Solving and plotting a piecewise ODE - matlab

Question

I have a function dφ/dt = γ - F(φ) (where F(φ) -- a is 2π-periodic function) and the graph of the function F(φ).
I need to create a program that outputs 6 plots of φ(t) for different values of γ (γ = 0.1, 0.5, 0.95, 1.05, 2, 5), and t∈[0,100].
Here is the definition of the F(φ) function:
-φ/a - π/a, if φ ∈ [-π, -π + a]
-1, if φ ∈ [-π + a, - a]
F(φ) = φ/a, if φ ∈ [- a, a]
1, if φ ∈ [a, π - a]
-φ/a + π/a, if φ ∈ [π - a, π]
^ F(φ)
|
|1 ______
| /| \
| / | \
| / | \ φ
__-π_______-a____|/___|________\π____>
\ | /|0 a
\ | / |
\ | / |
\ |/ |
¯¯¯¯¯¯ |-1
My problem is I don't know what inputs to give ode45 in terms of the bounds and the initial condition. What I do know is that the evolution of φ(t) must be continuous.
This is the code for case of γ = 0.1:
hold on;
df1dt = #(t,f1) 0.1 - f1 - 3.14;
df2dt = #(t,f2)- 1;
df3dt = #(t,f3) 0.1 + f3;
df4dt = #(t,f4)+1;
df5dt = #(t,f5) 0.1 - f5 + 3.14;
[T1,Y1] = ode45(df1dt, ...);
[T2,Y2] = ode45(df2dt, ...);
[T3,Y3] = ode45(df3dt, ...);
[T4,Y4] = ode45(df4dt, ...);
[T5,Y5] = ode45(df5dt, ...);
plot(T1,Y1);
plot(T2,Y2);
plot(T3,Y3);
plot(T4,Y4);
plot(T5,Y5);
hold off;
title('\gamma = 0.1')

Answer 1

Let us first define F(φ,a):
function out = F(p, a)
phi = mod(p,2*pi);
out = (0 <= phi & phi < a ).*(phi/a) ...
+ (a <= phi & phi < pi-a ).*(1) ...
+ (pi-a <= phi & phi < pi+a ).*(-phi/a + pi/a) ...
+ (pi+a <= phi & phi < 2*pi-a).*(-1) ...
+ (2*pi-a <= phi & phi < 2*pi ).*(phi/a - 2*pi/a);
end
Which for some example inputs gives:
using the plotting code:
x = linspace(-3*pi, 3*pi, 200);
a = pi/6;
figure(); plot(x,F(x, a));
xlim([-3*pi,3*pi]);
xticks(-3*pi:pi:3*pi);
xticklabels((-3:3)+ "\pi");
grid on; grid minor
ax = gca;
ax.XAxis.MinorTick = 'on';
ax.XAxis.MinorTickValues = ax.XAxis.Limits(1):pi/6:ax.XAxis.Limits(2);
From there you don't need to bother with the ranges anymore, and simply call ode45:
% Preparations:
a = pi/6;
g = [0.1, 0.5, 0.95, 1.05, 2, 5]; % γ
phi0 = 0; % you need to specify the correct initial condition (!)
tStart = 0;
tEnd = 100;
% Calling the solver:
[t, phi] = arrayfun(#(x)ode45(#(t,p)x-F(p,a), [tStart, tEnd], phi0), g, 'UniformOutput', false);
% Plotting:
plotData = [t; phi];
figure(); plot(plotData{:});
legend("γ=" + g, 'Location', 'northwest');
Resulting in: