Thus, structural stability breaks down and bifurcations can occur at those values
of l producing a system with one or more singular points having at least one
Jacobian eigenvalue with a zero real part.
Note that structural instability is a necessary condition for bifurcations to occur.
That is, a system must be structurally unstable for bifurcations to occur, but
structural instability does not automatically imply bifurcation.
For example, the unforced pendulum system possesses Jacobian eigenvalues
having zero eigenvalue when there is no damping, b = 0 (cf. Sect. 3.4.4). Hence for
b = 0 the system is structurally unstable, meaning that the addition of slight
damping (perturbation to b = 0) may change the qualitative properties of the system. Indeed, it was found in Sect. 3.4.4 that the addition of even infinitesimal
damping changes the singular points at h = p2p from centers to foci.
Assume now that we have located the bifurcation set of a given system fðx; lÞ.
Thus, we know those values of system parameters l for which structural stability
breaks down and bifurcations may occur that will change the qualitative behavior of
the system. Then how does the behavior of the system change? To answer this we
need to unfold the bifurcations, that is, to picture somehow the change in system
behavior in response to changes in l. We do this by drawing bifurcation diagrams.
In most cases this can be accomplished only locally, that is, for each individual
bifurcation value, and for small changes in l. Connecting local pictures of bifurcations together into a global bifurcation diagram can be a painstaking procedure, if
at all possible.
We next summarize the properties of the simplest of all local bifurcations, those
of codimension one. Though simple, codimension one bifurcations find applications for real systems of high dimension as well. This holds true when the essential
dynamics of a higher-dimensional system can be reduced to a one-dimensional
center manifold, as described in Sects. 5.4 and 5.5.
5.3 Codimension One Bifurcations of Equilibriums
As the name suggests, codimension one bifurcations can be described on a
one-dimensional manifold; it takes only a single parameter to unfold them. The
codimension of a bifurcation is the smallest dimension of a parameter space that
contains the bifurcation in a persistent way. The codimension is at least as large as
the number of Jacobian eigenvalues having a zero real part (a purely imaginary pair
counting as one). If there are no eigenvalues having a zero real part the codimension
is zero, and no bifurcations occur. If there is a simple zero eigenvalue or a purely
imaginary pair, the codimension is (usually) one, and bifurcations occur that can be
unfolded through a single parameter.
If there is more than one Jacobian eigenvalue having a zero real part, then
bifurcations of codimension two or higher can take place. These are more complicated, requiring at least two parameters for their unfolding. Hence, rather than the
actual dimension n of a system fðx; lÞ, it is the number of eigenvalues having zero
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5 Bifurcation Analysis
of l producing a system with one or more singular points having at least one
Jacobian eigenvalue with a zero real part.
Note that structural instability is a necessary condition for bifurcations to occur.
That is, a system must be structurally unstable for bifurcations to occur, but
structural instability does not automatically imply bifurcation.
For example, the unforced pendulum system possesses Jacobian eigenvalues
having zero eigenvalue when there is no damping, b = 0 (cf. Sect. 3.4.4). Hence for
b = 0 the system is structurally unstable, meaning that the addition of slight
damping (perturbation to b = 0) may change the qualitative properties of the system. Indeed, it was found in Sect. 3.4.4 that the addition of even infinitesimal
damping changes the singular points at h = p2p from centers to foci.
Assume now that we have located the bifurcation set of a given system fðx; lÞ.
Thus, we know those values of system parameters l for which structural stability
breaks down and bifurcations may occur that will change the qualitative behavior of
the system. Then how does the behavior of the system change? To answer this we
need to unfold the bifurcations, that is, to picture somehow the change in system
behavior in response to changes in l. We do this by drawing bifurcation diagrams.
In most cases this can be accomplished only locally, that is, for each individual
bifurcation value, and for small changes in l. Connecting local pictures of bifurcations together into a global bifurcation diagram can be a painstaking procedure, if
at all possible.
We next summarize the properties of the simplest of all local bifurcations, those
of codimension one. Though simple, codimension one bifurcations find applications for real systems of high dimension as well. This holds true when the essential
dynamics of a higher-dimensional system can be reduced to a one-dimensional
center manifold, as described in Sects. 5.4 and 5.5.
5.3 Codimension One Bifurcations of Equilibriums
As the name suggests, codimension one bifurcations can be described on a
one-dimensional manifold; it takes only a single parameter to unfold them. The
codimension of a bifurcation is the smallest dimension of a parameter space that
contains the bifurcation in a persistent way. The codimension is at least as large as
the number of Jacobian eigenvalues having a zero real part (a purely imaginary pair
counting as one). If there are no eigenvalues having a zero real part the codimension
is zero, and no bifurcations occur. If there is a simple zero eigenvalue or a purely
imaginary pair, the codimension is (usually) one, and bifurcations occur that can be
unfolded through a single parameter.
If there is more than one Jacobian eigenvalue having a zero real part, then
bifurcations of codimension two or higher can take place. These are more complicated, requiring at least two parameters for their unfolding. Hence, rather than the
actual dimension n of a system fðx; lÞ, it is the number of eigenvalues having zero
274
5 Bifurcation Analysis