Normal form reduction is a technique for systematically removing in-essential
nonlinear terms from a center manifold reduction. The technique involves a
near-identity nonlinear transformation of the critical variables, which is so chosen
as to make the transformed system as simple as possible. The result is a representation of the original system in so-called normal form. Normal forms exist and
are complete for codimension one bifurcations, and fairly complete for codimension
two problems.
The procedure for obtaining normal forms is considered outside the scope of this
introduction to bifurcation analysis. We refer instead to Guckenheimer and Holmes
(1983), Nayfeh (1993), Nayfeh and Balachandran (1995), and Troger and Steindl
(1991).
Except for the simplest cases, the computational burden associated with
obtaining normal forms may be overwhelming. Considering that the purpose of
normal form reduction is to simplify analysis, it could be worthwhile considering
alternatives, e.g., various perturbation methods. For example, the method of multiple scales (cf. Chaps. 3 and 4) also simplifies a system by eliminating ‘in-essential’
(non-secular) terms. Here, the identification of resonant terms can be viewed as a
search for nonlinear terms that are ‘essential’, in the context given. Again, the
double pendulum treated in Sect. 4.5 may serve to illustrate this: The original
system (4.52) is four-dimensional and highly complicated. However, a multiple
scales analysis (Sect. 4.5.3) reduces the system to the simple pair of Eqs. (4.73)–
(4.73), which in turn is similar to the generic system (5.10) describing a Hopf
bifurcation.
5.7 Bifurcating Periodic Solutions
Periodic solutions may bifurcate in response to changes in control parameters, just
as equilibrium solutions may do. Again, linearized eigenvalues govern the nature of
the bifurcations, and center manifold theory and normal forms can be employed for
studying particular unfoldings (e.g., Guckenheimer and Holmes 1983; Troger and
Steindl 1991; Nayfeh and Balachandran 1995). We now briefly mention some
frequently encountered bifurcations of this type.
Symmetry-breaking Bifurcation Plotting the amplitude of periodic oscillations
versus the control parameter, this bifurcation is similar to the pitchfork bifurcation
of an equilibrium (Fig. 5.1). Though, in this case the zero solution of the pitchfork
is replaced by a smooth curve in the upper half-plane. This curve corresponds to
symmetric oscillations, whereas the branches of the bifurcation correspond to
asymmetric oscillations. As with the pitchfork, the symmetry-breaking bifurcation
can be supercritical or subcritical.
Cyclic-fold Bifurcation Here the amplitude of periodic oscillations bifurcates in a
manner similar to the saddle-node bifurcation of equilibriums (Fig. 5.3), though
with the bifurcation point located in the upper half-plane. Thus, the cyclic-fold is
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5 Bifurcation Analysis
nonlinear terms from a center manifold reduction. The technique involves a
near-identity nonlinear transformation of the critical variables, which is so chosen
as to make the transformed system as simple as possible. The result is a representation of the original system in so-called normal form. Normal forms exist and
are complete for codimension one bifurcations, and fairly complete for codimension
two problems.
The procedure for obtaining normal forms is considered outside the scope of this
introduction to bifurcation analysis. We refer instead to Guckenheimer and Holmes
(1983), Nayfeh (1993), Nayfeh and Balachandran (1995), and Troger and Steindl
(1991).
Except for the simplest cases, the computational burden associated with
obtaining normal forms may be overwhelming. Considering that the purpose of
normal form reduction is to simplify analysis, it could be worthwhile considering
alternatives, e.g., various perturbation methods. For example, the method of multiple scales (cf. Chaps. 3 and 4) also simplifies a system by eliminating ‘in-essential’
(non-secular) terms. Here, the identification of resonant terms can be viewed as a
search for nonlinear terms that are ‘essential’, in the context given. Again, the
double pendulum treated in Sect. 4.5 may serve to illustrate this: The original
system (4.52) is four-dimensional and highly complicated. However, a multiple
scales analysis (Sect. 4.5.3) reduces the system to the simple pair of Eqs. (4.73)–
(4.73), which in turn is similar to the generic system (5.10) describing a Hopf
bifurcation.
5.7 Bifurcating Periodic Solutions
Periodic solutions may bifurcate in response to changes in control parameters, just
as equilibrium solutions may do. Again, linearized eigenvalues govern the nature of
the bifurcations, and center manifold theory and normal forms can be employed for
studying particular unfoldings (e.g., Guckenheimer and Holmes 1983; Troger and
Steindl 1991; Nayfeh and Balachandran 1995). We now briefly mention some
frequently encountered bifurcations of this type.
Symmetry-breaking Bifurcation Plotting the amplitude of periodic oscillations
versus the control parameter, this bifurcation is similar to the pitchfork bifurcation
of an equilibrium (Fig. 5.1). Though, in this case the zero solution of the pitchfork
is replaced by a smooth curve in the upper half-plane. This curve corresponds to
symmetric oscillations, whereas the branches of the bifurcation correspond to
asymmetric oscillations. As with the pitchfork, the symmetry-breaking bifurcation
can be supercritical or subcritical.
Cyclic-fold Bifurcation Here the amplitude of periodic oscillations bifurcates in a
manner similar to the saddle-node bifurcation of equilibriums (Fig. 5.3), though
with the bifurcation point located in the upper half-plane. Thus, the cyclic-fold is
292
5 Bifurcation Analysis