The Hopf theorem includes the notion of a center manifold, to which we shall
return in the next section. It appears that if a subsystem restricted to such a center
manifold (defined by x, c 0 , …, c 3 ) can be found, then we are able not only to prove
the existence of limit cycles, but also to compute their shape as defined by the
paraboloid equation r
2 = −(c 0 /c 3 )l.
However, the theorem provides no clues as to how this center manifold should
be computed; it merely says that there is one. For many systems the determination
of a suitable restriction to the center manifold can be a substantial undertaking, even
though only the coefficient c 3 is needed (c 0 is quite easy to obtain). Partly due to this
reason, the Hopf theorem is mostly used as a guide for where to look for periodic
solutions for a given system. Having located a set of system parameters causing
limit cycle behavior, a variety of other methods are at our disposal for determining
their shape, for example, the perturbation methods described in Chaps. 3 and 4.
5.5 Center Manifold Reduction
The center manifold theorem provides a basis for systematically reducing the
dimension of a system, while retaining its essential properties. For a nonlinear
system of arbitrary dimension the linearized Jacobian eigenvalues may have positive, negative and zero real parts. As has already been demonstrated the ‘interesting’
dynamics – i.e. the qualitative change in system behavior – is associated with the
subset of eigenvalues having zero real parts.
And that is what center manifold theory is all about: to throw away dimensions
that just serve to describe ‘uninteresting’, hyperbolic dynamics. This has nothing to
do with the common engineering practice of truncating discrete models to fit a
given level of accuracy or computational power. For example, to capture with
acceptable precision the behavior of a pinned-pinned column in response to a given
load, you may need a Galerkin discretization retaining five modes, or a finite
element model with 500 degrees of freedom. However, to capture the qualitative
behavior, i.e. the bifurcations to be expected, a single autonomous differential
equation may suffice (e.g., (5.2), (5.7) or (5.8)).
It is possible to proceed quite successfully through many kinds of nonlinear
analysis without even knowing the existence of center manifolds. However, since they
are so important in modern bifurcation analysis, and are entering into engineering
literature, one should have at least some basic knowledge of their applicability.
This section, and the subsequent one on normal forms, is intended only to
present the main ideas. Guckenheimer and Holmes (1983) provide a rather detailed
discussion of center manifolds, as do Nayfeh (1993), Nayfeh and Balachandran
(1995) and Troger and Steindl (1991). A pedagogical presentation with applications
(for flow-induced oscillations) was provided by Holmes (1977), specifically
addressing the engineering community. The exoticness of the subject in engineering
contexts is reflected by a tendency, in scientific papers, to summarize and explain
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5 Bifurcation Analysis
return in the next section. It appears that if a subsystem restricted to such a center
manifold (defined by x, c 0 , …, c 3 ) can be found, then we are able not only to prove
the existence of limit cycles, but also to compute their shape as defined by the
paraboloid equation r
2 = −(c 0 /c 3 )l.
However, the theorem provides no clues as to how this center manifold should
be computed; it merely says that there is one. For many systems the determination
of a suitable restriction to the center manifold can be a substantial undertaking, even
though only the coefficient c 3 is needed (c 0 is quite easy to obtain). Partly due to this
reason, the Hopf theorem is mostly used as a guide for where to look for periodic
solutions for a given system. Having located a set of system parameters causing
limit cycle behavior, a variety of other methods are at our disposal for determining
their shape, for example, the perturbation methods described in Chaps. 3 and 4.
5.5 Center Manifold Reduction
The center manifold theorem provides a basis for systematically reducing the
dimension of a system, while retaining its essential properties. For a nonlinear
system of arbitrary dimension the linearized Jacobian eigenvalues may have positive, negative and zero real parts. As has already been demonstrated the ‘interesting’
dynamics – i.e. the qualitative change in system behavior – is associated with the
subset of eigenvalues having zero real parts.
And that is what center manifold theory is all about: to throw away dimensions
that just serve to describe ‘uninteresting’, hyperbolic dynamics. This has nothing to
do with the common engineering practice of truncating discrete models to fit a
given level of accuracy or computational power. For example, to capture with
acceptable precision the behavior of a pinned-pinned column in response to a given
load, you may need a Galerkin discretization retaining five modes, or a finite
element model with 500 degrees of freedom. However, to capture the qualitative
behavior, i.e. the bifurcations to be expected, a single autonomous differential
equation may suffice (e.g., (5.2), (5.7) or (5.8)).
It is possible to proceed quite successfully through many kinds of nonlinear
analysis without even knowing the existence of center manifolds. However, since they
are so important in modern bifurcation analysis, and are entering into engineering
literature, one should have at least some basic knowledge of their applicability.
This section, and the subsequent one on normal forms, is intended only to
present the main ideas. Guckenheimer and Holmes (1983) provide a rather detailed
discussion of center manifolds, as do Nayfeh (1993), Nayfeh and Balachandran
(1995) and Troger and Steindl (1991). A pedagogical presentation with applications
(for flow-induced oscillations) was provided by Holmes (1977), specifically
addressing the engineering community. The exoticness of the subject in engineering
contexts is reflected by a tendency, in scientific papers, to summarize and explain
284
5 Bifurcation Analysis