the center manifold reduction (5.31): To second order this system is identical to the
generic system (5.8) for the transcritical bifurcation.
Seemingly, for this example we could just as well let y = 0 in the first of
Eqs. (5.27) for obtaining the center manifold reduction (5.31). However, since
x appears in the third equation of (5.27), one cannot deduce that y will tend to zero,
even though the eigenvalues have all negative real parts.
5.5.5 Summing Up on Center Manifold Reduction
The center manifold theorem constitutes a basis for systematically reducing the
dimension of a system of differential equations. The reduced system describes the
qualitative dynamics on a submanifold that is associated with linearized eigenvalues
having zero real parts. This can be used to examine local bifurcations of
higher-dimensional nonlinear systems. An example was given for illustrative purposes. More realistic application examples can be found in, e.g., Guckenheimer and
Holmes (1983), Holmes (1977), Païdoussis and Semler (1993), Li and Païdoussis
(1994), Troger and Steindl (1991), Nayfeh (1993), and Nayfeh and Balachandran
(1995).
The reduction process was described in terms of bifurcating equilibriums. It can
be extended to study also bifurcations of periodic solutions (e.g., Nayfeh and
Balachandran 1995). Though, for this purpose, one can instead employ a perturbation approach to obtain the modulation equations describing time evolution of
oscillation amplitudes (cf. Chaps. 3 and 4), and then examine the bifurcating
equilibriums of the modulation equations.
Center manifold reductions can alternatively be obtained by using perturbation
analysis. For example, the double pendulum treated in Sect. 4.5 obeys the highly
complicated four-dimensional set of equations (4.524.5). A multiple scales analysis
(Sect. 4.5.3) performed near the Hopf bifurcation set reduces the system to (4.73)–
(4.73), which is two-dimensional. This is the minimal dimension required for
capturing Hopf bifurcations. The reduced system is a center manifold reduction, as
indicated by the comment ‘disregarding damped terms’ just above (4.66), which
corresponds to ignoring dimensions needed only to describe motions on the stable
manifold.
5.6 Normal Form Reduction
Having reduced the essential dynamics of a system to a low-dimensional center
manifold, the reduced system may still be very complicated. Typically a high
number of nonlinear terms appear, of which only a few are essential for describing
bifurcations of the system.
5.5 Center Manifold Reduction
291
generic system (5.8) for the transcritical bifurcation.
Seemingly, for this example we could just as well let y = 0 in the first of
Eqs. (5.27) for obtaining the center manifold reduction (5.31). However, since
x appears in the third equation of (5.27), one cannot deduce that y will tend to zero,
even though the eigenvalues have all negative real parts.
5.5.5 Summing Up on Center Manifold Reduction
The center manifold theorem constitutes a basis for systematically reducing the
dimension of a system of differential equations. The reduced system describes the
qualitative dynamics on a submanifold that is associated with linearized eigenvalues
having zero real parts. This can be used to examine local bifurcations of
higher-dimensional nonlinear systems. An example was given for illustrative purposes. More realistic application examples can be found in, e.g., Guckenheimer and
Holmes (1983), Holmes (1977), Païdoussis and Semler (1993), Li and Païdoussis
(1994), Troger and Steindl (1991), Nayfeh (1993), and Nayfeh and Balachandran
(1995).
The reduction process was described in terms of bifurcating equilibriums. It can
be extended to study also bifurcations of periodic solutions (e.g., Nayfeh and
Balachandran 1995). Though, for this purpose, one can instead employ a perturbation approach to obtain the modulation equations describing time evolution of
oscillation amplitudes (cf. Chaps. 3 and 4), and then examine the bifurcating
equilibriums of the modulation equations.
Center manifold reductions can alternatively be obtained by using perturbation
analysis. For example, the double pendulum treated in Sect. 4.5 obeys the highly
complicated four-dimensional set of equations (4.524.5). A multiple scales analysis
(Sect. 4.5.3) performed near the Hopf bifurcation set reduces the system to (4.73)–
(4.73), which is two-dimensional. This is the minimal dimension required for
capturing Hopf bifurcations. The reduced system is a center manifold reduction, as
indicated by the comment ‘disregarding damped terms’ just above (4.66), which
corresponds to ignoring dimensions needed only to describe motions on the stable
manifold.
5.6 Normal Form Reduction
Having reduced the essential dynamics of a system to a low-dimensional center
manifold, the reduced system may still be very complicated. Typically a high
number of nonlinear terms appear, of which only a few are essential for describing
bifurcations of the system.
5.5 Center Manifold Reduction
291