International Symposium on Nonlinear Theory and its Applications
Xi'an, PRC, October 7{11, 2002
Multimode Oscillations in Two Coupled van der Pol Oscillatorswith Ninth-Power Nonlinear Characteristicsyz

y and Akio Ushiday

Mitsuhiro Yoneda , Yasuteru Hosokawa , Yoshifumi Nishioy Dept. of E. E. Eng., Tokushima University z Dept. of Information Sci., Shikoku University
Tokushima 770-8506, Japan
Tokushima 771-1192, Japan

yoneda@ee.tokushima-u.ac.jp

nonlinear function. In order to realize the complicatednonlinear characteristics by real circuit, we approximate
Coupled oscillators are good models to describe vari- the ninth-power nonlinear function by the nine-segment
ous nonlinear phenomena in the eld of natural science piecewise-liner function as shown in Fig. 2. The funcand a number of excellent studies on such systems have tion is described by using 4 parameters (B ; B1; B2 ; B3)
been carried out. Datardina and Linkens have investi- corresponding to the breakpoints and 50 parametersgated two van der Pol oscillators with hard nonlinearities (G0 ; G1 ; G2 ; G3; G4 ) corresponding to the resistances ascoupled by an inductor [1]. They have con rmed that follows.
nonresonant double-mode oscillations, namely simulta8 G4v + B3(G3 + G4 ) 0 B2(G2 + G3)
neous asynchronous oscillation of two periodic oscilla>> +B1 (G1 + G2) 0 B0(G0 + G1)
tions which could not occur for the case of third-power>>
nonlinearity, were stably excited in the coupled systems.
> 0G3 v 0 B2(G2 + G3 ) + B1 (G111 +1 (vG<2) 0B3)
They have also con rmed that four di erent modes co>> 0B (G + G )exist for a range of parameter values; zero, two single1 1 1 (0B3 < v < 0B2 )>> 0 0 1
modes, and a double-mode. Yoshinaga and Kawakami
have investigated a ring of van der Pol oscillators with>> G2v + B1(G1 + G2 ) 0 B0(G0 + G1)
1 1 1 (0B2 < v < 0B1 )hard nonlinearities and have con rmed interesting syn>>chronization phenomena of double-mode oscillations [2].
<0G1 v 0 B0 (G0 + G1 ) 1 1 1 (0B1 < v < 0B0 )In this study, we investigate multimode oscillations observed from two van der Pol oscillators with ninth-power i(v) = >>> G0G0v1 v + B0(G0 + G1 ) 1 1 1 (B1 101<(jvvj << BB10))
nonlinear characteristics coupled by an inductor. By cir>>
cuit experiments and computer calculations, we con rm>> G2v 0 B1(G1 + G2 ) + B0(1G1 10(+B G<1)v < B )that eight di erent modes coexist for a range of param1
2
eter values; zero, two in-phase single-modes, two anti>>> 0G3 v + B2(G2 + G3 ) 0 B1 (G1 + G2)
phase single-modes, and three di erent double-mode oscillations.
>> +B0 (G0 + G1)
1 1 1 (B2 < v < B3)>: G4v 0 B3(G3 + G4 ) + B2(G2 + G3)
2. van der Pol Oscillator with Ninth-Power Non0B1 (G1 + G2) + B0(G0 + G1) 1 1 1 (B3 < v)
1. Introduction(1)linear CharacteristicsiLCv

Figure 1: van der Pol oscillator.
Figure 1 shows van der Pol oscillator. We consider
the case that the v 0 i characteristics of the nonlinear
resistor in the oscillator are represented by ninth-power

Figure 2: Approximation of the ninth-power v 0 i characteristics of nonlinear negative resistor.

