.......................................................(4.2.25)

Now perturbing the above equations, one obtains

...................................................... (4.2.26)

where T is the transpose and is the Jacobian matrix whose eigenvalues will determine the stability and bifurcation of the system. The stability boundary for the linear system (i.e. the trival state) can be obtained from the eigenvalues of the matrix [ ] by letting . A detailed numerical results can be found in the work of Kar and Dwivedy (1999).

Exercise Problems

1.Study the stability of the following one dimensional systems by plotting the stable and unstable branches. Also, write about the type of bifurcation observed in the system.

(a) (b) (c) (d) (e) (f)

2. Develop the equation for steady state solution and the Jacobian matrix from the reduced equations obtained by solving the Duffing equation using method of multiple scales as given below. Plot the frequency response curve with stable and unstable branches with different colours.

Answer: Equation for frequency response is given by

The Jacobian matrix is