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 (…)

- Teoremas y ejemplos

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

Consider a system
$$ \begincases \dotx=f(x,y)\\ \doty=g(x,y) \endcases $$
where $f$ and $g$ have continuous partial derivatives, and such that solutions exist for all $t$. Let $R$ denote a closed, bounded region of the $xy$-plane which contains no fixed points. Suppose that no solution may leave $R$. Then the system has a periodic solution in the region $R$.

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Teorema de Lyapunov

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

Suppose that we can find a continuously differentiable, real-valued function $L(\boldsymbolx)$ such that $L(\boldsymbolx)>L(\bar\boldsymbolx)$ for all $\boldsymbolx$ in some neighborhood $U$ of $\bar\boldsymbolx$.
If $\dotL(\boldsymbolx)\leq 0$ for all $\boldsymbolx\in U$, then $\bar\boldsymbolx$ is stable.
If $\dotL(\boldsymbolx)< 0$ for all $\boldsymbolx\in U\backslash\\bar\boldsymbolx\$, then $\bar\boldsymbolx$ is asymptotically stable.
If $\dotL(\boldsymbolx)> 0\thinspace \textfor (…)

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Teorema de Hartman-Grobman

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

Suppose $(\bar x,\bar y)$ is a fixed point of a system
$$ \begincases \dotx = f(x,y)\\ \doty= g(x,y). \endcases $$ Assume that the real part of the eigenvalues of the Jacobian matrix
$$ A= \beginpmatrix \frac\partial f\partial x\scriptstyle(\bar x,\bar y) & \frac\partial f\partial y\scriptstyle(\bar x,\bar y)\\ \frac\partial g\partial x\scriptstyle(\bar x,\bar y) & \frac\partial g\partial x\scriptstyle(\bar x,\bar y) \endpmatrix $$ are nonzero. Then there is a small region around (…)

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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

Assume that, near $\mu=\mu^*$, the Jacobian matrix of the vector field evaluated at $(\barx(\mu),\bary(\mu))$ has eigenvalues of the form
$$ a(\mu)\pm i b(\mu) $$
with
$$ a(\mu^*)=0\quad\textand\quad b(\mu^*)\neq 0. $$
Also assume that the real parts of the eigenvalues change signs as $\mu$ is varied through $\mu^*$, i.e.,
$$ \frac\textda\textd\mu(\mu^*)\neq 0. $$
Given these hypotheses, the following possibilities arise: There is a range $\mu^*<\mu<\mu_1$ such (…)

- Teoremas y ejemplos