154
4 Nonlinear Dynamics and Bifurcations in Autonomous RLC Circuits
Fig. 4.22 Circuit with tunnel
diode exhibiting a Hopf
bifurcation
Fig. 4.23 Main parameters
of a tunnel diode
characteristic
Assume for simplicity C = 1, L = 1, and let x 1 = i L , x 2 = v C . Then the SEs
become
dx 1
dt
= x 2
dx 2
dt
= −x 1 + g(μ − x 2 )
where the battery voltage E is chosen as a parameter μ.
For any μ there is a unique EP ( ¯
x 1 , ¯
x 2 ) = (g(μ), 0). The change of variables
y 1 = x 1 − g(μ), y 2 = x 2 yields the SEs
dy 1
dt
= y 2
(4.9)
dy 2
dt
= −y 1 − g(μ) + g(μ − y 2 )
(4.10)
which have a unique EP at (0, 0) for any μ. The Jacobian of the vector field defining
the SEs evaluated at the EP is
J =
0
1
−1 −g (μ)
4 Nonlinear Dynamics and Bifurcations in Autonomous RLC Circuits
Fig. 4.22 Circuit with tunnel
diode exhibiting a Hopf
bifurcation
Fig. 4.23 Main parameters
of a tunnel diode
characteristic
Assume for simplicity C = 1, L = 1, and let x 1 = i L , x 2 = v C . Then the SEs
become
dx 1
dt
= x 2
dx 2
dt
= −x 1 + g(μ − x 2 )
where the battery voltage E is chosen as a parameter μ.
For any μ there is a unique EP ( ¯
x 1 , ¯
x 2 ) = (g(μ), 0). The change of variables
y 1 = x 1 − g(μ), y 2 = x 2 yields the SEs
dy 1
dt
= y 2
(4.9)
dy 2
dt
= −y 1 − g(μ) + g(μ − y 2 )
(4.10)
which have a unique EP at (0, 0) for any μ. The Jacobian of the vector field defining
the SEs evaluated at the EP is
J =
0
1
−1 −g (μ)