then the limit sets may change as well. Typically a small change in l produces only
small quantitative changes in a limit set. For example, a slight change of l could
cause the location of a singular point to be displaced somewhat, or modify the shape
of a periodic orbit.
However, for certain values of l even a slight change in l might set off more
drastic changes in limit behavior. For example, the number and/or stability of
singular points could change, a periodic orbit could appear, disappear or gain or
lose stability – or a chaotic attractor could appear, disappear or change character.
Such qualitative shifts in behavior are called bifurcations. A value of l for which a
bifurcation occurs is called a bifurcation value or critical value. If at some bifurcation value l = ~
l the stationary state is x = ~ x, then (~ l, ~ x) is a bifurcation point.
A bifurcation set consists of the union of all bifurcation values in l-space.
5.2.3 Bifurcation Conditions: Structural Instability
Bifurcations can occur only when a system is structurally unstable. A structurally
stable system retains its qualitative properties even if its parameters are slightly
perturbed. A structurally unstable system does not.
Structural stability has nothing to do with ‘engineering stability’, ‘Lyapunov
stability’, or other common descriptors of the attracting or repelling properties of
given limit sets. These notions of stability characterize properties of specific
solutions ~ x in response to slight perturbations in x-space. Structural stability, by
contrast, characterizes properties of the limit sets of a system fðx; lÞ in response to
slight perturbations in l-space. Thus, a structurally stable system may well possess
unstable limit sets, insofar as these do not change in number, character or stability
when system parameters are slightly varied.
Consider, as an example, a lightly damped pendulum in gravity, but otherwise
unforced. The state of upside-down equilibrium is unstable in the sense of
Lyapunov, since a small perturbation causes the pendulum to escape from that state.
The system is nevertheless structurally stable, since slight perturbations to the
parameters of the system (natural frequency and damping) do not modify the
qualitative behavior of the pendulum. However, with no damping the pendulum
system becomes structurally unstable, since the dynamics of an undamped pendulum differ qualitatively from that of a pendulum subjected to even infinitesimal
amounts of damping.
A system is structurally unstable if any of its singular points are non-hyperbolic.
A singular point ~ x is hyperbolic (or non-degenerate) if the Jacobian evaluated at the
point, Jð~ x; lÞ ¼ ð@f=@xÞj x¼~ x , has no eigenvalues with zero real part. So, any singular point for which the Jacobian has an eigenvalue with zero real part is
non-hyperbolic (or degenerate).
5.2 Systems, Bifurcations, and Bifurcation Conditions
273
small quantitative changes in a limit set. For example, a slight change of l could
cause the location of a singular point to be displaced somewhat, or modify the shape
of a periodic orbit.
However, for certain values of l even a slight change in l might set off more
drastic changes in limit behavior. For example, the number and/or stability of
singular points could change, a periodic orbit could appear, disappear or gain or
lose stability – or a chaotic attractor could appear, disappear or change character.
Such qualitative shifts in behavior are called bifurcations. A value of l for which a
bifurcation occurs is called a bifurcation value or critical value. If at some bifurcation value l = ~
l the stationary state is x = ~ x, then (~ l, ~ x) is a bifurcation point.
A bifurcation set consists of the union of all bifurcation values in l-space.
5.2.3 Bifurcation Conditions: Structural Instability
Bifurcations can occur only when a system is structurally unstable. A structurally
stable system retains its qualitative properties even if its parameters are slightly
perturbed. A structurally unstable system does not.
Structural stability has nothing to do with ‘engineering stability’, ‘Lyapunov
stability’, or other common descriptors of the attracting or repelling properties of
given limit sets. These notions of stability characterize properties of specific
solutions ~ x in response to slight perturbations in x-space. Structural stability, by
contrast, characterizes properties of the limit sets of a system fðx; lÞ in response to
slight perturbations in l-space. Thus, a structurally stable system may well possess
unstable limit sets, insofar as these do not change in number, character or stability
when system parameters are slightly varied.
Consider, as an example, a lightly damped pendulum in gravity, but otherwise
unforced. The state of upside-down equilibrium is unstable in the sense of
Lyapunov, since a small perturbation causes the pendulum to escape from that state.
The system is nevertheless structurally stable, since slight perturbations to the
parameters of the system (natural frequency and damping) do not modify the
qualitative behavior of the pendulum. However, with no damping the pendulum
system becomes structurally unstable, since the dynamics of an undamped pendulum differ qualitatively from that of a pendulum subjected to even infinitesimal
amounts of damping.
A system is structurally unstable if any of its singular points are non-hyperbolic.
A singular point ~ x is hyperbolic (or non-degenerate) if the Jacobian evaluated at the
point, Jð~ x; lÞ ¼ ð@f=@xÞj x¼~ x , has no eigenvalues with zero real part. So, any singular point for which the Jacobian has an eigenvalue with zero real part is
non-hyperbolic (or degenerate).
5.2 Systems, Bifurcations, and Bifurcation Conditions
273