2004 RISP International Workshop on Nonlinear
Circuit and Signal Processing (NCSP'04)
Hawaii, USA, Mar. 5-7, 2004.

Multimode Asynchronous Oscillations in Coupled Multi–State Chaotic Circuits
Masahiro WADA† and Yoshifumi NISHIO‡†

Dept. of Information Science and Systems Engineering, Konan University
8–9–1 Okamoto, Kobe 658–8501, Japan.
Phone/FAX: +81–78–435–2531, E-mail: wada–m@konan–u.ac.jp‡
Dept. of Electrical and Electronic Engineering, Tokushima Universityv
Abstract— In this paper, novel types of multimode asyni L1i L2chronous oscillations in the coupled multi–state chaotic cirV
L2+cuits are investigated. Each chaotic circuit can behave asynchronously both chaotic or periodic oscillation at the samer / 2 = rs
L1NRv
Cparameters. Because it has a hard nonlinearity of the resisD
tor with five segments piecewise linear regions, the both difi
ferent oscillation modes are separated. A design scheme ofthe piecewise linear elements constructed by using electrical-Vparts is proposed. Further results of some types of multimodeoscillations in numerical simulation and circuit experiment Figure 1: Schematic figure for the chaotic circuit, and a typiare also shown.
cal characteristic of the diode model with polarities.
dddd1. Introduction
The dynamics of chaotic multimode oscillations is still
considerable interest from the viewpoint of both naturalscientific fields and several applications. They have been
confirmed in several systems, e.g., coupled van der Pol
oscillators[1], laser systems[2], and so on. On the other hand,
many types of chaotic systems and circuits have already been
proposed and investigated in detail. If the active elements including in the systems have complexity constructed by compound nonlinear elements, it can be easily consider that theyyield several interesting features. There are famous chaotic
attractors such a double-scroll family[3], n-double scroll[4]–[6] and scroll grid attractors[7]. The purpose of our study isto clarify coexistence of both chaotic and non–chaotic behavior in the chaotic systems. Further complex behavior in thelarge scale network of the coupled chaotic circuits are also
investigated.
In this study, we investigate a novel type of multimodeasynchronous oscillations on the coupled multi–state chaoticcircuits. There is a typical three dimensional autonomouschaotic system proposed by Inaba[8], which consists of threememory elements, some diodes and designed negative resistors. It is well known that it can behave as Rössler typechaotic motions. We substitute a symmetrical continuous
five segments piecewise linear resistor for the negative active resistor including in the chaotic circuit. This proposed
circuit can behave both chaotic and periodic motions at thesame parameters when we supply with different initial conditions. Although it may be considered that this is similar
to the chaotic systems[9][10] at a glance, those systems aredifferent from stability at the origin. Hence it is excited at
the origin because of negative resistance. Generally it canbe constructed a concept for complicated structure of chaotic
attractors to make many equilibrium points. In the past ourworks, we had presented and confirmed that multi-state os-

NR
R1

R4+-C

L1

L2+R7-

R2R6R5R3E2

E1Figure 2: Proposed chaotic circuit with piecewise linear resistors.
cillations both chaotic and non–chaotic (limit cycle) can begenerated asynchronously at the same parameters on the computer simulations [11] and that realization on the real circuits[12]. In this paper, firstly the design scheme of piecewise
nonlinear segments constructed by electrical elements is explained. Secondary both chaotic and periodic oscillations at
the same parameters which can be confirmed in numericalsimulations and circuit experiment are shown. Further, theresults for several types of multimode oscillations on the coupled chaotic circuits are also shown.
2. Model DescriptionA well–known chaotic circuit proposed by Inaba andSaito[8] is shown in Fig. 1 and i–v characteristic of a typical
diode model with polarities also is also shown. The variable
vd (iL2 ) is a function depending on the current through their
diodes D in Fig. 1, which determines their chaotic dynamics.
In this study, we substitute a symmetrical continuous fivesegments piecewise linear resistor for the negative active re-- 535 -iim satii

