140
4 Nonlinear Dynamics and Bifurcations in Autonomous RLC Circuits
Note that the resistor is locally active, i.e., ˆ
v (i) < 0 for small |i|, while it is
eventually passive, i.e., vi = i ˆ
v(i) > 0 for large |i|.
The SEs are easily derived as
dv C
dt
= −
i L
C
di L
dt
=
v C − ˆ
v(i L )
L
.
It is seen that the circuit has a unique EP at the origin, i.e., ( ¯
v C , ¯
i L ) = (0, 0), and
that the Jacobian of the vector field defining the SEs at the EP is given by
J =
0 −
1
C
1
L −
ˆ
v (0)
L
.
We have
T = trJ = −
ˆ
v (0)
L
and
Δ = det J =
1
LC
.
Because T > 0 and Δ > 0, it follows that the eigenvalues of J are real positive,
or they are complex conjugate with positive real part. In any case the unique EP is
unstable and it repels nearby solutions [5].
Consider now the nonlinear resistor characteristic. Since ˆ
v(0) = 0 and ˆ
v (0) < 0,
for small |i| we have vi < 0 so that the resistor supplies electric energy to the L − C
circuit. This physically explains why solutions can depart from the EP at the origin
and head to infinity. However, due to (4.3), for large |i| we have vi < 0, i.e., the
resistor absorbs electric energy from the L − C circuit. Then, the initial outward
motion of the trajectory will be damped out by losses due to power dissipated inside
the resistor when the trajectory is sufficiently far out. Soon, the trajectory must
“grind to a halt” and start “falling” back toward the origin. Now, since there are
no EPs other than the unstable EP at the origin, there is no way for a trajectory to
come at rest and then it must set into an oscillation.
Summing up, two physical mechanisms are necessary for producing oscillations
using the circuit structure in Fig. 4.9:
1. An unstable EP which repels nearby trajectories. This in turn requires that the
nonlinear resistor must be active at least in a small neighborhood around the EP.
2. A dissipative mechanism which restrains the trajectories from running away to
infinity. This in turn requires that the nonlinear resistor is eventually passive.
4 Nonlinear Dynamics and Bifurcations in Autonomous RLC Circuits
Note that the resistor is locally active, i.e., ˆ
v (i) < 0 for small |i|, while it is
eventually passive, i.e., vi = i ˆ
v(i) > 0 for large |i|.
The SEs are easily derived as
dv C
dt
= −
i L
C
di L
dt
=
v C − ˆ
v(i L )
L
.
It is seen that the circuit has a unique EP at the origin, i.e., ( ¯
v C , ¯
i L ) = (0, 0), and
that the Jacobian of the vector field defining the SEs at the EP is given by
J =
0 −
1
C
1
L −
ˆ
v (0)
L
.
We have
T = trJ = −
ˆ
v (0)
L
and
Δ = det J =
1
LC
.
Because T > 0 and Δ > 0, it follows that the eigenvalues of J are real positive,
or they are complex conjugate with positive real part. In any case the unique EP is
unstable and it repels nearby solutions [5].
Consider now the nonlinear resistor characteristic. Since ˆ
v(0) = 0 and ˆ
v (0) < 0,
for small |i| we have vi < 0 so that the resistor supplies electric energy to the L − C
circuit. This physically explains why solutions can depart from the EP at the origin
and head to infinity. However, due to (4.3), for large |i| we have vi < 0, i.e., the
resistor absorbs electric energy from the L − C circuit. Then, the initial outward
motion of the trajectory will be damped out by losses due to power dissipated inside
the resistor when the trajectory is sufficiently far out. Soon, the trajectory must
“grind to a halt” and start “falling” back toward the origin. Now, since there are
no EPs other than the unstable EP at the origin, there is no way for a trajectory to
come at rest and then it must set into an oscillation.
Summing up, two physical mechanisms are necessary for producing oscillations
using the circuit structure in Fig. 4.9:
1. An unstable EP which repels nearby trajectories. This in turn requires that the
nonlinear resistor must be active at least in a small neighborhood around the EP.
2. A dissipative mechanism which restrains the trajectories from running away to
infinity. This in turn requires that the nonlinear resistor is eventually passive.