-459-The equation governing the circuit in Fig. 1 is de-scribed as follows:

di=v
dtdv
C = 0i 0 i(v)
dt

L(2)

By changing the variable and the parameters,rC yp
di=; v = B0x; t = LC; \ 1 " = ;LadrLBBB; = 1; = 2; = 3;"=aC 1 B0 2 B0 3 B0
0 = B0 G0 ; 1 = B0 G1 ; 2 = B0 G2 ;3 = B0 G3 ; 4 = B0 G4 ;(2) is normalized as(b)(a)(3) Figure 3: Two di erent sizes of oscillations obtained bycomputer calculations. Horizontal axis is x and verticalaxis is y . = 0.05, 1 = 2.3, 2 = 4.18, 3 = 6.27, 0= 0.17, 1 = 0.35, 2 = 0.34, 3 = 0.37, 4 = 0.60, " =0.3. (a) Small oscillation. (b) Large oscillation.

y_ = xx_ = 0y 0 "f (x)Figure 5 shows the circuit realization of the negative(4) resistor
with the nine-segment piecewise-linear v 0i characteristics. Figure 6 shows the measured v 0 i characwhere the function f (x) corresponds to i(v ) and is rep- teristics of the negative resistor.
resented as follows.
8 4x + 3 4 + ( 3 0 2 ) 3 0 ( 2 0 1) 2>> +( 1 0 1) 1 0 0
1 1 1 (x < 0 3)
i L>> 0 3x 0 2 3 0 ( 2 0 1) 2ii>> +( 1 0 1) 1 0 0 1 1 1 (0 3 < x < 0 2)
LCLC>> 2x + 1 2 + ( 1 0 1) 1 0 0vv>>>> 0 1x 0 ( 0 + 1) 1 1111(10(02 1<<xx<<001)1)<1 1 1 (jxj < 1) (5)f (x) =>> 00x x + ( + )1 1 1 (1 < x < 1)>> 1 0 1
Figure 4: Two van der Pol oscillators coupled by an>> 2x 0 1 2 0 ( 1 0 1) 1 + 0
inductor.
1 1 1 ( 1 < x < 2)>>>> 0 3x + 2 3 + ( 2 0 1) 2
1 1 1 ( 2 < x < 3)>> 0( 1 0 1) 1 + 0>: 4x 0 3 4 0 ( 3 0 2 ) 3 + ( 2 0 1) 2iR0( 1 0 1) 1 + 0
1 1 1 ( 3 < x)+R
By circuit experiments and computer calculations, weR
con rmed that this circuit have two sizes of oscillations+Rand that the origin is stable for the same parameter valRues. Figure 3 shows two sizes of oscillations observedv
Rfrom the circuit. Figure 3(a) shows the small oscillation.
RIn this case, the solution uses the inside negative resistance regions between 6B0 and 6B1 , namely the solution does not reach the regions beyond the breakpoints
6B2 . Figure 3(b) shows the large oscillation. In thiscase, the solution uses the outside negative resistanceregions between 6B2 and 6B3 in Fig. 2.
Figure 5: Circuit realization of the negative resistor
with the nine-segment piecewise-linear v 0 i character3. Two Coupled Oscillators
istics. R1 = R2=47k , R3 =1.2k , R4=6.2k , R5 =The main object of this paper is to investigate two van R6 =820 , and R7 =1.2k .
der Pol oscillators with ninth-power hard nonlinearities
coupled by a inductor. The coupled model is shown inFig. 4.
The equations governing the circuit in Fig. 4 are given
001212514723-460-6

(1)Figure 6: Measured v 0 i characteristics of the negativeresistor.
as follows:

y_1 = x1
x_1 = 0y1 + (y2 0 y1 ) 0 "f (x)y_2 = x2
x_2 = 0y2 + (y1 0 y2 ) 0 "f (x)

(2)

