Question

2. The following ODE model (for the Duffing oscillator) describes the motion of a damped spring driven by a periodic force: r(0) = zo (a) Rewrite the second order non-autonomous system in one independent variable above as an autonomous system in three independent variables: x, y and r, where: y-r ano T 1, with T(0)-0 (b) Fix the parameter values of α = 1, β-0, δ 0.05, w-1. Additionally, fix the initial conditions 2(0)-10, z'(0) . For the values of γ below, plot the the behavior of x(t) againstt over the time interval [0,400]. Upload your graphs to Sakai. I recommend using Python (and the scipy. integrate.solve.ivp function) to produce the graphs. You do not need to upload your code if you choose to code the solution to this problem. Produce your graphs using the following values for γ: γ-0.0, 1.5, 4.0, 7.5 (c) What appears to be the long term behavior of the solution for each value of γ?

Answer 1

Matlab code for solving ode clear all close all tAnswering question a %Initial conditions for ode x0-[10:0]; gall parameter values a-1; b-0; d- 0.05; w-1; %a11 gamma values gamma-[0 1.5 4 7.5]; for ii-1:length (gamma) g-gamma(ii); %minimum and maximum time span tspan [0 400]; Solution of ODEs using ode45 matlab function sol- ode45 ( (t,x) odefcnl(t,x,a, b,g,d,w), tspan, x0); %Equally splitting time into .02 sec interval for 0 to 50 tllinspace (tspan(1),tspan (end),10001); %yy is the corresponding x y v and z 1 deva 1 ( sol , ti); %plotting x(t) vs t figure(ii) plot (tl,yyl(l,:)) title (sprintf('Plot for x (t) vs. t for gamma-$2.2f',g)) xlabel('time') ylabel('x(t)) end %Function for evaluating the ODE function dudt odefcnl(t, x, a,b,g,d,w) here a-alpha; b-beta; grgamma; d- delta; w-w eql-x (2); eq2-9"cos ( w* t )-d-x(2)-br x ( 1 )-a*(x(1),-3; Evaluate the ODE for our present problem

Plot for x(t) vs. t for gamma-0.00 10 10 50 100 150 200 time 250 300 350 400 Plat for x(t) vs. t for gamma-1.50 10 10 50 100 150 250 350 400 time

Plot for x(t) vs. t for gamma 4.00 10 10 350 50 100 150 200 250 300 400 Plat for x(t) vs. t for gamma-7.50 10 10 150 350 50 100 250 300 400