m0
m1m2

Esat

Bp3
Bp1
-Esat

-Bp1

Esatv

Bp2

-Bp1
Bp1v

-Bp3

-Bp2

Esatv

-Bp2
-Bp3

-Bp1

Bp3

Bp1v

Bp2

-Esat
-Esat

(a)
(b)
(c)
(d)
Figure 3: Design method for v–i characteristic of a piecewise linear resistor in the chaotic circuit. Characteristics of (a) Nr1 ,(b) diodes and one resistor, (c) Nr2 and ideal diodes with polarities, (d) compound characteristics of NR connected by (a) to(c) in parallel.
sistor including in the chaotic circuit. The designed chaotic
circuit is shown in Fig. 2. The piecewise linear resistor canbe easily constructed by a connection of some components inparallel as shown in Fig. 3(a)–(c), and then compound characteristics of NR can be illustrated as shown in (d).
The circuit equation can be normalized when we chose Vdfor a threshold voltage value of the diodes by changing thefollowing variables and parameters as follows.
rrCC
Vd x , iL2 =Vd y ,iL1 =L1
L1pdv = Vd z , t = L1 C dτ , “ · ” =,(1)rr dτL1L1Cβ=, γ=g, δ = rsL2CL1
where g is a linear negative conductance value of NR if weconsider the negative resistor as an ideal. Further let us consider that the part of negative resistance in Fig. 1 replaces toh(z) as a function of voltage source z, then the circuit equations can be rewritten by ẋ = z ¡¢(2)ẏ = β z − f (y)ż = −(x + y) − h(z)f (y) =o1n|δy + 1| − |δy − 1| .
2(3)where f (y) is a function of the current y and h(z) is afunction of the voltage z, respectively. The function h(z)which can be designed by symmetrical five segments piecewise linear with respect to the origin for the parameters
four breakpoints at {±Bp1 , ±Bp2 } and five slopes by{m0 , m1 , m2 , m1 , m0 } is described with a canonical form asfollows.
¡¢γ∗ nh(z) , m0 γ ∗ z+
(m0 −m1 ) |z−Bp2 |−|z+Bp2 |2(4)¡¢o+(m1 −m2 ) |z−Bp1 |−|z +Bp1 |where γ ∗ is a basic variable parameter, hence values mk (k =0,1,2) mean the ratio to the value γ ∗ .

3. Circuit Experiment and Simulation Results
The circuit experiment results by using real implemented
circuit are shown. In order to realize the nonlinear characteristic of NR , we designed a piecewise linear resistor constructed by using some OP amps(TL082CP) and resistors.
The details technique is explained in Ref. [13]. The nonlinear resistor is realized by two OP amps, some diodes and DC
voltage which is used for setting the threshold voltage strictly,
and resistors. The circuit parameters chosen in this simulation
are as follows.
L1 = 123.1[mH], L2 = 10.2[mH], C = 68.7[nF],R1 = 33.2 [kΩ], R2 = 21.7 [kΩ], R3 = 1.22 [kΩ],R4 = 196 [Ω], R5 = 333 [Ω], R6 = 1.47 [kΩ],R7 = 10.3 [kΩ], E1 = 2.78 [V], E2 = 4.80 [V].
Normally, the voltage v of the circuit oscillates in the area ofthe threshold voltage between around ± Vd (Vd ' ±5.6[V])because of a breakpoint at the part of L2 and diodes. Chaotic
and periodic attractors are shown in Fig. 4. Both chaotic andperiodic attractors can be observed in the same circuit parameters.
Some numerical simulation results are also shown. The
other parameters are β = 10.0, γ ∗ = 0.76, δ = 100, and construction of the piecewise linear characteristics form are realized by Bp1 = 0.30, Bp2 = 0.56, m0 = −1.0, m1 = 1.0 and
m2 = −0.5. It is noticed that values of these breakpoints arenormalized by threshold voltage Vd of the diodes with polarities connected to L2 . If we chose Vd between breakpoints
Bp2 and Bp3 and it is assumed that the value of breakpointBp3 is enough larger than Vd , it can be ignored the area overthe voltage ±Bp3 in the computer calculation. Therefore fivesegments regions drawn in Fig. 3(d) are used dynamically andthis canonical form of five segments piecewise linear function
was also only described in Eq. (4). In order to be accuracy numerical calculation, the boundary of switching of the diodes
is calculated by using bisection method. Figure 4(d) and (e)show attractors for γ ∗ = 0.76 when the initial conditions arechanged differently. Limit cycle (e) and chaotic attractor (d)can be confirmed coexistence at the same parameters. Whenγ ∗ increases, this oscillation mode bifurcates to chaos fromone periodic state via period doubling bifurcation in the outside region the following route while keeping the limit cyclein the inside region. Oscillation of symmetrical 1-period →asymmetrical 1-period → bifurcates to 2n period → asym-- 536 -i L0
L0i L11i+6-v

v1

L 21+i L22
L 22v2
C1-

L 11NR1(a)i L12i L21

Circuit-1

D1C2-L 12NR2

D2

Circuit-2

Figure 5: Circuit model of two MSCCs coupled by an inductor.
(b)(c)(d)(e)
Figure 4: Some snapshots of the circuit experiment and numerical simulation. (a) v–i characteristics of the designed
piecewise linear resistor, horizontal: 5[V/div]. (b) chaotic attractor, (c) limit cycle, horizontal: 2[V/div]. Threshold voltage of one diode vth ' 0.78 [V]. Corresponding results (d)and (e) obtained by numerical calculation onto the z–x planefor the parameters β = 10.0, γ ∗ = 0.76, δ = 100, Bp1 =0.30, Bp2 = 0.56, m0 = −1.0, m1 = 1.0 and m2 = −0.5.
metrical slight chaos → symmetrical fluttered chaos. We canobserve that both two oscillation modes exist separately at thesame parameters. Hereby we call this this circuit a multi–statechaotic circuit(abbr. MSCC).

