# Questions tagged [dynamical-systems]

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 models of diverse fields such as classical mechanics, economics, traffic modelling, population dynamics, and biological feedback.

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### Counterexample for a weakened definition of contraction mapping

I was trying to solve this exercise form a book on dynamical systems. There, after introducing the notion of contraction mapping, it is shown that, given a complete metric space $(X,d)$ and a ...
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### Why large linear systems of saddle point type are indefiniteness and often poor spectral properties?

I'm reading the paper "Numerical solution of saddle point problems" by Michele Benzi. In the abstract, he states that these types of large linear systems of saddle point are challenging due ...
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### Suspension of Diffeomorphisms

There is no doubt that Suspension of Diffeomorphisms is a special case of Suspension of a Representation (that is also known as Suspension of Diffeomorphism Group). Therefore, I would be very grateful ...
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### About Riesz's proof of the Birkhoff Ergodic Theorem

I'm trying to write a fine-grained version of Riesz's proof of the Pointwise Ergodic Theorem (sometimes called Birkhoff Ergodic Theorem) and I'm struggling with a couple of details. You can read that ...
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### How to increase firs order system dynamic to second order system [closed]

So, I have system dynamic equations, and I don't know how to increase system order 1 to order second
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### Can I use $||J||_2$ as a “gradient descent” for the system $J = b - Ax$?

One quick question. Let's assume that I want to solve $Ax = b$ but I want to do that in a special way. My idea is that I first find the difference between $b$ and $Ax$, we call it $J$. $$J = b-Ax$$ ...
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### Persistence of homoclinic points - the noncompact case

It is well known that transverse homoclinic points of $C^1$-diffeomorphisms of compact manifolds $M$ persist under small $C^1$ perturbations. Does the same hold for non-compact manifolds (with the ...
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### Intersection between dynamical systems and stochastics?

Are there any subjects that deal with the intersection between dynamical systems and stochastics? (Specifically in pure mathematics.) I know about random dynamical systems, but there does not seem to ...
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### Highly Recommended References on Harmonic and Stationary Measures

A harmonic or stationary measure is said to be Regular if it is Absolutely Continuous with respect to Lebesgue Measure and Singular if it is not. I need to understand the Regularity and Singularity of ...
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### System of differential equation of the form $\dot x = (A x) \circ x$

I am interested in systems of first-order differential equations of the general form $\dot x=(A x) \circ x$, where $A$ is a square, constant, real matrix and $\circ$ is the elementwise product (...
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### Canonical transformation and generating function

I am stuck on an exercise of which I do not understand the solution. The exercise is the following: Consider the Hamiltonian system $$H(\theta, p, t) = \dfrac{(p-\omega t)^2}{2} - k\cos(\theta)$$ ...
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### If $F\in\mathcal D \implies \left\lvert \frac{F^n(0)}{n} - F(0)\right\rvert\le\frac{n-1}{n}\, ,\forall \,n\in\mathbb N$

Let $F\in\mathcal D$, $\phi=F-id$, show that $\left\lvert \cfrac{F^n(0)}{n} - F(0)\right\rvert\le\cfrac{n-1}{n} ,\forall\, n\in\mathbb N$ Note: We define $\mathcal D$ as the set of increasing ...
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### Two questions on the existence of sets associated with LaSalle invariance principle

I am copying in verbatim a standard reference on Lasalle's theorem (which is Hassan Khalil's textbook, page 128) Theorem: Let $\Omega \subset D$ be a compact set that is positively invariant with ...
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### Multidimensional obstacle avoidance in ODE. Part II

Multidimensional obstacle avoidance in ODE For some time, I studied this question more closely and came to the conclusion that it is best to assign its own constraint for each selected coordinate. ...
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### Every hyperbolic matrix $A\in SL(3,\mathbb{Z})$ has 3 distinct eigenvalues

Let $A\in SL(3,\mathbb{Z})$ be a hyperbolic matrix (i.e., the absolute value of each eigenvalue is not $1$), then all its eigenvalues are real and different. I read this result here, Proposition 4.12. ...
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### Trying to understand random dynamical systems [duplicate]

Let $\{\theta_t : \Omega \mapsto \Omega \ | \ t \in T \}$ be a family of measure preserving transformations on the probability space $(\Omega, \mathcal{F}, \mathbb{P}).$ A measurable random ...
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### What is the best ODE solver for stability and speed?

I'm going to build a ODE solver in C-code and I'm looking for an ODE solver that are robust against stiff ODE:s and very fast. But still, it also need to have a very good accuracy. I would be happy if ...
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### If $h$ is a diffeomorphism, $F,G$ vector fields, such that $dh(x)F(x)=G(h(x))$, so $h(\omega_F(x))=\omega_G(h(x))$

Let $F:U\rightarrow \Bbb{R}^d,G:V\rightarrow \Bbb{R}^d$ smooth vector fields, where $U,V$ are open sets in $\Bbb{R}^d$. Let $h:U\rightarrow V$ a diffeomorphism such that dh(x)F(x)=G(h(x))\quad\...