I. INTRODUCTION
Chaos has extensive possibility of engineering applica¬ tions.  For instance, neural network systems, chaotic communication systems, spatio-temporal behavior in coupled chaotic systems and so o n. For realizing these applications, an integration of chaotic circuits is very important subject[1].  We proposed a chaotic circuit suitable for the integration in NOLTA'99[2]. The cir- ciut consists of two ring oscillators, two diodes, two resistors and one capacitor. A CMOS ring oscillator is used for the CMOS process performance test. This os- cillator consists of a series of inverter circuits. Diodes are realized simply on IC chip.  It has very simple structure. Therefore, we consider that the proposed circuit is realized very easily. In this study, we investigate chaotic phenomena gen- erated by the proposed circiut. The exact solutions are derived, the Lyapunov exponents are calculated and bifurcation scenario is explained.
II.  CIRCUIT MODEL
A. Ring Oscillator Fig. 1 shows a three-stage ring oscillator. This os¬ cillator consists of three CMOS inverters. Fig 2 shows SPICE simulation results of the ring oscillator in Fig. 1. Channel width of p-MOS is 60[ mm] and length is 4 [μm]. Channel width of n-MOS is 20[μm] and length is 4 [ μ m]. A standard 0.5 μ m layout design rule is used for the transistor model.
B. Designed Circuit
We design a chaotic circuit using this ring oscillator. Fig. 3 shows the designed circuit. In order to con- trol amplitudes of the oscillators, R1 and R2 are ∞n- nected. The frequency of the upper side oscillator are controlled by C1. Fig. 4 shows SPICE simulation re- sults of the circuit in Fig.  3.  Parameter values are W /L(pMOS) = 60/4, W /L(nM S) = 20/4, R 1 = 2000[Ω], R 2 = 200[Ω], C 1 = 9.00[p F], V ad = 2.70[V], V s  = 2.30[V]. Maximum time step is 50.0 [ps], and ob- servation time is 10.0-100.0 [ s ]. The main oscillation frequen cy band width is about 100-250[MHz].
C. Circuit Model
We approximate M1, M2 and CMOS inverters in the circuit as follows. M 1 and M 2 have a character- istic of the same as the coupling diodes shown in Fig. 5. Therefore, M 1 and M 2 are approximated as the following three-regions piecewise-linear function.
Fig. 6 shows the approximation ofthe ring oscillator’s inverter. The circuit elements C, R and Gm represent all parasitic capacitors associated with the input and the output nodes, the input and the output resistances and the inverter’s gain, respectively.  By the reason that the inverter is connected to the previous-stage and the next-stage inverters, the device values of the input and the output nodes are put together in this approximation. Circuit equations are described as follows:
where
By substituting the variables and the parameters,
( 2 ) and (3 ) are normalized as
where
By using the exact solutions of Eq. (5), we can calculate attractors and represent the Poincaré map and its Jacobian matrix for calculating the Lyapnov exponents. We define three piecewise linear regions as Table 1.
We calculate the eigenvalues in each region and the equilibrium points in D+ and D=. The equilibrium points are calculated by putting the right side of Eq. (5) to be equal to zero. Then, we can describe the exact solutions in each linear region.
III.  ANALY SIS
A.  Poincaré Map
In order to confirm the generation of chaos and to investigate bifurcation scenario, we derive the Poin caré map. Let us  define the following five-dimensional subspace
where
dition between D+ and D0, while the subspace S2 corresponds to the condition x a 3 − x a 3 > 0. Namely, S corresponds to the transitional condition from Do to D +. Let us consider the solution starting form an ini- tial point on S. The solution returns back to S again after wandering as shown in Fig. 7. Hence, we can derive the Poincaré map as a five- Dimensional map as follows.
where xn is an initial point on S, while T(xn) is the point at which the solution starting from xn hits S again. T(x n ) is given using the exact solutions of Eq. (5). The Jacobian matrix DT of the Poincaré map can be also derived rigorously from the exact solutions. We can calculate the largest Lyapunov exponent by
where e , is a norm alized base
B. Computer Calculation
One-parameter bifurcation diagram of the Poincaré map T and the calculated largest Lyapunov exponent are shown in Figs.  8 (a)(b). By using these results, we can say the generation of chaos is confirmed numerically and we can describe detailed bifurcation scenario as follows. As ε decreases, one-periodic orbit bifurcates to torus around ε = 4.00. For 3.13 < ε < 4.00, we can observe several phase-locked states in torus region.  For 1.81 < ε < 3.13, the largest Lyapunov exponent becomes positive .  Namely, chaos is generated.
IV. CONCLUSIONS