#desolve

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In the first part of our series (http://searchvoidstar.tumblr.com/post/158949147558/studying-chaos-starting-with-an-overview-of-ode), we started reading Non-linear Dynamics and Chaos by Strogatz and found that there are a number of mathematical concepts being introduced right off the bat. We took it slowly and introduced the idea of the differential equation and how to simulate one using R and the deSolve package.

We also looked at a simple system, a damped oscillator, or basically a spring type system that experiences some damping like friction. The last part of the problem our figure had something called a phase plot or “portrait” which is a good way to illustrate the relationship between position and velocity of our oscillator https://en.wikipedia.org/wiki/Phase_portrait

In this post, we will look at a “damped driven oscillator”, introduced in chapter 1 of the book as well. 

This system is similar to the simple “damped oscillator” but it adds an extra “forcing” so that instead of just decaying, the system receives an extra inputted force that goes back and forth. This system is well studied because it can lead to some “sporadic resonance” in systems that should not be resonating, e.g. bridge designs, and cause collapse. Here is a good demo of a damped driven oscillator on a table https://www.youtube.com/watch?v=OINsClhiUyo

Here is the equation:

We can now implement this as code by altering our first program slightly (the plot function is switched out for ggplot2 here for fun)

library(deSolve) library(ggplot2) library(cowplot) # x1 = initial position # x2 = initial velocity # x3 = initial "time", note that it is used by cos, and you can give it a multiplier to increase frequency state <- c(x1 = 0, x2 = 0, x3 = 0) # b = damping constant (e.g. friction) # m = mass # F = oscillating force # k = spring constant for oscillator parameters <- c(b = 0.1, F = 10, k = 30, m = 1) DampedDrivenOscillator<-function(t, state, parameters) {  with(as.list(c(state, parameters)), {    dX <- x2    dY <- 1/m*(-k*x1-b*x2+F*cos(2*x3))    dZ <- 1    list(c(dX, dY, dZ))  }) } # go from time 0 to 100, iterating by 0.01 seconds times <- seq(0, 100, by = 0.001) # run simulation for each timestep out <- ode(y = state, times = times, func = DampedDrivenOscillator, parms = parameters) # plot of position over time out2=data.frame(position=out[,'x1'],velocity=out[,'x2'],t=out[,'x3'],time=out[,'time']) # plot of variable over time p1<-ggplot(out2, aes(time, position))+geom_line() # plot of position vs velocity p2<-ggplot(out2, aes(time, velocity))+geom_line() # phase plot p3<-ggplot(out2, aes(position, velocity))+geom_path() # plotting our fake "time" variable p4<-ggplot(out2, aes(time, t))+geom_line() # phase plot plot_grid(p1, p2, p3, p4, align='hv', labels=c('A', 'B', 'C', 'D'))

In this example, we can start our system at position 0, velocity 0, and the forcing will start moving our system. We can see from this that it begins a little chaotically but it will then fall into a “resonant” mode

If the forcing becomes very large (increase F) then it will take over the system entirely

state <- c(x1 = 0, x2 = 0, x3 = 0) parameters <- c(b = 10, F = 100, k = 30, m = 1)

Now let’s bring F back down to a small value, and then also change the 2*x3 to 0.1*x3, then we can see that the forcing is slower and will have less influence on the system

state <- c(x1 = 0, x2 = 0, x3 = 0) parameters <- c(b = 0.1, F = 10, k = 30, m = 10)

With these examples, you can see that the forcing takes over some of the oscillations, but that the damping will chaotically work against the resonance. 

There are some good take homes here. On the applied side, this system can be applied by engineers to modelling dynamics of a structure, and we can use our program to simulate outcomes.

thank you so much!!!!!!!

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Integración de ecuaciones diferenciales ordinarias acompladas

En el siguiente código se utiliza el paquete deSolve para resolver un sistema dinámico, representado por ecuaciones diferenciales ordinarias (ODE) acompladas.

En este ejemplo, se tiene un sistema dinámico entre una presa (\( u \)) y un depredador (\( v \)), del tipo Lotka-Volterra, con una interacción tipo II de Holling, i.e,

\[ \alpha uv/(u+d) \]

donde \( \alpha \) es la tasa de ataque máxima por unidad de presa del depredador, y \( d \) es la abundancia de presa en la cual la tasa de ataque es la mitad del nivel de saciedad. Esta versión del tipo II de Holling es la versión tipo Michaelis-Menten.

El siguiente sistema de EDO acopladas es el siguiente:

\[ du/dt = u(1-u) - \alpha uv/(u+d) \] \[ dv/dt = bv(1-v/u) \]

Las condiciones iniciales al tiempo 0 son \( u(0) = 1.2 \) y \( v(0)= 0.2 \), y los parámetros son \( b=0.2 \), \( \alpha = 1 \), y \( d=0.1 \).

El código es el siguiente. Primero Se debe llamar al programa de Solve, i.e.

library(deSolve)

Luego se debe escribir la función para la relación de Holling tipo II, i.e.

f <- function(x, y, d) { x * y/(x + d) } # Holling tipo II

Ahora el sistema de EDO:

model <- function(t, xx, parms) { u <- xx[1] v <- xx[2] with(as.list(parms), { du <- u * (1 - u) - alpha * f(u, v, d) dv <- b * v * (1 - v/u) list(c(du, dv)) }) } # La secuencia temporal que se integrará (100 años): times <- seq(0, 100, 0.5) # Los parámetros del sistema: parms <- c(b = 0.2, alpha = 1, d = 0.2) # El estado inicial del sistema: xstart <- c(u = 1.2, v = 0.2) # La solución del sistema dinámico se realiza con ode y Runge-Kutta out <- as.data.frame(ode(xstart, times, model, parms, method = "rk4"))

El comando 'ode' permite resolver el sistema de ecuaciones diferenciales, en este caso se usó el método Rung-Kutta.

El sistema en el tiempo muestra el típico comportamiento depredador-presa, i.e.,

El comportamiento dinámico, también se puede observar al graficar la abundancia de la presa en función del depredador, i.e.,

Curso Herramientas Cuantitativas en Ecología

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