(b)
(a)
Figure 7: Circuit experimental results of single-mode
oscillations. (a) In-phase single-mode. (b) Anti-phase(1) Small oscillation. (2) Large oscillation.
where = L=L0 is a new parameter corresponding to single-mode.
Horizontalaxis
is v1 [2V/div.] and vertical axis is v2the coupling.
[2V/div.].
In the following circuit experiments, the circuit parameters are xed as L = 200mH, L0 = 1.15H, and C= 100nF. While for the computer calculations, the parameter values are as = 0.05, 1 = 2.3, 2 = 3.8, 3 =5.75, 0 = 0.052, 1 = 0.087, 2 = 0.087, 3 = 0.108, 4= 1.86, " = 0.3. Further, (6) is integrated by using theRunge-Kutta method with step size h = 0.005.
From the coupled circuit in Fig. 4, we con rmedthat eight di erent oscillation modes coexist; zero, twoin-phase single-modes, two anti-phase single-modes andthree double-modes. Zero means that both of two oscil- (1)lators are not excited. This state is always stable andrelatively small initial values are attracted to this state.
In the following we do not show any gures of this state,because it is trivial. In-phase single-mode means thatthe two oscillators are synchronized at the in-phase. Twodi erent sizes of the in-phase single-mode oscillations areshown in Figs. 7(a) and 8(a). Anti-phase single-modemeans that the two oscillators are synchronized at theanti-phase. Two di erent sizes of the anti-phase singlemode oscillations are shown in Figs. 7(b) and 8(b).
(2)
4. Double-mode Oscillation(6)

Double-mode means that above two single-mode oscillations, namely in-phase and anti-phase single-modes,
are simultaneously excited. We have found three different types of double-mode oscillations from our cir(b)(a)
cuit model. The circuit experimental results and thecorresponding computer calculated results are shown in Figure 8: Computer calculated results of single-modeFigs. 9 and 10, respectively. We can characterize the oscillations. (a) In-phase single-mode. (b) Anti-phase
three double-mode oscillations as follows. In Figs. 9(1) single-mode. (1) Small oscillation. (2) Large oscillation.
and 10(1), the solution moves from zero to small oscillation alternately. In other words, the double-mode oscillation consists of zero and small oscillation. In Figs. 9(2)

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(1)

(1)

(2)

(2)

(3)

(3)

(b)
(a)
(a)
(b)
Figure 9: Circuit experimental results of double-mode Figure 10: Computer calculated results of double-mode
oscillations. (a) Time waveforms. Horizontal axis is t oscillations. (a) Time waveforms. Horizontal axis is [2ms/div.] and vertical axis is v1 [5V/div.]. (b) Attrac- and vertical axis is x1. (b) Attractors. Horizontal axis
tors. Horizontal axis is v1 [2V/div.] and vertical axis is is x1 and vertical axis is x2. (1) Double-mode oscillav2 [2v/div.]. (1) Double-mode oscillation consisting of tion consisting of zero and small oscillation. (2) Doublezero and small oscillation. (2) Double-mode oscillation mode oscillation consisting of small and large oscillaconsisting of small and large oscillations. (3) Double- tions. (3) Double-mode oscillation consisting of zero andmode oscillation consisting of zero and large oscillation. large oscillation.
and 10(2), the solution moves from small oscillation to in-phase single-modes, two anti-phase single-modes, andlarge oscillation alternately. In other words, the double- three double-modes.
mode oscillation consists of small and large oscillations. We consider that there remain some interesting probFinally, in Figs. 9(3) and 10(3), the solution moves from lems. In the present circuit model, we could not nd thezero to large oscillation alternately. In other words, the multimode oscillation consisting of three possible stabledouble-mode oscillation consists of zero and large oscil- states; zero, small oscillation and large oscillation. The
lations. Namely, we can characterize the double-mode oscillation could be a triple-mode oscillation. Further,oscillations by using the stable states included in the theoretical analysis should be carried out to investigate
oscillations. This would be an interesting approach to the stability of the various modes of oscillations.
analyze multimode oscillations observed from complex
nonlinear circuits.
References
5. Concluding Remarks

In this study, we have investigated multimode oscillations observed from two van der Pol oscillators with
ninth-power nonlinear characteristics coupled by an inductor. By both circuit experiments and computer calculations, we con rmed that eight di erent oscillation
modes coexist for a range of parameter values; zero, two[1] S. P. Datardina and D. A. Linkens, \Multimode oscillations in mutually coupled van der pol type oscillatorswith fth-power nonlinear characteristics," IEEE Trans.
Circuits Syst., vol. 25, no. 5, pp. 308-315, May 1978.
[2] T. Yoshinaga and H. Kawakami, \Synchronized quasiperiodic oscillations in a ring of coupled oscillators withhard characteristics," Trans. IEICE, vol. J75-A, no. 12,
pp. 1811{1818, Dec. 1992.

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