3. Find the Jacobian matrix for the system given in equations (4.2.6) and (4.2.7).

4. Find the Jacobian matrix for the system given by the following equations.

,

.

Answer: The Jacobian matrix is given by

5. Write a Matlab code to plot time response and phase portrait of a periodic response (a) (b) .

Solution

t=0:0.01:10;
y=5*sin(5*t);
yt=25*cos(5*t);
y1=5*(sin(5*t)+sin(10*t));
yt1=5*(5*cos(5*t)+10*cos(10*t));
figure(1)
subplot(2,2,1) plot(t,y)
xlabel('t')
ylabel('y')
subplot(2,2,2)
plot(y,yt)
xlabel('t')
ylabel('y')
subplot(2,2,3)
plot(t,y1)
xlabel('t')
ylabel('y1')
subplot(2,2,4) plot(y1,yt1)
xlabel('y') ylabel('yt1')

 

Reference:

1.  A. H. Nayfeh, and B. Balachandran, Applied Nonlinear Dynamics , Wiley, 1995.

2.  R. C. Kar and S. K. Dwivedy, Non-linear dynamics of a slender beam carrying a lumped mass with principal parametric and internal resonances. International Journal of Nonlinear Mechanics, 34, ( 3), 515-529, 1999.

3.  J.E. Marsden and M. McCracken, The Hopf Bifurcation and its Applications, Springer-Verlag, New York, 1976