Solving The Roessler Oscillator using Runge-Kutta algorithm python

Question

I am having some trouble with this question. I am given this system of equations

dx / dt = -y -z
dy / dt = x + a * y
dz / dt = b + z * (x - c)

and default values a=0.1, b=0.1, c=14 and also the Runge-Kutta algorithm:

def rk4(f, xvinit, Tmax, N):
        T = np.linspace(0,Tmax,N+1)
        xv = np.zeros( (len(T), len(xvinit)) )
        xv[0] = xvinit
        h = Tmax / N
        for i in range(N):
            k1 = f(xv[i])
            k2 = f(xv[i] + h/2.0*k1)
            k3 = f(xv[i] + h/2.0*k2)
            k4 = f(xv[i] + h*k3)
            xv[i+1] = xv[i] + h/6.0 *( k1 + 2*k2 + 2*k3 + k4)
        return T, xv

I need to solve this system from t=0 to t=100 in time steps of 0.1 and using initial conditions (????0,????0,????0)=(0,0,0) at ????=0 I'm not really sure where to begin on this, I've tried defining a function to give the Oscillator:

def roessler(xyx, a=0.1, b=0.1, c=14):
    xyx=(x,y,x)
    dxdt=-y-z
    dydt=x+a*y
    dzdt=b+z*(x-c)
    return dxdt ,dydt ,dzdt

which returns the right side of the equation with default values, i've then tried to solve by replacing f with roessler and filling in values for xvinit,Tmax and N with values i'm given but it's not working. Any help is appreciated sorry if some of this is formatted wrong i'm new here.

Peter Meisrimel Peter Meisrimel · Accepted Answer · 2020-08-03T17:02:03

Well, you almost got it already. Changing your roessler function to the following

def roessler(xyx, a=0.1, b=0.1, c=14):
    x, y, z = xyx
    dxdt=-y-z
    dydt=x+a*y
    dzdt=b+z*(x-c)
    return np.array([dxdt, dydt, dzdt])

and then calling

T, sol = rk4(roessler, np.array([0, 0, 0]), 100, 1000)

makes it work.

Taking aside the typo in the first line of your roessler function, the key to solving this is to understand that you have a system of differential equations, i.e., you need to work with vectors. While you already had the input as the vector correct, you also need to make the output of roessler a vector and put in the initial value with the appropriate shape.

Solving The Roessler Oscillator using Runge-Kutta algorithm python

1 Answers