Investigation of Nonlinear Characteristics of Oscillators Using Gyrator
Yoko Uwate and Yoshifumi Nishio
Dept. of Electrical and Electronic Engineering, Tokushima University
2-1 Minami-Josanjima, Tokushima, Japan
Email: {uwate, nishio}@ee.tokushima-u.ac.jp

Abstracta gyrator and a capacitor, and investigate the nonlinear characteristics when the parameters are chenged.

In this study, we investigate the change in nonlinear features
when the coil of van der Pol oscillator is replaced by gyrator
and capacitor. The oscillator with a gyrator has two moreparameters than the standard oscillator. We investigate thenonlinearity features when those two parameters are changed.
And we discuss whether the oscillator using a gyrator exhibits
the same phenomena as the standard oscillator, and whether
new phenomena can be observed.

2. van der Pol oscillator using Gyrator
Figure 1 shows a standard van der Pol oscillator which iscomposed of one inductor, one capacitor and one negativeresister. The circuit equation is expredded by the following
equation.

iiR
1. Introduction

Cv

L

The coupled oscillator system is an excellent model for
representing high-dimensional nonlinear phenomena. The
synchronization phenomena observed in the coupled oscillator system depend on the topology of the network and theoscillators at the nodes, and a variety of oscillator networks
Figure 1: A standard van der Pol oscillator.
have been proposed and interesting synchronization phenomena have been observed.
In our research group, we have been focusing on the syndi L =vchronization of coupled oscillaotrs and chaotic circuits as sevdt(1)eral types of network topologies. In Ref. [1], we have re C dv = −i − iR
ported the synchronization phenomena of van der Pol oscilladt
tors observed from polygonal oscillatory networks. The obwhere
tained synchronization were explained by using the theoretiiR = −g1 v + g3 v 3 .
(2)
cal analysis of power consumptions. There are several engineering applications of the coupled polygonal oscillatory netBy using the variables and the parameters,
works, for example, synchronization problem of complex netr
√works [2] including the Internet, the World Wide Web, sensor
g1t = LCτ , v =x,networks, and power grid networks, and generation and con3g3trol of gait patterns of multi-legged robots [3], [4].
rrr
In recent years, advances in semiconductor technology
g1C
L
i=y, ε = g1,
have led to the incorporation of various circuits as integrated
3g3 L
Ccircuits. However, the van der Pol oscillation circuit contains
a coil. It is extremely difficult to create an inductor using a the normalized circuit equations are obteined as follows.
coil in an integrated circuit, so there is a method of expressdx1= εx(1 − x2 ) − ying the inductor characteristics using a gyrator and a capacidτ
3(3)tor. In this study, we use a circuit in which the inductor of ady=xconventional van der Pol oscillation circuit is represented bydτ

129

r
r
g1
C2
For the computer simulations, we calculate Eq. (3) using
v2 =y, ε = g1,
the fourth-order Runge-Kutta method with the step size h =
3g3
C1
r
0.01.
C2
1
In Eq. (3), the parameter ε denotes the nonlinearlity of osα=, β= .
CR1cillator. When the value of the parameter ε is close to zero,
the nonlinearity of the capacitor voltage waveform is weak, the normalized circuit equations are obteined as follows.
and the waveform closely resembles a sine wave. In contrast,dx
1
as the value of ε increases, the nonlinearity becomes stronger= εx(1 − x2 ) − αβydτ3(6)and the waveform becomes similar to a square wave.
βdy


= x
Figure 2 shows the how to change the time wave form bydταchanging the nonlinearity parameter ε.
The oscillator with a gyrator has two more parameters (αand β) than the standard van der Poloscillator. If the valuesof the two parameters are both zero (α = β = 1.0), then the
differential equation is exactly the same as that of the standardoscillator. The time series waveforms of the changing theparameter ε are shown in Fig. 4. This result is the same as forthe standard van der Pol oscillator (Fig. 2).

Figure 2: Nonlinear Characteristcis of time wave form obtained from a starndard van der Pol oscillator.
Next, we consider the oscillator which is composed of one
negative resistor, two capacitors and one gyrator as shown inFig. 3. A gyrator is a circuit network that inverts impedance.
By using a gyrator, it is possible to convert capacitance intoinductance.
Figure 4: Nonlinear Characteristcis of time wave form obtained from oscillator using Gyrator.

