152
4 Nonlinear Dynamics and Bifurcations in Autonomous RLC Circuits
Clearly, by increasing μ, there is a bifurcation with the appearance of two EPs at
the critical parameter value μ = g(x
0 ) and a bifurcation with the disappearance of
two EPs at the second critical parameter value μ = g(x 0 ).
Let us study in more detail the latter bifurcation. Note that g(x 0 ) > 0, g (x 0 ) = 0,
and g (x 0 ) < 0. By developing in Taylor series g in a neighborhood of x 0 we have
dx
dt
= μ − g(x 0 ) − g
(x 0 )(x − x 0 ) −
1
2
g
(x 0 )(x − x 0 )
2
= r −
1
2
g
(x 0 )(x − x 0 )
2
(4.8)
where we introduced the new parameter r = μ − g(x 0 ) and we omitted higher-order
terms of the expansion.
The situation in a small neighborhood of x 0 , and for three different values of
r, namely, r < 0 (μ < g(x 0 )), r = 0 (μ = g(x 0 )), and r > 0 (μ > g(x 0 )) is
represented in Fig. 4.21. When r < 0 there are two distinct EPs in a neighborhood
of x 0 and, from the dynamic route, it is seen that one is asymptotically stable while
the other is unstable. When r = 0 the two EPs collide and then annihilate each
other when r becomes positive. This situation corresponds to a typical saddle-node
bifurcation of EPs [3, Ch. 3]. The critical parameter value at which a saddle-node
bifurcation occurs is given by μ = g(x 0 ) (r = 0). The form in (4.8) is typical for a
first-order system undergoing a saddle-node bifurcation. In the bifurcation literature
this is referred to as a normal form for the saddle-node bifurcation.
4.4.2 Hopf Bifurcations
In this case we are dealing with an autonomous nonlinear system depending on a
parameter μ whose variation causes an EP to change its local stability properties.
The main issue is whether the change in stability of the EP can be associated
with the appearance of a periodic solution (a limit cycle). Roughly speaking, the
Hopf bifurcation theory says that if a pair of complex conjugate eigenvalues of
the linearization about the EP cross the imaginary axis as μ varies through certain
critical values, then for near-critical values of μ there are limit cycles close to the EP.
Example 4.9 Consider the second-order nonlinear autonomous circuit with a tunnel
diode shown in Fig. 4.22. The nonmonotone diode characteristic I D = g(V D ) is
shown in Fig. 4.23. The SEs of the circuit are easily written as
di L
dt
=
v C
L
dv C
dt
= −
i L
C
+
1
C
g(E − v C ).
4 Nonlinear Dynamics and Bifurcations in Autonomous RLC Circuits
Clearly, by increasing μ, there is a bifurcation with the appearance of two EPs at
the critical parameter value μ = g(x
0 ) and a bifurcation with the disappearance of
two EPs at the second critical parameter value μ = g(x 0 ).
Let us study in more detail the latter bifurcation. Note that g(x 0 ) > 0, g (x 0 ) = 0,
and g (x 0 ) < 0. By developing in Taylor series g in a neighborhood of x 0 we have
dx
dt
= μ − g(x 0 ) − g
(x 0 )(x − x 0 ) −
1
2
g
(x 0 )(x − x 0 )
2
= r −
1
2
g
(x 0 )(x − x 0 )
2
(4.8)
where we introduced the new parameter r = μ − g(x 0 ) and we omitted higher-order
terms of the expansion.
The situation in a small neighborhood of x 0 , and for three different values of
r, namely, r < 0 (μ < g(x 0 )), r = 0 (μ = g(x 0 )), and r > 0 (μ > g(x 0 )) is
represented in Fig. 4.21. When r < 0 there are two distinct EPs in a neighborhood
of x 0 and, from the dynamic route, it is seen that one is asymptotically stable while
the other is unstable. When r = 0 the two EPs collide and then annihilate each
other when r becomes positive. This situation corresponds to a typical saddle-node
bifurcation of EPs [3, Ch. 3]. The critical parameter value at which a saddle-node
bifurcation occurs is given by μ = g(x 0 ) (r = 0). The form in (4.8) is typical for a
first-order system undergoing a saddle-node bifurcation. In the bifurcation literature
this is referred to as a normal form for the saddle-node bifurcation.
4.4.2 Hopf Bifurcations
In this case we are dealing with an autonomous nonlinear system depending on a
parameter μ whose variation causes an EP to change its local stability properties.
The main issue is whether the change in stability of the EP can be associated
with the appearance of a periodic solution (a limit cycle). Roughly speaking, the
Hopf bifurcation theory says that if a pair of complex conjugate eigenvalues of
the linearization about the EP cross the imaginary axis as μ varies through certain
critical values, then for near-critical values of μ there are limit cycles close to the EP.
Example 4.9 Consider the second-order nonlinear autonomous circuit with a tunnel
diode shown in Fig. 4.22. The nonmonotone diode characteristic I D = g(V D ) is
shown in Fig. 4.23. The SEs of the circuit are easily written as
di L
dt
=
v C
L
dv C
dt
= −
i L
C
+
1
C
g(E − v C ).