Neimark), with supercritical and subcritical variants of the symmetry-breaking and
cyclic-fold bifurcations.
These bifurcations account for many of the qualitative changes in behavior
experienced with physical systems, but not for all. There are codimension two and
higher bifurcations, bifurcations of periodic solutions other than those already
mentioned, and global bifurcations. Below we summarize (from Nayfeh and
Balachandran 1995) some commonly encountered notions of bifurcations,
describing their general effects rather than their particular unfolding.
Static bifurcations involve only singular points, i.e. equilibrium solutions,
whereas dynamic bifurcations involve at least one branch representing motion of
the system. The pitchfork, saddle-node and transcritical bifurcations are static
bifurcations, whereas the Hopf bifurcation is dynamic.
Some bifurcations are continuous whereas other are discontinuous (or catastrophic), dependent on whether the states of the system vary continuously or discontinuously with the control parameters. The pitchfork bifurcation is continuous
whereas the saddle-node is discontinuous.
During a dangerous bifurcation the current attractor (point, periodic or chaotic)
of a system suddenly disappears, and the state jumps to a remote disconnected
attractor. Dangerous bifurcations, also called blue sky catastrophes, are always
discontinuous. The remote attractor can be bounded (point, periodic or chaotic) or
unbounded. Whatever the case, such non-smooth changes in behavior may represent dangers to the life of real physical systems and their environment. Reversing
the change in control parameters, the state may remain on the remote attractor well
below the critical value, thus giving rise to hysteresis (different outcome dependent
on the direction of parameter change). Saddle-node bifurcations and all kinds of
subcritical bifurcations are usually considered to be dangerous.
During an explosive bifurcation, which is too discontinuous, the current attractor
(point, periodic or chaotic) ‘explodes’ into a larger attractor. The new attractor
includes the old one in form of a ghost or phantom attractor. Reversing the change
in control parameters, the large attractor implodes into the original one.
5.9 On the Stability of Bifurcations to Perturbations
A bifurcation is said to be stable to a specified perturbation if this perturbation does
not cause qualitative changes to the bifurcation diagram. We consider two
examples.
5.9.1 Stability of a Saddle-node Bifurcation
We first consider the stability of the saddle-node bifurcation for the generic system
(5.7). By adding to (5.7) a finite perturbation ex one obtains:
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5 Bifurcation Analysis
cyclic-fold bifurcations.
These bifurcations account for many of the qualitative changes in behavior
experienced with physical systems, but not for all. There are codimension two and
higher bifurcations, bifurcations of periodic solutions other than those already
mentioned, and global bifurcations. Below we summarize (from Nayfeh and
Balachandran 1995) some commonly encountered notions of bifurcations,
describing their general effects rather than their particular unfolding.
Static bifurcations involve only singular points, i.e. equilibrium solutions,
whereas dynamic bifurcations involve at least one branch representing motion of
the system. The pitchfork, saddle-node and transcritical bifurcations are static
bifurcations, whereas the Hopf bifurcation is dynamic.
Some bifurcations are continuous whereas other are discontinuous (or catastrophic), dependent on whether the states of the system vary continuously or discontinuously with the control parameters. The pitchfork bifurcation is continuous
whereas the saddle-node is discontinuous.
During a dangerous bifurcation the current attractor (point, periodic or chaotic)
of a system suddenly disappears, and the state jumps to a remote disconnected
attractor. Dangerous bifurcations, also called blue sky catastrophes, are always
discontinuous. The remote attractor can be bounded (point, periodic or chaotic) or
unbounded. Whatever the case, such non-smooth changes in behavior may represent dangers to the life of real physical systems and their environment. Reversing
the change in control parameters, the state may remain on the remote attractor well
below the critical value, thus giving rise to hysteresis (different outcome dependent
on the direction of parameter change). Saddle-node bifurcations and all kinds of
subcritical bifurcations are usually considered to be dangerous.
During an explosive bifurcation, which is too discontinuous, the current attractor
(point, periodic or chaotic) ‘explodes’ into a larger attractor. The new attractor
includes the old one in form of a ghost or phantom attractor. Reversing the change
in control parameters, the large attractor implodes into the original one.
5.9 On the Stability of Bifurcations to Perturbations
A bifurcation is said to be stable to a specified perturbation if this perturbation does
not cause qualitative changes to the bifurcation diagram. We consider two
examples.
5.9.1 Stability of a Saddle-node Bifurcation
We first consider the stability of the saddle-node bifurcation for the generic system
(5.7). By adding to (5.7) a finite perturbation ex one obtains:
294
5 Bifurcation Analysis