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44.1 Introduction to dynamics | ||

44.2 Functions and Variables for dynamics |

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The additional package `dynamics`

includes several
functions to create various graphical representations of discrete
dynamical systems and fractals, and an implementation of the Runge-Kutta
4th-order numerical method for solving systems of differential equations.

To use the functions in this package you must first load it with
`load("dynamics")`

.

Starting with Maxima 5.12, the dynamics package now uses the function
`plot2d`

to do the graphs. The commands that produce graphics
(with the exception of `julia`

and `mandelbrot`

) now accept
any options of `plot2d`

, including the option to change among the
various graphical interfaces, using different plot styles and colors,
and representing one or both axes in a logarithmic scale. The old
options `domain`, `pointsize`, `xcenter`, `xradius`,
`ycenter`, `yradius`, `xaxislabel` and `yaxislabel`
are not accepted in this new version.

All programs will now accept any variables names, and not just `x`
and `y` as in the older versions. Two required parameters have
changes in two of the programs: `evolution2d`

now requires a list
naming explicitely the two independent variables, and the horizontal
range for `orbits`

no longer requires a step size; the range
should only specify the variable name, and the minimum and maximum
values; the number of steps can now be changed with the option
`nticks`.

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__Function:__**chaosgame***(*`[[`

`x1`,`y1``]`

, …,`[`

`xm`,`ym``]]`

,`[`

`x0`,`y0``]`

,`b`,`n`, …, options, …)Implements the so-called chaos game: the initial point (

`x0`,`y0`) is plotted and then one of the`m`points`[`

`x1`,`y1``]`

, ...,`[`

`xm`,`ym``]`

will be selected at random. The next point plotted will be on the segment from the previous point plotted to the point chosen randomly, at a distance from the random point which will be`b`times that segment's length. The procedure is repeated`n`times.

__Function:__**evolution***(*`F`,`y0`,`n`, …, options, …)Draws

`n+1`points in a two-dimensional graph, where the horizontal coordinates of the points are the integers 0, 1, 2, …,`n`, and the vertical coordinates are the corresponding values`y(n)`of the sequence defined by the recurrence relationy(n+1) = F(y(n))

With initial value

`y(0)`equal to`y0`.`F`must be an expression that depends only on one variable (in the example, it depend on`y`, but any other variable can be used),`y0`must be a real number and`n`must be a positive integer.

__Function:__**evolution2d***(*`[`

`F`,`G``]`

,`[`

`u`,`v``]`

,`[`

`u0`,`y0``]`

,`n`, …, options, …)Shows, in a two-dimensional plot, the first

`n+1`points in the sequence of points defined by the two-dimensional discrete dynamical system with recurrence relationsu(n+1) = F(u(n), v(n)) v(n+1) = G(u(n), v(n))

With initial values

`u0`and`v0`.`F`and`G`must be two expressions that depend only on two variables,`u`and`v`, which must be named explicitely in a list.

__Function:__**ifs***(*`[`

`r1`, …,`rm``]`

,`[`

`A1`, …,`Am``]`

,`[[`

`x1`,`y1``]`

, …,`[`

`xm`,`ym``]]`

,`[`

`x0`,`y0``]`

,`n`, …, options, …)Implements the Iterated Function System method. This method is similar to the method described in the function

`chaosgame`

, but instead of shrinking the segment from the current point to the randomly chosen point, the 2 components of that segment will be multiplied by the 2 by 2 matrix`Ai`that corresponds to the point chosen randomly.The random choice of one of the

`m`attractive points can be made with a non-uniform probability distribution defined by the weights`r1`, …,`rm`. Those weights are given in cumulative form; for instance if there are 3 points with probabilities 0.2, 0.5 and 0.3, the weights`r1`,`r2`and`r3`could be 2, 7 and 10.

__Function:__**orbits***(*`F`,`y0`,`n1`,`n2`, [`x`,`x0`,`xf`,`xstep`], …, options, …)Draws the orbits diagram for a family of one-dimensional discrete dynamical systems, with one parameter

`x`; that kind of diagram is used to study the bifurcations of a one-dimensional discrete system.The function

`F(y)`defines a sequence with a starting value of`y0`, as in the case of the function`evolution`

, but in this case that function will also depend on a parameter`x`that will take values in the interval from`x0`to`xf`with increments of`xstep`. Each value used for the parameter`x`is shown on the horizontal axis. The vertical axis will show the`n2`values of the sequence`y(n1+1)`,...,`y(n1+n2+1)`obtained after letting the sequence evolve`n1`iterations.

__Function:__**rk***(*`ODE`,`var`,`initial`,`domain`)__Function:__**rk***([*`ODE1`, …,`ODEm`], [`v1`, …,`vm`], [`init1`, …,`initm`],`domain`)The first form solves numerically one first-order ordinary differential equation, and the second form solves a system of m of those equations, using the 4th order Runge-Kutta method.

`var`represents the dependent variable.`ODE`must be an expression that depends only on the independent and dependent variables and defines the derivative of the dependent variable with respect to the independent variable.The independent variable is specified with

`domain`

, which must be a list of four elements as, for instance:[t, 0, 10, 0.1]

the first element of the list identifies the independent variable, the second and third elements are the initial and final values for that variable, and the last element sets the increments that should be used within that interval.