in Fig. 6 for the case of the parameter α = 0.80, γ ∗ = 0.480,
and other parameters are the same used in the section 3, fromthe left attractor of z1 –x1 , attractor of z2 –x2 and synchronization state of z1 –z2 projection respectively. Figure 6(a) shows
an in-phase synchronous state of small size periodic attractors. Figure 6(b) shows an anti-phase synchronous state ofsmall size periodic attractors. Figure 6(c) shows an anti-phasesynchronous state of large size chaotic attractors. Figure 6(d)shows a double-mode asynchronous state of large size chaotic
and small size periodic motions. These all simulation results
are confirmed at the same parameters. Multimode oscillationmeans that it consists several types of oscillation simultaneously. The multimode oscillations were confirmed in the proposed coupled MSCCs.
Some circuit experimental results are shown in Fig. 7. Attractors between v1 and iL11 , synchronization state between
v1 and v2 are shown. Several types of oscillation modes canbe observed. Figure 8 shows a time waveform of v1 and its
FFT spectrum in the case of double-mode oscillations. This
one MSCC oscillates around 1.73[kHz] as a fundamental frequency. It can be easily calculated from the circuit parameters. We can confirm that two peaks and wide band frequencies are observed from the FFT result. The highest peakis certainly located at around this point. Therefore chaotic
and non-chaotic oscillations in the two circuits are generated
alternatively, we can observe the double-mode oscillationsin the coupled circuits. Besides we can also confirm several types of multimode oscillations, i.e., in–phase and anti–
phase synchronization of periodic oscillations, double-mode
chaotic oscillations, unfortunately except for an anti–phasechaotic synchronization mode from the circuit experiment.

4. Multimode Oscillations in Coupled MSCCs
Here we consider the case of the coupled two MSCCs byan inductor as shown in Fig. 5. By using KCL at the loop ofthe inductors L0 , L11 and L21 , the circuit equations can be reduced to the six–dimensional differential equations. Further
we use a new parameter as α = L1 /L0 , by changing parameters and variables similar to Eq. (2), the circuit equation canbe described as follows.
5. Conclusions
 ẋk = zk¡
In this study, we have investigated multimode oscilla¢ẏk = β zk − f (yk )(5) tions in the coupled multi–state chaotic circuits by numeri
cal simulation and circuit experiment. Further coexistence ofżk = (−1)k α(x1 − x2 ) − (xk + yk ) − h(zk ).
four types oscillation modes, i.e., in-phase limit cycle, antiwhere k is a number of the circuit(k = 1,2.), and α corre- phase limit cycle, anti-phase chaotic synchronization, andsponds to the strength of coupling.
double-mode asynchronous of both chaotic and periodic osSome numerical and circuit experiment results are shown. cillations have been observed in coupled MSCCs by an inWe can change values γ ∗ and mk as control parameters. In ductor. These chaotic behavior observed in this study arethis case, four asynchronous oscillation modes could be con- expected to yield new chaotic phenomena in several types
firmed consequently by numerical simulations when the ini- of coupled chaotic systems, e.g., chaotic itinerancy, spatio–
tial conditions are varied. These simulation results are shown temporal chaos, multi–agent systems, and inherent emergent- 537 -(a)iL11v26-(a–1)

v1

6
-

(a–2)

v1(b)

(c)

(b–1)

(b–2)

(c–1)

(c–2)

(d)

Figure 6: Four oscillation modes exist at the same parameters: α = 0.80, β = 10.0 and γ ∗ = 0.480. (a) in-phase synchronous limit cycle, (b) anti-phase synchronous limit cycle,(c) anti-phase chaotic synchronous attractor, (d) double-mode
oscillations both of two different attractors chaotic and small
size limit cycle.

Figure 7: Some snapshots from circuit experiment results forthe following parameters; L0 = 183.3[mH], L1 = 123.1[mH],L2 = 10.2[mH], C = 68.7[nF], R1 = 33.2 [kΩ], R2 = 21.7 [kΩ],R3 = 1.22 [kΩ], R4 = 196 [Ω], R5 = 333 [Ω], R6 = 1.47 [kΩ],R7 = 10.3 [kΩ], E1 = 2.78 [V], E2 = 4.80 [V], horizontal:
5[V/div]. (a) in-phase synchronous limit cycle, (b) anti-phasesynchronous limit cycle, (c) double-mode oscillations.

property, in which concerned with current topics.
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(a)

(b)

Figure 8: Snapshots of a time waveform and a FFT spectrumat the double-mode chaotic oscillations. (a) time waveform,horizontal: 5[msec/div], vertical: 2.5[V/div], (b) FFT, horizontal: 500[Hz/div], vertical: 400[mV/div].

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