RiR
C1

v1

C2v2

3. Nonlinear Characteristics of Oscillator using Gyrator

In this section, we investigate the change in the nonlinearity of the voltage waveform of the capacitor with the valuesof parameters α and β. Figure 5 shows the simulation results
when the parameter α is set to 0.5, 2.0, and 5.0. This result
Figure 3: van der Pol using Gyrator.
shows that different alphas have no effect on the change innonlinearity. It was found that as α increases, the amplitude
The circuit equation is expredded by the following equa- of the time series increases.
tion.
Figure 8 shows the simulation results when the parameterdv1
 C1 1 = − v2 − iR
β is set to 0.5, 2.0, and 5.0. In this case, we can see thatdtR(4) the change in nonlinearity varies greatly with the value of β.
dv1
2 C2= v1
First, it can be said that the oscillation frequency of the oscildtRlator depends on the value of β. When the value of β is 0.5,
wherethe oscillation frequency is small, and as the beta increases,
iR = −g1 v1 + g3 v13 .
(5) the oscillation frequency increases.
Furthermore, if we look at the change in nonlinearity, weBy using the variables and the parameters,
can see that when β is 0.5, the nonlinearity becomes strongerrpas ǫ increases. However, when β is 2.0 and 5.0, the nonling1
t = C1 C2 τ , v1 =x,
earity is not so strong even when ǫ is increased.
3g3130(a) α=0.5.

(a) β=0.5.

(b) α=2.0.

(b) β=2.0.

(c) α=5.0.

(c) β=5.0.

Figure 5: Time wave forms depending on α.

Figure 6: Time wave forms depending on β.

131

4. Oscillations using Circuit Simulator
Finally, the oscillation phenomenon of the gyrator-type oscillator is investigated using a circuit simulator (LTspice).
The circuit model of the gyrator-type oscillator is shown inFig. 7. The gyrator is modeled by the voltage controlled current source.
(a) G2 =1.0.

(b) G2 =0.5.

Figure 7: Gyrator type of oscillator for LTspice.
The simulation results of LTspice are shown in Fig. 8. Bychanging the value of the conductance (G2 ) of the voltagecontrolled current source, it corresponds to the change in β.
From these results, it can be said that the oscillation frequency
increases as the value of conductance increases, and the nonlinearity does not become stronger, which is similar to thesimulation results described above. However, when G2 =0.5, the nonlinearity is stronger than in the simulation results
above. The investigation of this difference will be the subjectof future research.
In addition, we investigated oscillation in a gyrator circuit
using an operational amplifier, but we could not successfully
observe the oscillation phenomenon.

(c) G2 =2.0.

5. Conclusions(d) G2 =5.0.

In this study, we have investigated the change in nonlinear
features when the coil of van der Pol oscillator is replacedby gyrator and capacitor. By changing the parameters of thegyrator, we were able to observe changes in the nonlinearity features that could not be observed with the standard vander Pol oscillator. Oscillators using gyrators do not use coils,
which makes them suitable for integrated circuits.
The investigation of synchronization phenomena in a network of coupled gyrator oscillators is our future work.
References[1] Y. Uwate and Y. Nishio, “Synchronization in Several
Types of Coupled Polygonal Oscillatory Networks,”
IEEE Transactions on Circuits and Systems I, vol. 59,
no. 5, pp. 1042-1050, May 2012.
[2] X. Li and G. Chen, “Synchronization and desynchronization of complex dynamical networks: An engineer-132

Figure 8: Simulation retuls of LTspice.
ing viewpoint,” IEEE Trans. Circuits Syst. I, Fundam.
Theory Appl., vol. 50, no. 11, pp. 13811390, Nov. 2003.
[3] C. Liu, Q. Chen, and J. Zhang, “Coupled Van Der
Pol Oscillators Utilised as Central Pattern Generators for Quadruped Locomotion,” Proc. CCDC09,pp.36773682, Jun. 2009.
[4] S. Mondal, A. Nandy, Chandrapal, P. Chakraborty, andG. C. Nandi, “A Central Pattern Generator based Nonlinear Controller to Simulate Biped Locomotion with aStable Human Gait Oscillation,” Int. J. Robot. Autom.,vol. 2, no. 2, pp. 93106, 2011.