Interpretation:
To show
To show the Jacobian matrix for the linearization has the form
Here, A and B are
By considering trace and determinant of matrix
To show that depending on the sizes of g and T, the determinant of matrix
By using computer, show that Hopf bifurcation can be supercritical.
Concept Introduction:
To find fixed point of the system, put
Jacobian matrix is used to check the stability of the fixed points, and it is given as:
The system which is settling down to equilibrium by exponentially damping and its decay rate depends on a control parameter
Answer to Problem 17E
Solution:
It is shown that the Jacobian matrix for the linearization has the form
It is shown that all eigenvalues of the matrix
It is shown that values of g and T determine whether the determinant of matrix
It is shown that the system goes through supercritical Hopf bifurcation.
Explanation of Solution
a)
The system equations are:
Assuming that
Here, the condition of
In above equation, L.H.S. and R.H.S. are zero and positive respectively when
Thus, the curves intersect and have unique symmetric solution.
b)
The Jacobian matrix is:
To prove determinant law,
Expanding along last column,
Again expanding along bottom row,
Therefore,
This result is used for calculation of eigenvalues of the
c)
Using determinant is product of the eigenvalues and trace is sum of eigenvalues,
But
Here,
From the above result,
The trace
Thus,
d)
For the matrix
The determinant
And trace
From the above results, the determinant is positive if
Trace can negative or positive depending upon values of b and T.
From the above conclusions of eigenvalues, pitchfork or Hopf bifurcation can occur at the point
By adjusting parameter g, if initially trace is negative, and by varying parameter g, its value is shifting towards positive value, then the pitchfork bifurcation occurs at
By adjusting parameter T, if the determinant is positive, by changing the parameter T, the trace of the matrix changes its sign, then the Hopf bifurcation occurs at a
e)
The plots of
For T being very small, the system is nearly settled down to fixed point
Increasing the value of T, the stable limit cycles just appear.
Further increasing the value of T, the size of the limit cycle increases.
Therefore, the system goes through supercritical Hopf bifurcation.
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Chapter 8 Solutions
Nonlinear Dynamics and Chaos
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