Experiments with the Duffing Oscillator from Nonlinear dynamics

Experiments with the Duffing Oscillator from Nonlinear dynamics

Experiments with the Duffing Oscillator
from Nonlinear dynamics and Chaos by
J.M.T. Thompson and H.B. Stewart
The 2nd order ODE is
x”+.05x’+x3=7.5cos(t)
Such an equation might model the motion of
a forced structure undergoing large elastic
deflections.
Here we chose the parameters so as to
see chaos thanks to work of Ueda in 1980.
Turn this into a system of 2 ODEs
with
x’(t)= y
y’(t)= 7.5cos(t) - .05x’ + x3
We use 2 Matlab m-files:
1
function dz=duffing(t,z)
dz=[z(2);(7.5*cos(t)-.05*z(2)-z(1)^3)];
%duffing oscillator
[t,z]=ode45('duffing',[0,100],[3,4]);
hold off
figure
plot(t,z(:,1))
hold on
[t,z]=ode45('duffing',[0,100],[3.01,4.01]);
plot(t,z(:,1),'r')
hold off
figure
[t,z]=ode45('duffing',[0,100],[3,4]);
plot(z(:,1),z(:,2))
hold on
[t,z]=ode45('duffing',[0,100],[3.01,4.01]);
plot(z(:,1),z(:,2),'r')
2
In the pictures below we first plot both
(x(t),y(t)) first starting at the point (3,4)
which is the blue curve and then starting
at the point (3.01,4.01) which is the red
curve. The butterfly effect is quite visible
in this example. There is divergence from
adjacent starts.
3
Then we plot the x-coordinates (the actual
solution x(t) to the original 2nd order ODE
first for the initial conditions
x(0)=3,
x’(0)=4, in blue and then for the initial
conditions x(0)=3.01
and x’(0)=4.01
in
red.
4
function dz=duffing(t,z)
dz=[z(2);(7.5*cos(t)-.05*z(2)-z(1)^3)];
%duffchaos.m
[t,z]=ode45('duffing',[0,100],[0,0]);
hold off
figure
plot(t,z(:,1))
hold on
[t,z]=ode45('duffing',[0,100],[0.01,0.01]);
plot(t,z(:,1),'r')
hold off
figure
[t,z]=ode45('duffing',[0,100],[0,0]);
plot(z(:,1),z(:,2))
hold on
[t,z]=ode45('duffing',[0,100],[0.01,0.01]);
plot(z(:,1),z(:,2),'r')
Here we redo the plots with the initial
point (x,y)=(0,0) meaning the initial
conditions x(0)=0=x’(0).
5
6
7