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Fig. 2.15
Phase diagram of
the Van der Pol oscillator for
D 0:2
2.5
2
1.5
1
0.5
0
−0.5
−1
−1.5
−2
−2.5
−2.5
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5
x
1
Fig. 2.16
Phase diagram of
the Van der Pol oscillator for
D 1
3
2
1
0
−1
−2
−3
−2.5
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5
x
1
Theorem 1 (Poincaré-Bendixson).
If a trajectory of the nonlinear system of
Eq. (
2.65
) remains in a finite region
˝
, then one of the following is true: (i) the
trajectory goes to an equilibrium point, (ii) the trajectory tends to a limit cycle, (iii)
the trajectory is itself a limit cycle.
Moreover, the following theorem provides a sufficient condition for the nonexis-
tence of limit cycles:
Theorem 2 (Bendixson).
For the nonlinear system of Eq. (
2.65
), no limit cycle can
exist in a region
˝
of the phase plane in which
@f
1
@f
2
@x
1
C
@x
2
does not vanish and does
not change sign.
