5. Hopf-Andronov-Poincare bifurcation
In this section, we shall show that the system (2) undergoes a Hopf-Andronov-Poincare bifurcation by using as a bifurcation real parameter. Without loss of generality, suppose that is a function of and . Then system (2) becomes
(29) with .
System (29) can be written as
(30)
where , and is the bifurcation real parameter. The function is a on an open set in . Let be the set of equilibria of system (29). Suppose that be a sufficiently large open set containing such that for some . We are now interested to vary in order to observe the qualitative changes of solution trajectories near .
Corollary 1. Assume that Theorems [3-7] are hold. Then there is …show more content…
For the application of Hopf’s bifurcation theory to the system (29) (see [Marsden J.E., M. Mckracken, 1976]), it is required to satisfy the following transversality condition
(34)
Substituting , and into (31), and calculating the derivatives with respect to , we obtain
(35) where Since , we have , and .
Hence there is a Hopf bifurcation at . We have the following result:
Theorem 13. Suppose holds. Then the system (29) undergoes a Hopf bifurcation when passes through emanating from the steady state leading to periodic solutions trajectories for either (super-critical bifurcation), or (sub-critical-bifurcation) or at .
Next, Hopf bifurcation at :
The Jacobian matrix corresponding to is given by
(36)
where and its nonzero elements are defined as in (14) and (15) with & replaced by & respectively.
The characteristic equation corresponding to obeys
(37)
where the coefficients are defined as in (16).
By the Routh–Hurwitz criteria, all the roots of (37) will have negative real parts if , and
It is required for a Hopf bifurcation that either or must be violated. Suppose such
In this section, we shall show that the system (2) undergoes a Hopf-Andronov-Poincare bifurcation by using as a bifurcation real parameter. Without loss of generality, suppose that is a function of and . Then system (2) becomes
(29) with .
System (29) can be written as
(30)
where , and is the bifurcation real parameter. The function is a on an open set in . Let be the set of equilibria of system (29). Suppose that be a sufficiently large open set containing such that for some . We are now interested to vary in order to observe the qualitative changes of solution trajectories near .
Corollary 1. Assume that Theorems [3-7] are hold. Then there is …show more content…
For the application of Hopf’s bifurcation theory to the system (29) (see [Marsden J.E., M. Mckracken, 1976]), it is required to satisfy the following transversality condition
(34)
Substituting , and into (31), and calculating the derivatives with respect to , we obtain
(35) where Since , we have , and .
Hence there is a Hopf bifurcation at . We have the following result:
Theorem 13. Suppose holds. Then the system (29) undergoes a Hopf bifurcation when passes through emanating from the steady state leading to periodic solutions trajectories for either (super-critical bifurcation), or (sub-critical-bifurcation) or at .
Next, Hopf bifurcation at :
The Jacobian matrix corresponding to is given by
(36)
where and its nonzero elements are defined as in (14) and (15) with & replaced by & respectively.
The characteristic equation corresponding to obeys
(37)
where the coefficients are defined as in (16).
By the Routh–Hurwitz criteria, all the roots of (37) will have negative real parts if , and
It is required for a Hopf bifurcation that either or must be violated. Suppose such