Design of an analog chaos-generating circuit using piecewise-constant dynamics

One of the major concerns related to the analysis of chaos in nonlinear continuous-time dynamics is the difﬁculty involved in demonstrating chaotic behavior in a rigorous sense, and this problem has attracted intensive research interest in the last three decades. Many researchers have attempted to solve this problem by adopting simpler dynamics. In this study, we propose an extremely simple third-order chaos-generating circuit with a diode whose governing equation is represented by a piecewise-constant dynamical system. Note that the memory elements of our circuit are capacitors only. We consider the idealized case where the diode is assumed to oper-ate as a switch. In this case, the Poincar´e return map is constructed from a piecewise-linear 1D map. We analytically prove the generation of chaos with a positive Lyapunov exponent through a simple systematic procedure, and observe the attractor through laboratory measurements. We strongly believe that the explicit simplicity of the proposed circuit dynamics could signiﬁcantly affect the architecture of chaos-generating circuit design in application because the procedure for proving chaos is systematic,

PTEP 2016, 053A01 T. Tsubone et al. chaotic behavior in a rigorous sense [12,13]. Many researchers have attempted to solve this problem by employing simpler dynamics. Chua et al. attempted to do this by adopting piecewise-linear dynamics [12]. They succeeded in demonstrating chaos in Shil'nikov's sense [21] in the double scroll circuit [8,9], which indicates the existence of an infinite number of horseshoes [22]. Horseshoes imply the existence of an uncountable number of nonperiodic orbits. However, the existence of horseshoes does not indicate that the nonperiodic orbits will become an attractor. In general, in piecewise-linear dynamics, one has to solve implicit equations to determine when the solution hits each boundary of the piecewise-linear branches [10,11,19,20]. Therefore, solutions in piecewise-linear systems are usually connected by solving the implicit equations computationally. This makes it difficult to prove the existence of chaos.
Other researchers have attempted to solve this problem using slow/fast dynamics, which includes a small parameter ε. Slow/fast dynamics refers to ordinary differential equations (ODEs) in which one of the variables changes faster than the others. Slow/fast 3D chaos-generating dynamics was first proposed by Rössler [23,24]. Levi explained the theory behind the existence of nonperiodic orbits in a slow/fast driven van der Pol oscillator using modern mathematical techniques [13].
Saito analyzed some novel chaos-generating circuits by adopting a piecewise-linear technique combined with a singular perturbation [10,11]. Saito considered a limit of ε → 0 in his circuits. In this case, the nonlinear term in his circuits operated as an idealized hysteresis, and, under this idealization, he succeeded in proving the existence of chaos with the largest positive Lyapunov exponent [10,11]. It is remarkable that the chaos in his circuits is also observable in laboratory measurements [10,11]. However, the chaos-generating circuits that Saito proposed were limited to just a few examples, and, furthermore, the procedure required proving that the existence of the positive Lyapunov exponent was not systematic.
In this study, we propose an extremely simplified chaos-generating circuit utilizing a diode, for which the dynamics represents a piecewise-constant equation. We systematize the procedure for conducting a bifurcation analysis of autonomous piecewise-constant oscillators and apply this procedure to the proposed circuit. We analytically prove the generation of chaos with a positive Lyapunov exponent in this oscillator. The idea of piecewise-constant dynamics was proposed by the authors to discuss chaos [28] and quasiperiodic bifurcations [29,30]. However, the procedures for demonstrating bifurcations were ad hoc in our previous studies. The chaos-generating circuit proposed in the present study comprises three capacitors, three voltage-controlled current sources (VCCSs), and one diode. Since the circuit does not include inductors, it is suitable for implementation in integrated circuits. The analysis of piecewise-constant dynamics is much simpler than that of piecewise-linear dynamics. Note that the trajectory of piecewise-constant dynamics is piecewise-linear in state space. We employ an ideal diode in this circuit to demonstrate chaos generation [14,15]. From this circuit, the Poincaré return map is rigorously derived as a piecewise-linear 1D map. By analyzing the Poincaré return map, it is demonstrated that the return map is ergodic [27] and has a positive Lyapunov exponent. Furthermore, the generation of chaos is verified through experimental measurements. The experimental results agree well with the theoretical ones. The procedures used for the analysis of bifurcations and chaos in this study are applicable to a large class of piecewise-constant autonomous circuits.
Recently, it has been demonstrated that simple and convenient secure communications can be realized using chaos-generating circuits. Furthermore, chaos computing has attracted the attention of many researchers [31,32]. The circuit architecture discussed in the present study could contribute to an artificial design of chaos-generating circuits, because the circuit construction of the present study 2 is quite simple and it is expected that our circuit could be naturally extended to higher-dimensional chaos generators.

