# 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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### Measure preserving ergodic map commutes with complementation?

This is probably trivial (in which case I apologize), but it's late and I would really like a quick proof/counterexample for this (for a different problem that I'm doing): if $(X,\mathcal{M},\mu,T)$ ...
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### 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 ...
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### Existence of periodic orbit of the ODE system $\dot{r}=r-r^3 \cos^2(\theta),\,\dot{\theta}= 1$

Consider the system of ODEs (in polar coordinates): $$\dot{r}=r-r^3 \cos^2(\theta)$$ $$\dot{\theta}= 1$$ If we take $r_1 = \frac{1}{4}$ then $\dot{r}> 0$, and if we take $r_2 = \frac{3}{2}$ ...
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### If $f$ and $g$ are bounded, then every solution of the autonomous system of differential equations is defined for $t \in \mathbb R$.

Consider the system of autonomous differential equations (autonomous system of differential equations?) $$x' = f(x,y)$$ $$y' = g(x,y)$$ where $x=x(t)$ and $y=y(t)$ Let $f$ and $g$ have first ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ergodic....
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### 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 ...
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### Proof that $M_ {\boldsymbol f}$ has a neighbourood diffeomorphic to the product $T^n\times D^n$

I'm reading Arnold's proof of Liouville's theorem and got stuck with the following problem in subsection §50, A. Here the manifold $M_{\boldsymbol f}$ is defined as $\boldsymbol F^{-1}(\boldsymbol f)$,...
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### Estimation for entropy

Let $T\colon X\to X$ be continuous and $X$ compact and $K\subset X$ compact. By $s_n(2^{-k},K,T)$ denote the maximal cardinality of any $(n,2^{-k})$ separated subset of $K$. Suppose, we know for ...
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### 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 ...
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### 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 ...
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### 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!
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### 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 ...
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