Generative eBooks

El modelo Rosenzweig-McArthur

Generative eBooks — 2023-12-06T15:36:02Z

We come back to preys and predators. We previously studied a basic model for the interaction of sharks and sardines, namely $$ \begincases \dotx= x(1-x) - xy\ \doty=\beta y ( x-\alpha) \endcases $$ where $x$ represents the density of sardines, $y$ the density of sharks, and $\alpha,\beta$ are positive parameters. We observed and proved that there cannot be limit cycles in this model, which means that the populations of sharks and sardines cannot oscillate periodically in a robust way.
A (…)

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El péndulo idealizado

Generative eBooks — 2023-12-06T15:35:38Z

Consider the motion of the following idealized pendulum: a bob of mass $m$ is attached to one end of a massless rigid rod. The other end of the rod is pivoted so that the mass may swing in a vertical plane. We neglect both the friction of the pivot and air drag.
The swinging of the pendulum is governed by $$ \ddot\theta=-\fracgL \sin \theta $$ where $\theta$ is the angle between the rod and the downward vertical. This equation is derived in all textbooks of classical mechanics.
Phase (…)

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Teorema de Poincaré-Bendixson

Generative eBooks — 2023-12-06T15:35:14Z

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Teorema de Lyapunov

Generative eBooks — 2023-12-06T15:34:44Z

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Teorema de Hartman-Grobman

Generative eBooks — 2023-12-06T15:34:06Z

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Construir un ciclo límite desde cero

Generative eBooks — 2023-12-06T15:33:27Z

We want to cook up a simple example of a limit cycle, some kind of toy model. Let us look for a system having the circle of radius one centered at $(0,0)$ as an attracting limit cycle. The trick is to think in terms of polar coordinates $(r,\theta)$. The simplest situation would be to have uncoupled equations for the radial and angular motions: $$ \begincases \dotr=f(r)\ \dot\theta=g(\theta). \endcases $$ In the $r$-direction, we want $f(1)=0$, that is $r=1$ to be a fixed point. We also want (…)

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Teorema de bifurcación de Andronov-Hopf

Generative eBooks — 2023-12-06T15:32:40Z

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