When a 6 ¼ 0 the second Eq. in (3.246) gives _
u ¼ 0; with a constant solution
u(t) = u 0 , while letting _
a ¼ 0 in the first equation gives a nontrivial stationary
solution a = a 0 = 2. Substituting this into (3.244), the approximate solution to
(3.243) for small e then becomes:
uðtÞ ¼ 2 sinðt þ u 0 Þ; _
uðtÞ ¼ 2 cosðt þ u 0 Þ; e ( 1;
ð3:247Þ
which is seen to be independent on e (but still assumes small e), and is stable for all
positive values of e. It forms a limit cycle in the ðu; _
uÞ phase plane, to which orbits
starting from any nonzero initial condition will be attracted, the quicker the higher
the value of e. It appears from (3.247) that u
2
þ _
u
2
¼ 2
2
; meaning that in the ðu; _
uÞ
phase plane the limit cycle forms a circle of radius 2.
Fig. 3.25 shows solutions to the van der Pol equation, as obtained by direct
numerical simulation of (3.243). The upper left figure depicts a post-transient time
series for a small value of e = 0.1. It has a time-harmonic/sinusoidal shape with
amplitude 2, while the corresponding stationary phase plane orbits in the upper
right figure approaches the close-to-circular limit cycle of radius 2, i.e. as predicted
by the approximate perturbation solution (3.247) for small e.
The two lower figures in Fig. 3.25 shows numerical solutions when e = 1, i.e.
not small. With this much stronger level of damping nonlinearity the time series
appears with clear harmonic distortion, i.e. not as a pure sinusoidal; a Fourier
transform of the time series would show non-zero components not only at the
fundamental linear oscillation frequency x = 1, but as well at the higher harmonics,
x = 3, 5, … (odd harmonics only, due to the u-symmetry of (3.243), which is
invariant to the transformation u ! –u). The effect of nonlinearity is even more
visible in the phase plane plot (lower right figure), where the stable limit cycle
(thick line) is clearly non-circular. The nonlinear distortion, increasing with
increasing e, is completely absent in the approximate perturbation solution (3.247).
Nevertheless, the approximate predictions, with an oscillation amplitude of 2, are at
least roughly agreeing with numerical simulation even for the evidently non-small
value of e = 1.
For even larger values of the damping parameter, e ) 1, the oscillations change
shape towards resembling a train of rectangular pulses (not shown), that is: For
most of the oscillation period u changes very slowly, while the change from positive to negative values occurs almost instantaneously; this is called relaxation
oscillations.
3.8.2 The Rayleigh Oscillator
A nondimensional form of the Rayleigh oscillator equation u = u(t) is:
€ u À eð1 À
1
3
_
u
2
Þ _
u þ u ¼ 0;
ð3:248Þ
3.8 Two More Classical Nonlinear Oscillators
173
u ¼ 0; with a constant solution
u(t) = u 0 , while letting _
a ¼ 0 in the first equation gives a nontrivial stationary
solution a = a 0 = 2. Substituting this into (3.244), the approximate solution to
(3.243) for small e then becomes:
uðtÞ ¼ 2 sinðt þ u 0 Þ; _
uðtÞ ¼ 2 cosðt þ u 0 Þ; e ( 1;
ð3:247Þ
which is seen to be independent on e (but still assumes small e), and is stable for all
positive values of e. It forms a limit cycle in the ðu; _
uÞ phase plane, to which orbits
starting from any nonzero initial condition will be attracted, the quicker the higher
the value of e. It appears from (3.247) that u
2
þ _
u
2
¼ 2
2
; meaning that in the ðu; _
uÞ
phase plane the limit cycle forms a circle of radius 2.
Fig. 3.25 shows solutions to the van der Pol equation, as obtained by direct
numerical simulation of (3.243). The upper left figure depicts a post-transient time
series for a small value of e = 0.1. It has a time-harmonic/sinusoidal shape with
amplitude 2, while the corresponding stationary phase plane orbits in the upper
right figure approaches the close-to-circular limit cycle of radius 2, i.e. as predicted
by the approximate perturbation solution (3.247) for small e.
The two lower figures in Fig. 3.25 shows numerical solutions when e = 1, i.e.
not small. With this much stronger level of damping nonlinearity the time series
appears with clear harmonic distortion, i.e. not as a pure sinusoidal; a Fourier
transform of the time series would show non-zero components not only at the
fundamental linear oscillation frequency x = 1, but as well at the higher harmonics,
x = 3, 5, … (odd harmonics only, due to the u-symmetry of (3.243), which is
invariant to the transformation u ! –u). The effect of nonlinearity is even more
visible in the phase plane plot (lower right figure), where the stable limit cycle
(thick line) is clearly non-circular. The nonlinear distortion, increasing with
increasing e, is completely absent in the approximate perturbation solution (3.247).
Nevertheless, the approximate predictions, with an oscillation amplitude of 2, are at
least roughly agreeing with numerical simulation even for the evidently non-small
value of e = 1.
For even larger values of the damping parameter, e ) 1, the oscillations change
shape towards resembling a train of rectangular pulses (not shown), that is: For
most of the oscillation period u changes very slowly, while the change from positive to negative values occurs almost instantaneously; this is called relaxation
oscillations.
3.8.2 The Rayleigh Oscillator
A nondimensional form of the Rayleigh oscillator equation u = u(t) is:
€ u À eð1 À
1
3
_
u
2
Þ _
u þ u ¼ 0;
ð3:248Þ
3.8 Two More Classical Nonlinear Oscillators
173