unperturbed system. Thus, the supercritical pitchfork bifurcation associated with
(5.2) is unstable to the perturbation e.
Real physical systems typically possess imperfections of some kind, which can
be modeled as perturbations to the corresponding ideal system. For example, in the
system (5.33), x could describe the deviation of the centerline of a column from the
straight configuration, in response to a load l. The perturbation e would then
describe an imperfection of the column, say, an initial curvature or localized buckle.
An initially straight column (e = 0) responds to increased loading as depicted in
Fig. 5.10(b). That is, beyond the critical loading (l = 0 in this case) the straight
configuration loses stability in favor a symmetric pair of buckled configurations.
However, with an initial deviation from the straight configuration (e 6 ¼ 0) the
column responds as in Fig. 5.10(c) (or (a)), that is: Increased compressive loading
just causes the column to deform further in the direction already determined by the
initial imperfection, without the sudden loss of stability characterizing the ideal case
e = 0. Beyond the bifurcation value there are two possible stable configurations
which, however, are not symmetric as for the case e = 0. With small imperfections
(e ( 1) the response is close to that of the perfect case, with the difference that one
of stable branches is favored over the other.
5.10 Summing Up on Different Notions of Stability
By now we have encountered the term ‘stability’ in so many contexts, and with so
many different meanings, that clearing up the concepts seems appropriate. The
below list summarizes the different notions and their interpretation for continuous
systems of differential equations, _
x = f(t,x,l), where x(t) is a vector of state variables and l a vector of control parameters. Rigorous mathematical definitions can
be found elsewhere, e.g., in the references given in the chapter introduction.
Bounded (or Lagrange) stability Requires a solution just to remain within finite
limits, however large.
Lyapunov (or marginal or neutral or meta) stability Any solution x(t) coming
near a Lyapunov stable solution ~ x(t) stays near at all (future) times.
Uniform stability Same as Lyapunov stability, though only used with autonomous
systems.
Asymptotic (or strict) stability Requires x(t) ! ~ x(t) for t ! ∞ (and implies
Lyapunov stability).
Engineering stability Same as asymptotic stability (because that is what engineering safety is very much about: keeping structures and machines where they are
put, in the state for which they were designed).
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5 Bifurcation Analysis
(5.2) is unstable to the perturbation e.
Real physical systems typically possess imperfections of some kind, which can
be modeled as perturbations to the corresponding ideal system. For example, in the
system (5.33), x could describe the deviation of the centerline of a column from the
straight configuration, in response to a load l. The perturbation e would then
describe an imperfection of the column, say, an initial curvature or localized buckle.
An initially straight column (e = 0) responds to increased loading as depicted in
Fig. 5.10(b). That is, beyond the critical loading (l = 0 in this case) the straight
configuration loses stability in favor a symmetric pair of buckled configurations.
However, with an initial deviation from the straight configuration (e 6 ¼ 0) the
column responds as in Fig. 5.10(c) (or (a)), that is: Increased compressive loading
just causes the column to deform further in the direction already determined by the
initial imperfection, without the sudden loss of stability characterizing the ideal case
e = 0. Beyond the bifurcation value there are two possible stable configurations
which, however, are not symmetric as for the case e = 0. With small imperfections
(e ( 1) the response is close to that of the perfect case, with the difference that one
of stable branches is favored over the other.
5.10 Summing Up on Different Notions of Stability
By now we have encountered the term ‘stability’ in so many contexts, and with so
many different meanings, that clearing up the concepts seems appropriate. The
below list summarizes the different notions and their interpretation for continuous
systems of differential equations, _
x = f(t,x,l), where x(t) is a vector of state variables and l a vector of control parameters. Rigorous mathematical definitions can
be found elsewhere, e.g., in the references given in the chapter introduction.
Bounded (or Lagrange) stability Requires a solution just to remain within finite
limits, however large.
Lyapunov (or marginal or neutral or meta) stability Any solution x(t) coming
near a Lyapunov stable solution ~ x(t) stays near at all (future) times.
Uniform stability Same as Lyapunov stability, though only used with autonomous
systems.
Asymptotic (or strict) stability Requires x(t) ! ~ x(t) for t ! ∞ (and implies
Lyapunov stability).
Engineering stability Same as asymptotic stability (because that is what engineering safety is very much about: keeping structures and machines where they are
put, in the state for which they were designed).
296
5 Bifurcation Analysis