**Asignatura**:
Métodos de la Física Matemática

**Curso**:
**3º de la Licenciatura de Física**
**
**

**Profesores:
Santos Bravo Yuste**
y **Vicente
Garzó Puertos**

**Some didactic
resources**

This Demonstration plots the phase portrait (or phase plane) and the vector field of directions around the fixed point (0,0) of the two-dimensional linear system of first-order ordinary differential equations

dx/dt = a

_{1 }x + b_{1}ydy/dt = a

_{2 }x + b_{2}y

Drag the four locators (the big circles) to see the trajectories of four solutions of the system that go through them. The position of these points can be chosen by clicking on wherever you like inside the graphics. Thick orange lines parallel to the eigenvectors are shown if the eigenvalues of the system are real

**
►Phase portrait and vector field of directions of some nonlinear
first-order differential equations**

You can study (and play with) the phase portrait (or phase plane) and the vector field of directions of the systems

__of__ the book "Métodos
Mátemáticos Avanzados para Científicos e Ingenieros" by means of the *Mathematica* notebooks that you can download from the corresponding hyperlinks.

For using these notebooks you need
*Mathematica 7 or *the* Mathematica Player *program* *(which can
be downloaded from
http://www.wolfram.com/products/player/.) You can choose the
region of the phase plane where the solutions and vector field of directions are
shown, and, also, the values of the parameters "a" and " b" that describe the
actual system. Trajectories of several solutions of the system that go
through the "locators" (the big circles) are shown. The position of these
locators can be chosen by clicking on wherever you like inside the graphics, or,
simply, by dragging them to the location you like.

**
►Phase portrait and vector field of directions of
the Lotka-Volterra equations**

You can study (and play with) the phase portrait (or phase plane) and the vector field of directions of the Lotka-Volterra equations given in the

__of__ the book "Métodos
Mátemáticos Avanzados para Científicos e Ingenieros" by means of the *Mathematica*
notebook that you can download from the previous hyperlink.

For using this notebook you need
*Mathematica 7 or *the* Mathematica Player *program* *(which can
be downloaded from
http://www.wolfram.com/products/player/.) You can choose the
region of the phase plane where the solutions and vector field of directions are
shown, and, also, the values of the parameters "a" , " b" ,
"α" and "β"
that describe the
actual system. Trajectories of several solutions of the system that go
through the "locators" (the big circles) are shown. The position of these
locators can be chosen by clicking on wherever you like inside the graphics, or,
simply,
by dragging them to the location you like.