also referred to as a tangent or turning point bifurcation. The lack of solutions on
the one side of this bifurcation does not imply that no solutions exist for the
corresponding values of the control parameter, but only that such solutions cannot
be described locally. To describe the post-bifurcation state one must usually rely on
numerical solutions. This may reveal the existence of a remote disconnected attractor (point, periodic, chaotic), an unbounded solution, or a ‘large’ attractor
corresponding to state variables much larger in magnitude than the pre-bifurcational
values.
Transcritical Bifurcation The amplitude of periodic oscillations here bifurcates as
for the transcritical bifurcation of equilibriums (Fig. 5.4), though, with the bifurcation point located in the upper half-plane. At transcritical bifurcations two
solutions meet and ‘exchange stability’.
Period-doubling (or flip) Bifurcation As with the symmetry-breaking bifurcation,
a branch corresponding to periodic oscillations gains or loses stability upon continuously passing through the point of bifurcation. In addition the bifurcation creates a new state of periodic oscillations in which the period of oscillations is
doubled. The period-doubling bifurcation can be supercritical (with a stable
period-doubled solution) or subcritical (with an unstable period-doubled solution).
Period-doublings frequently occur as precursors to chaos (cf. Chap. 6).
Secondary Hopf (or Neimark) Bifurcation Recall from Sect. 5.3.4 that a Hopf
bifurcation corresponds to the creation of a periodic solution from an equilibrium
(Fig. 5.5(a)). The amplitudes of oscillations can then be perceived as new equilibriums (for the equations governing modulation of amplitudes), as illustrated in
Fig. 5.5(b). Now, in response to further variations of the control parameter, such an
amplitude equilibrium can also experience a Hopf bifurcations. Since the equilibrium itself was created by a Hopf bifurcation, the new bifurcation is naturally called
secondary. The outcome of a secondary Hopf bifurcation take the form of oscillations at a single frequency x 1 (determined by the first Hopf bifurcation), with
amplitudes that vary periodically with another frequency x 2 (as determined by the
secondary bifurcation). Typically the two frequencies x 1 and x 2 are incommensurate (i.e. x 1 /x 2 is an irrational number) and the oscillations are then quasiperiodic. Quasiperiodic oscillations caused by secondary Hopf bifurcations often occur
as precursors to chaos (cf. Chap. 6).
5.8 Grouping Bifurcations According to their Effect
At this stage we have encountered all possible local codimension one bifurcations
of equilibriums: pitchfork, saddle-node, transcritical and Hopf. The pitchfork and
Hopf bifurcations come in two variants: supercritical and subcritical. Further, we
have mentioned some local bifurcations of periodic solutions: Symmetry-breaking,
cyclic-fold, transcritical, period-doubling (or flip) and secondary Hopf (or
5.7 Bifurcating Periodic Solutions
293
the one side of this bifurcation does not imply that no solutions exist for the
corresponding values of the control parameter, but only that such solutions cannot
be described locally. To describe the post-bifurcation state one must usually rely on
numerical solutions. This may reveal the existence of a remote disconnected attractor (point, periodic, chaotic), an unbounded solution, or a ‘large’ attractor
corresponding to state variables much larger in magnitude than the pre-bifurcational
values.
Transcritical Bifurcation The amplitude of periodic oscillations here bifurcates as
for the transcritical bifurcation of equilibriums (Fig. 5.4), though, with the bifurcation point located in the upper half-plane. At transcritical bifurcations two
solutions meet and ‘exchange stability’.
Period-doubling (or flip) Bifurcation As with the symmetry-breaking bifurcation,
a branch corresponding to periodic oscillations gains or loses stability upon continuously passing through the point of bifurcation. In addition the bifurcation creates a new state of periodic oscillations in which the period of oscillations is
doubled. The period-doubling bifurcation can be supercritical (with a stable
period-doubled solution) or subcritical (with an unstable period-doubled solution).
Period-doublings frequently occur as precursors to chaos (cf. Chap. 6).
Secondary Hopf (or Neimark) Bifurcation Recall from Sect. 5.3.4 that a Hopf
bifurcation corresponds to the creation of a periodic solution from an equilibrium
(Fig. 5.5(a)). The amplitudes of oscillations can then be perceived as new equilibriums (for the equations governing modulation of amplitudes), as illustrated in
Fig. 5.5(b). Now, in response to further variations of the control parameter, such an
amplitude equilibrium can also experience a Hopf bifurcations. Since the equilibrium itself was created by a Hopf bifurcation, the new bifurcation is naturally called
secondary. The outcome of a secondary Hopf bifurcation take the form of oscillations at a single frequency x 1 (determined by the first Hopf bifurcation), with
amplitudes that vary periodically with another frequency x 2 (as determined by the
secondary bifurcation). Typically the two frequencies x 1 and x 2 are incommensurate (i.e. x 1 /x 2 is an irrational number) and the oscillations are then quasiperiodic. Quasiperiodic oscillations caused by secondary Hopf bifurcations often occur
as precursors to chaos (cf. Chap. 6).
5.8 Grouping Bifurcations According to their Effect
At this stage we have encountered all possible local codimension one bifurcations
of equilibriums: pitchfork, saddle-node, transcritical and Hopf. The pitchfork and
Hopf bifurcations come in two variants: supercritical and subcritical. Further, we
have mentioned some local bifurcations of periodic solutions: Symmetry-breaking,
cyclic-fold, transcritical, period-doubling (or flip) and secondary Hopf (or
5.7 Bifurcating Periodic Solutions
293