Preliminary study for realization of expanding piecewise-constant oscillation
In this section, we explain the mechanism that causes the expanding oscillation of the piecewiseconstant dynamics and its implementation in a circuit. The fundamental idea of piecewise-constant dynamics was not demonstrated in our previous work [28][29][30]. One can grasp the essence of piecewise-constant dynamics by understanding the behavior of the solution of a 2D piecewiseconstant dynamic system.
Let us consider the following second-order linear equations: (1) has a pair of complex-conjugate eigenvalues, and the solution is oscillatory and expanding. Let the eigenvalues be denoted by δ ± iω. Then, δ = 1/2a and ω = √ 1 − δ 2 , and the solution when the initial conditions are (τ ; x, y) = (τ 0 ; x 0 , y 0 ) is explicitly given by the following equation: where f (τ ) = e δτ sin ωτ e δτ cos ωτ .
The positive δ value realizes expanding oscillations, as shown in Fig. 1. Thus, the explicit solutions are obtained in piecewise-linear dynamics in general in each piecewiselinear branch. However, one encounters a serious difficulty when the solution hits the boundaries of each piecewise-linear branch. For example, let us consider the case where the solution of Eq. (1) hits the boundary y = 1 at τ = τ 1 . This situation is illustrated in Fig. 1. Although the explicit solutions are obtained in each piecewise-linear region, we must solve the following implicit equation to derive the time τ 1 computationally: i.e., Through numerical calculation of τ 1 , x(τ 1 ) is usually obtained as follows: In general, chaos can be generated in three or more dimensional autonomous ODEs and in two or more dimensional nonautonomous ODEs. However, to connect the explicit solutions of piecewiselinear ODEs at each boundary of the piecewise-linear branches, inevitably, the implicit equations   that correspond to Eq. (5) must be solved numerically. These implicit equations make it significantly more difficult to theoretically prove the generation of chaos, even if the chaos-generating piecewiselinear dynamics is very simple. Chua et al. proved that the strange attractor called double scroll [8] is chaos in a Shil'nikov sense and Saito succeeded in proving the existence of chaos in some circuits with a positive Lyapunov exponent using a piecewise-linear technique combined with a singular perturbation [10,11]. However, their approaches are ad hoc. Furthermore, they provided no method for extending their circuits to higher-order chaos-generating dynamics.
We will now explain the fundamental concept of piecewise-constant dynamics. Let us consider the following equations:ẋ = Sgn(y) where Sgn(·) is a signum function defined as follows: The characteristic form of the signum function is presented in Fig. 2. Note that Eq. (7) is a significantly simplified form of the dynamics of Eq. (1). Becauseẋ andẏ take values of 1 or −1, we call such dynamics piecewise-constant. It is noteworthy that the piecewise-constant equations can be realized using very simple circuits. For example, the dynamics of Eq. (7) is realized using the circuit shown in Fig. 3. It comprises only two capacitors and two VCCSs with signum characteristics (see Fig. 2). The VCCS outputs a current PTEP 2016, 053A01 T. Tsubone et al.
as a function of the input voltage. In the circuit shown in Fig. 3, i 1 is determined by v 2 , and i 2 is determined by v 2 − v 1 . They are assumed as signum functions, as shown on the right-hand side of Fig. 3. The VCCSs can be conveniently realized using operational transconductance amplifiers [28]. From Kirchhoff's law, the circuit dynamics is represented by Via rescaling Eq. (7) is obtained, where E is a dummy variable that is introduced to make the voltage variables dimensionless.
For simplicity, we rewrite Eq. (7) as follows:ẋ where x = (x, y) and a i is given by Table 1.
It is easy to note that the solution of Eq. (11), for which the initial condition (τ ; x) = (0; x 0 ), is explicitly given as follows: The solution of Eq. (11) is illustrated in Fig. 4. Thus, a > 1 guarantees piecewise-constant-expanding oscillations. Note that the trajectory of Eq. (11) is piecewise-linear in vector fields. Furthermore, when τ increases, the explicit solution given by Eq. (12)   can be expressed in the following form: where n is a normal vector for the boundary, and D is a scalar. For example, if the flow hits y = 0, n = (0 1) and D = 0 represent the boundary condition. Therefore, substituting Eq. (12) into Eq. (13), the time of the arrival, which is denoted by τ 1 , satisfies and, thus, we obtain Hence, the solution on the boundary, which is denoted by x 1 , is represented by Furthermore, note that the solution of the variational equation can also be explicitly derived in piecewise-constant dynamics. For example, it is clear from Eq. (16) that the Jacobian matrix is represented as follows:

