Orbital (or Poincaré) stability Any solution x(t) coming near in state space to an
orbital stable solution stays near at all (future) times. Poincaré introduced this
notion of stability to remedy the rather awkward fact that, for nonlinear systems,
even steady periodic oscillations can be unstable in the sense of Lyapunov.
2
Structural (or system) stability Structurally stable systems (of differential equations) retain their qualitative properties as the control parameters l are slightly perturbed (cf. Sect. 5.2.3). Bifurcations can occur only for structurally unstable systems.
Be careful to note that structural stability refers to perturbations of control parameters
l, whereas most other notions of stability refers to perturbations of solutions x(t).
Bifurcational stability A bifurcation is considered stable to a specified perturbation if
this perturbation does not qualitatively change the bifurcation diagram (cf. Sect. 5.9).
Some confusion may arise when comparing with certain texts dealing exclusively with linear physical systems and differential equations. There one may find
notions of stability similar to those given above, though with a quite different
meaning, as well as notions that have meaning only for linear systems. For linear
systems many features are implicitly assumed that do not hold for nonlinear systems. For example, linear systems cannot possess multiple solutions and finite
post-bifurcation behavior. There is at most one solution or state to be concerned
about (e.g., an equilibrium position), and thus concepts such as ‘system’, ‘equilibrium’ and ‘structure’ can be mixed or left undefined without causing confusion.
For example, some authors (e.g., Ziegler 1968) use the notion of structural
stability to characterize properties of real engineering structures, rather than the
structural properties of mathematical models. In this sense a column can be
‘structurally stable’, meaning that its design configuration is robust to small
perturbations.
Also, some use the term ‘system’ to mean a real physical system, rather than a
system of differential equations. In this sense ‘system stability’ implies robustness
to perturbations of some particular state associated with the physical system
concerned.
Further, for linear systems, concepts such as static and dynamic stability are used
to describe what we would call static and dynamic bifurcations respectively.
The relevant interpretation of terms usually appears from the context, and should
cause only little confusion. It is, however, desirable to employ concepts of stability
in a consistent manner. In this respect the notions given above are recommended.
These seem to be widely accepted across engineering and scientific disciplines, and
hold generally for linear and nonlinear systems.
2
Example: For an undamped pendulum the frequency of oscillation depends on the amplitude, if
this is finite. Hence two solutions initiated at slightly different amplitudes will oscillate at slightly
different frequencies. With time the two solutions will be running increasingly out of step, and are
thus unstable in the sense of Lyapunov, even though the corresponding phase plane orbits stay
close at all times.
5.10 Summing Up on Different Notions of Stability
297
orbital stable solution stays near at all (future) times. Poincaré introduced this
notion of stability to remedy the rather awkward fact that, for nonlinear systems,
even steady periodic oscillations can be unstable in the sense of Lyapunov.
2
Structural (or system) stability Structurally stable systems (of differential equations) retain their qualitative properties as the control parameters l are slightly perturbed (cf. Sect. 5.2.3). Bifurcations can occur only for structurally unstable systems.
Be careful to note that structural stability refers to perturbations of control parameters
l, whereas most other notions of stability refers to perturbations of solutions x(t).
Bifurcational stability A bifurcation is considered stable to a specified perturbation if
this perturbation does not qualitatively change the bifurcation diagram (cf. Sect. 5.9).
Some confusion may arise when comparing with certain texts dealing exclusively with linear physical systems and differential equations. There one may find
notions of stability similar to those given above, though with a quite different
meaning, as well as notions that have meaning only for linear systems. For linear
systems many features are implicitly assumed that do not hold for nonlinear systems. For example, linear systems cannot possess multiple solutions and finite
post-bifurcation behavior. There is at most one solution or state to be concerned
about (e.g., an equilibrium position), and thus concepts such as ‘system’, ‘equilibrium’ and ‘structure’ can be mixed or left undefined without causing confusion.
For example, some authors (e.g., Ziegler 1968) use the notion of structural
stability to characterize properties of real engineering structures, rather than the
structural properties of mathematical models. In this sense a column can be
‘structurally stable’, meaning that its design configuration is robust to small
perturbations.
Also, some use the term ‘system’ to mean a real physical system, rather than a
system of differential equations. In this sense ‘system stability’ implies robustness
to perturbations of some particular state associated with the physical system
concerned.
Further, for linear systems, concepts such as static and dynamic stability are used
to describe what we would call static and dynamic bifurcations respectively.
The relevant interpretation of terms usually appears from the context, and should
cause only little confusion. It is, however, desirable to employ concepts of stability
in a consistent manner. In this respect the notions given above are recommended.
These seem to be widely accepted across engineering and scientific disciplines, and
hold generally for linear and nonlinear systems.
2
Example: For an undamped pendulum the frequency of oscillation depends on the amplitude, if
this is finite. Hence two solutions initiated at slightly different amplitudes will oscillate at slightly
different frequencies. With time the two solutions will be running increasingly out of step, and are
thus unstable in the sense of Lyapunov, even though the corresponding phase plane orbits stay
close at all times.
5.10 Summing Up on Different Notions of Stability
297