2. The Jacobian matrix has a pair of purely imaginary eigenvalues while all of its other eigenvalues have non zero real parts at . Hence, the point is a nonhyperbolic fixed point

3. For , let the analytical continuation of the pair of imaginary eigenvalues be . Then, at . This condition is known as the transversality condition as the eigenvalue crosses the imaginary axis with nonzero speed.

When all the above three conditions are satisfied, a periodic solution of period is developed at . Such bifurcations are called Hopf bifurcation or Poincare'-Andronov Hopf bifurcation.

The normal form for a generic Hopf bifurcation of a fixed point is given as follows.

...............................................................................(4.2.13)

...............................................................................(4.2.14)

One can obtain the fixed point at (0,0) for all values of μ. The Jacobian matrix corresponding to this equilibrium point can be given by

...............................................................................................................(4.2.15)

Hence, the eigenvalues are obtained by solving the equation . So, the eigenvalues are and . For control parameter , a pair of complex conjugate eigenvalues cross the imaginary axis at non-zero values as and . Hence, the three conditions for the Hopf bifurcations are satisfied at the fixed point (0,0). The period of the bifurcating periodic solution at (0,0;0) is .

In polar form these two equations can be written by substituting and as follows.

.........................................................................................(4.2.16)

For steady state . Substituting , it can be observed that the system has both trivial and non-trivial solution. The trivial solution corresponds to the equilibrium point or the fixed-point response ( ) and the nontrivial fixed point corresponds to a periodic solution with amplitude r and frequency which is created due to the Hopf bifurcation.