There are other ways in which nonlinearities may create periodic oscillations out
of linearly unbounded motions. One is associated with dry friction, which is
responsible for the unpleasant squeals from, e.g., activated disk-breaks, train-wheels
in curved tracks, the teacher’s chalk across the blackboard or rusty door hinges – or
the more pleasant sounds from a bow and violin. And there is flutter and galloping,
that is, structural oscillations caused by fluids flowing across plates or strings, inside
tubes, etc. Here, linear theory predicts oscillations of the structure to grow
unbounded when the flow-speed exceeds a critical value. Typically however,
nonlinearity will limit the response – and you hear the ringing of airborne power
cables, the sound of an oboe or a human voice, see vibrations of the aircraft wing,
or have difficulties in controlling self-sustained oscillations of the garden water
hose. Nature disposes on a rich source of nonlinearities for turning nonperiodic
input into periodic oscillations.
Bifurcations As parameters are varied in a dynamical system, the stability of
equilibrium solutions may change, as may the number and character of solutions.
Such changes, representing marked qualitative shifts in system behavior, are called
bifurcations. The specific values of system parameters at which bifurcations take
place are then bifurcation values, or critical values. Approaching a bifurcation
value, the system approaches a critical state, at which arbitrarily small changes in
the parameters of the system may cause the magnitude and/or character of the
response to change drastically.
For example, in Fig. 3.10 the frequencies x A = x B , x C , and x D = x E are
bifurcation values. Marked changes of possible responses take place upon crossing
any of these frequencies, whereas in between only trivial changes in
response-magnitude occur. The corresponding points (x A , 0), (x C , 0), and (x D , a D )
at the response curve are called bifurcation points.
In decreasing x from above x C in Fig. 3.10, a stable limit cycle is born at x C
and the zero solution becomes unstable. This type of bifurcation is rather common
in problems of nonlinear dynamics. It is the Hopf bifurcation, named after the
famous mathematician who established conditions for its existence.
The Hopf bifurcation at (x C ,0) involves the transition stable equilibrium !
unstable equilibrium + stable limit cycle. This is called a supercritical or soft Hopf
bifurcation. The term ‘soft’ here serves to indicates that no drastic change of
response occurs across the bifurcation value. For the present case the amplitude
simply changes from zero to a non-zero but small value. Supercritical/soft bifurcations represent smooth and peaceable changes in system behavior.
The bifurcation at (x A ,0), by contrast, involves the transition unstable equilibrium ! stable equilibrium + unstable limit cycle. This too is a Hopf bifurcation,
since a limit cycle is created from an equilibrium, though, the Hopf bifurcation is in
this case subcritical or hard. Subcritical/hard Hopf bifurcations are more spectacular than their supercritical/soft counterparts, since a finite disturbance to the stable
equilibrium may throw the system beyond the unstable limit cycle. A real structure
might not survive the ‘hard’ effect of a subcritical Hopf bifurcation. We shall return
to Hopf- and several other bifurcations in Chap. 5.
3.6 The Forced Response – Multiple Scales Analysis
145
of linearly unbounded motions. One is associated with dry friction, which is
responsible for the unpleasant squeals from, e.g., activated disk-breaks, train-wheels
in curved tracks, the teacher’s chalk across the blackboard or rusty door hinges – or
the more pleasant sounds from a bow and violin. And there is flutter and galloping,
that is, structural oscillations caused by fluids flowing across plates or strings, inside
tubes, etc. Here, linear theory predicts oscillations of the structure to grow
unbounded when the flow-speed exceeds a critical value. Typically however,
nonlinearity will limit the response – and you hear the ringing of airborne power
cables, the sound of an oboe or a human voice, see vibrations of the aircraft wing,
or have difficulties in controlling self-sustained oscillations of the garden water
hose. Nature disposes on a rich source of nonlinearities for turning nonperiodic
input into periodic oscillations.
Bifurcations As parameters are varied in a dynamical system, the stability of
equilibrium solutions may change, as may the number and character of solutions.
Such changes, representing marked qualitative shifts in system behavior, are called
bifurcations. The specific values of system parameters at which bifurcations take
place are then bifurcation values, or critical values. Approaching a bifurcation
value, the system approaches a critical state, at which arbitrarily small changes in
the parameters of the system may cause the magnitude and/or character of the
response to change drastically.
For example, in Fig. 3.10 the frequencies x A = x B , x C , and x D = x E are
bifurcation values. Marked changes of possible responses take place upon crossing
any of these frequencies, whereas in between only trivial changes in
response-magnitude occur. The corresponding points (x A , 0), (x C , 0), and (x D , a D )
at the response curve are called bifurcation points.
In decreasing x from above x C in Fig. 3.10, a stable limit cycle is born at x C
and the zero solution becomes unstable. This type of bifurcation is rather common
in problems of nonlinear dynamics. It is the Hopf bifurcation, named after the
famous mathematician who established conditions for its existence.
The Hopf bifurcation at (x C ,0) involves the transition stable equilibrium !
unstable equilibrium + stable limit cycle. This is called a supercritical or soft Hopf
bifurcation. The term ‘soft’ here serves to indicates that no drastic change of
response occurs across the bifurcation value. For the present case the amplitude
simply changes from zero to a non-zero but small value. Supercritical/soft bifurcations represent smooth and peaceable changes in system behavior.
The bifurcation at (x A ,0), by contrast, involves the transition unstable equilibrium ! stable equilibrium + unstable limit cycle. This too is a Hopf bifurcation,
since a limit cycle is created from an equilibrium, though, the Hopf bifurcation is in
this case subcritical or hard. Subcritical/hard Hopf bifurcations are more spectacular than their supercritical/soft counterparts, since a finite disturbance to the stable
equilibrium may throw the system beyond the unstable limit cycle. A real structure
might not survive the ‘hard’ effect of a subcritical Hopf bifurcation. We shall return
to Hopf- and several other bifurcations in Chap. 5.
3.6 The Forced Response – Multiple Scales Analysis
145