potential research problem that I am learning about. Let's say I am studying a physical system that is governed by N objects. At each time, each object is either "active" and ...
46 views

2D Bifurcation Classification

Given the system with m as a varying parameter: $\dot x = mx^2-y$ and $\dot y = m+y - x$ Determine any bifurcations that occur Attempt: x nullcline $y=mx^2$ y nullcline $y=x-m$ Fixed ...