5 Bifurcation Analysis
5.1 Introduction
Bifurcations mark the qualitative changes in system behavior that may occur when
the parameters of a system are varied. For example, for a damped pendulum a
bifurcation occurs when the damping parameter is changed from zero to a small
positive value, because an undamped pendulum behaves in a qualitatively different
way from a damped pendulum.
Bifurcation analysis is then concerned with why, when and how bifurcations
occur. Bifurcation analysis can be cumbersome, though mostly worth the effort. The
real strength of it comes in at an early stage of analysis, the one at which the case to
analyze is decided upon. Bifurcation analysis reveals the critical cases and the kinds
of qualitative shifts in behavior to be expected, and are thus particularly relevant
where these are not immediately obvious. Further, even a basic knowledge on
bifurcations helps in recognizing and understanding phenomena that occur universally across a variety of mathematical models. For example, the Hopf bifurcation
(introduced already in Chap. 3) creates periodic oscillations out of equilibriums,
and appears universally for physical, chemical, biological, economical and other
systems. It is of course desirable to know the conditions under which such a
dramatic change can take place.
The full story of bifurcation theory is long, mathematically involved, and far
from complete. Initiated by Poincaré (1899), it remains an area of highly active
research. This chapter is intended to provide only a brief introduction to notions and
tools. We restrict ourselves to a rather simple and well-defined class of bifurcations,
the so-called codimension one bifurcations of equilibriums. Besides being simple,
they appear with so many systems that one should be able to recognize them when
they occur (which they do for most examples of this book).
A rather pragmatic approach is chosen, and readers with a preference for
mathematical rigor are referred to, e.g., Guckenheimer and Holmes (1983), Hale
and Kocak (1991), Jackson (1991) or Troger and Steindl (1991). Nayfeh and
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
J. J. Thomsen, Vibrations and Stability,
https://doi.org/10.1007/978-3-030-68045-9_5
271
5.1 Introduction
Bifurcations mark the qualitative changes in system behavior that may occur when
the parameters of a system are varied. For example, for a damped pendulum a
bifurcation occurs when the damping parameter is changed from zero to a small
positive value, because an undamped pendulum behaves in a qualitatively different
way from a damped pendulum.
Bifurcation analysis is then concerned with why, when and how bifurcations
occur. Bifurcation analysis can be cumbersome, though mostly worth the effort. The
real strength of it comes in at an early stage of analysis, the one at which the case to
analyze is decided upon. Bifurcation analysis reveals the critical cases and the kinds
of qualitative shifts in behavior to be expected, and are thus particularly relevant
where these are not immediately obvious. Further, even a basic knowledge on
bifurcations helps in recognizing and understanding phenomena that occur universally across a variety of mathematical models. For example, the Hopf bifurcation
(introduced already in Chap. 3) creates periodic oscillations out of equilibriums,
and appears universally for physical, chemical, biological, economical and other
systems. It is of course desirable to know the conditions under which such a
dramatic change can take place.
The full story of bifurcation theory is long, mathematically involved, and far
from complete. Initiated by Poincaré (1899), it remains an area of highly active
research. This chapter is intended to provide only a brief introduction to notions and
tools. We restrict ourselves to a rather simple and well-defined class of bifurcations,
the so-called codimension one bifurcations of equilibriums. Besides being simple,
they appear with so many systems that one should be able to recognize them when
they occur (which they do for most examples of this book).
A rather pragmatic approach is chosen, and readers with a preference for
mathematical rigor are referred to, e.g., Guckenheimer and Holmes (1983), Hale
and Kocak (1991), Jackson (1991) or Troger and Steindl (1991). Nayfeh and
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
J. J. Thomsen, Vibrations and Stability,
https://doi.org/10.1007/978-3-030-68045-9_5
271