If

`m`equations are going to be solved, there should be`m`dependent variables`v1`,`v2`, …,`vm`. The initial values for those variables will be`init1`,`init2`, …,`initm`. There will still be just one independent variable defined by`domain`

, as in the previous case.`ODE1`, …,`ODEm`are the expressions that define the derivatives of each dependent variable in terms of the independent variable. The only variables that may appear in those expressions are the independent variable and any of the dependent variables. It is important to give the derivatives`ODE1`, …,`ODEm`in the list in exactly the same order used for the dependent variables; for instance, the third element in the list will be interpreted as the derivative of the third dependent variable.The program will try to integrate the equations from the initial value of the independent variable until its last value, using constant increments. If at some step one of the dependent variables takes an absolute value too large, the integration will be interrupted at that point. The result will be a list with as many elements as the number of iterations made. Each element in the results list is itself another list with

`m`+1 elements: the value of the independent variable, followed by the values of the dependent variables corresponding to that point.

__Function:__**staircase***(*`F`,`y0`,`n`, …, options, …)Draws a staircase diagram for the sequence defined by the recurrence relation

y(n+1) = F(y(n))

The interpretation and allowed values of the input parameters is the same as for the function

`evolution`

. A staircase diagram consists of a plot of the function`F(y)`, together with the line`G(y)``=`

`y`. A vertical segment is drawn from the point (`y0`,`y0`) on that line until the point where it intersects the function`F`. From that point a horizontal segment is drawn until it reaches the point (`y1`,`y1`) on the line, and the procedure is repeated`n`times until the point (`yn`,`yn`) is reached.

**Options**

Each option is a list of two or more items. The first item is the name of the option, and the remainder comprises the arguments for the option.

The options accepted by the functions `evolution`

, `evolution2d`

,
`staircase`

, `orbits`

, `ifs`

and `chaosgame`

are the same
as the options for `plot2d`

. In addition to those options, `orbits`

accepts and extra option `pixels` that sets up the maximum number of
different points that will be represented in the vertical direction.

**Examples**

Graphical representation and staircase diagram for the sequence: 2, cos(2), cos(cos(2)),...

(%i1) load("dynamics")$ (%i2) evolution(cos(y), 2, 11); (%i3) staircase(cos(y), 1, 11, [y, 0, 1.2]);

If your system is slow, you'll have to reduce the number of iterations in
the following examples. And if the dots appear too small in your
monitor, you might want to try a different style, such as
`[`

`style`,`[`

`points`,0.8`]]`

.

Orbits diagram for the quadratic map, with a parameter `a`.

x(n+1) = a + x(n)^2

(%i4) orbits(x^2+a, 0, 50, 200, [a, -2, 0.25], [style, dots]);

To enlarge the region around the lower bifurcation near x `=`

-1.25 use:

(%i5) orbits(x^2+a, 0, 100, 400, [a,-1,-1.53], [x,-1.6,-0.8], [nticks, 400], [style,dots]);

Evolution of a two-dimensional system that leads to a fractal:

(%i6) f: 0.6*x*(1+2*x)+0.8*y*(x-1)-y^2-0.9$ (%i7) g: 0.1*x*(1-6*x+4*y)+0.1*y*(1+9*y)-0.4$ (%i8) evolution2d([f,g], [x,y], [-0.5,0], 50000, [style,dots]);

And an enlargement of a small region in that fractal:

(%i9) evolution2d([f,g], [x,y], [-0.5,0], 300000, [x,-0.8,-0.6], [y,-0.4,-0.2], [style, dots]);

A plot of Sierpinsky's triangle, obtained with the chaos game:

(%i9) chaosgame([[0, 0], [1, 0], [0.5, sqrt(3)/2]], [0.1, 0.1], 1/2, 30000, [style, dots]);

Barnsley's fern, obtained with an Iterated Function System:

(%i10) a1: matrix([0.85,0.04],[-0.04,0.85])$ (%i11) a2: matrix([0.2,-0.26],[0.23,0.22])$ (%i12) a3: matrix([-0.15,0.28],[0.26,0.24])$ (%i13) a4: matrix([0,0],[0,0.16])$ (%i14) p1: [0,1.6]$ (%i15) p2: [0,1.6]$ (%i16) p3: [0,0.44]$ (%i17) p4: [0,0]$ (%i18) w: [85,92,99,100]$ (%i19) ifs(w, [a1,a2,a3,a4], [p1,p2,p3,p4], [5,0], 50000, [style,dots]);

To solve numerically the differential equation

dx/dt = t - x^2

With initial value x(t=0) = 1, in the interval of t from 0 to 8 and with increments of 0.1 for t, use:

(%i20) results: rk(t-x^2,x,1,[t,0,8,0.1])$

the results will be saved in the list `results`

.

To solve numerically the system:

dx/dt = 4-x^2-4*y^2 dy/dt = y^2-x^2+1

for t between 0 and 4, and with values of -1.25 and 0.75 for x and y at t=0:

(%i21) sol: rk([4-x^2-4*y^2,y^2-x^2+1],[x,y],[-1.25,0.75],[t,0,4,0.02])$

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