foci, whereas those corresponding to odd k are saddles. Again, one could sketch the
phase plane orbits of Fig. 3.7 by using this information (and perhaps a bit of
imagination and experience).
In Summary, Singular points for n = 2 The singular points of
two-dimensional systems divide themselves into nodes, saddles, centers and foci.
To classify a particular singular point, one computes the Jacobian of the system,
inserting the singular point. The eigenvalues of this Jacobian determine the type of
the singular point, as summarized in Fig. 3.8. If the eigenvalues are real and equal,
it may be necessary to compute also the Jacobian eigenvectors, or at least to
determine whether these are linearly dependent or not.
Linear systems have at most one singular point. Classifying the type and stability
of this point, the qualitative flow of orbits is revealed for the entire phase plane.
Nonlinear systems may have any number of singular points. These points can be
classified through linearization, that is, by evaluating eigenvalues of the Jacobian (if
non-singular) at each singular point in turn. This will reveal the local arrangement
of orbits in the immediate vicinity of the singular points.
For systems of arbitrary dimension, linear or nonlinear, the stability of a singular
point can be determined by examining the Jacobian eigenvalue having the largest
real part. If the largest real part is positive (negative), then the point is unstable
(stable). For nonlinear systems, this method yields a local measure of stability,
describing the consequences of small perturbations to orbits near a singular point.
3.5 Quantitative Analysis
3.5.1 Approximate Methods
This section presents three commonly applied methods for approximately solving
nonlinear systems: the method of multiple scales, the method of averaging, and the
method of harmonic balance. The two first of these are perturbation methods, that
is, they work by applying small nonlinear perturbations to linearized solutions.
Their application is restricted to weakly nonlinear systems, so the nonlinear terms
should be small compared to linear terms
3 . Usually this is the case when motions
are finite, but not very large. The correctness of perturbation solutions typically
decreases for growing amplitudes of motion.
Other perturbation methods exist than those described here. To some extent, the
choice of perturbation method for a particular problem is a matter of personal taste
and habit. None of the existing perturbation methods appear to be superior in
general. The methods to be presented seem to be those most widely used; they will
3
Sometimes this assumption can be relaxed, e.g. Lakrad and Belhaq (2002) shows how to employ
the multiple scales perturbation method for strongly nonlinear systems, by expressing the solutions
in terms of Jacobian elliptic functions; see also Burton and Rahman (1986), Thomsen (2008b).
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3 Nonlinear Vibrations: Classical Local Theory
phase plane orbits of Fig. 3.7 by using this information (and perhaps a bit of
imagination and experience).
In Summary, Singular points for n = 2 The singular points of
two-dimensional systems divide themselves into nodes, saddles, centers and foci.
To classify a particular singular point, one computes the Jacobian of the system,
inserting the singular point. The eigenvalues of this Jacobian determine the type of
the singular point, as summarized in Fig. 3.8. If the eigenvalues are real and equal,
it may be necessary to compute also the Jacobian eigenvectors, or at least to
determine whether these are linearly dependent or not.
Linear systems have at most one singular point. Classifying the type and stability
of this point, the qualitative flow of orbits is revealed for the entire phase plane.
Nonlinear systems may have any number of singular points. These points can be
classified through linearization, that is, by evaluating eigenvalues of the Jacobian (if
non-singular) at each singular point in turn. This will reveal the local arrangement
of orbits in the immediate vicinity of the singular points.
For systems of arbitrary dimension, linear or nonlinear, the stability of a singular
point can be determined by examining the Jacobian eigenvalue having the largest
real part. If the largest real part is positive (negative), then the point is unstable
(stable). For nonlinear systems, this method yields a local measure of stability,
describing the consequences of small perturbations to orbits near a singular point.
3.5 Quantitative Analysis
3.5.1 Approximate Methods
This section presents three commonly applied methods for approximately solving
nonlinear systems: the method of multiple scales, the method of averaging, and the
method of harmonic balance. The two first of these are perturbation methods, that
is, they work by applying small nonlinear perturbations to linearized solutions.
Their application is restricted to weakly nonlinear systems, so the nonlinear terms
should be small compared to linear terms
3 . Usually this is the case when motions
are finite, but not very large. The correctness of perturbation solutions typically
decreases for growing amplitudes of motion.
Other perturbation methods exist than those described here. To some extent, the
choice of perturbation method for a particular problem is a matter of personal taste
and habit. None of the existing perturbation methods appear to be superior in
general. The methods to be presented seem to be those most widely used; they will
3
Sometimes this assumption can be relaxed, e.g. Lakrad and Belhaq (2002) shows how to employ
the multiple scales perturbation method for strongly nonlinear systems, by expressing the solutions
in terms of Jacobian elliptic functions; see also Burton and Rahman (1986), Thomsen (2008b).
116
3 Nonlinear Vibrations: Classical Local Theory