Chaos-generating piecewise-constant oscillator with an idealized diode
To generate chaos in a piecewise-constant circuit, for which the fundamental idea is explained in the previous section, we must increase the dimension of the oscillator shown in Fig. 3, because secondorder autonomous ODEs cannot generate chaos. Furthermore, we must add an energy-consuming element to the circuit because the amplitude of the oscillation of the circuit of Fig. 3 is increasing. Therefore, we adopt another set of a capacitor and a VCCS to increase the dimension of the oscillator, and we add a diode in series with the capacitor as an energy-consuming element. The chaos-generating piecewise-constant oscillator as modified is shown in Fig. 5. Chaos occurs when the dynamics has a stretching and folding mechanism. In this circuit, the VCCSs are considered to constitute the stretching mechanism, and the diode realizes the folding mechanism.
If the v-i characteristics of the diode are represented by a function of v 3 , the governing equation is represented by a third-order autonomous differential equation. For simplicity, we assume that the diode operates as an idealized switch, as shown in Fig. 6. Let the current through the diode be denoted   1. when the diode is off (v 3 < E holds):

when the diode is on
where Sgn i (i = 1, 2, 3) is a signum function defined by Eq. (8), and I s1 , I s2 , and I s3 are the amplitudes of the current sources. Equation (19)  is on because v 3 = E (const). Therefore, the current through the diode when it is on is I s3 · Sgn(v 1 ). These transition conditions are expressed as follows: Via rescaling the normalized equation is derived as follows: 1. when the diode is off: 2. when the diode is on: Note that 1 ab , 1 b , 1 is the equilibrium point when the diode is on, and the solution is divergently rotating around the equilibrium point constrained to the plane z = 1 when the diode is on. The details are explained below.
Furthermore, the transition condition is represented as follows: We consider the case where a > 1, and b > 0.
The condition a > 1 guarantees that the oscillation is expanding in the local systems Eqs. (22) and (23). So, b > 0 is obvious, because C 2 , C 3 , I s2 , and I s3 are positive. Equations (22), (23), and (24) can be rewritten similar to Eq. (11) as follows: where D i is a subset in the state space defined by Tables 2 and 3. Figure 7 shows a realization circuit diagram of the circuit shown in Fig. 5    Let us define a plane and a line segment as follows: where n 0 = (0, 0, 1) and α 0 = (0, 0, 1) . π is the plane that the diode is on. When the governing equation is represented by Eq. (23), the solution is piecewise-constantly divergent on π around 1 ab , 1 b , 1 (see Fig. 10) until the solution hits the line L 0 , which represents the transition condition 2 ON → 1 OFF. When hitting L 0 , the solution enters the region 1 OFF, where the equation is governed by Eq. (22). After wandering through the diode-off region, the solution returns to the plane π again.
Although it is possible to clarify all the bifurcation mechanisms, we focus on the proof of the generation of chaos with a positive Lyapunov exponent. We define a line segment L 1 and a half-line L  on the plane π as follows: where L 1 is a subset of L, and T h and T h 1 are given as follows: , The meanings of T h and T h 1 are explained below. Let us define two points P 1 and P on L as follows: P is a point on L such that the solution leaving P hits the singular point (0, 0, 1), as shown by the dashed line in Fig. 10. is located between P and (0, 0, 1) in Fig. 10 does not diverge and necessarily returns to L for any parameter set. Fig. 10 such that the solution leaving L 1 enters the diode-off region (z < 1), wanders through D 4 , D 1 , and D 0 , and hits π on L. Let us consider the solution leaving a point on L 1 .
the solution leaving L 1 returns to L.
The proof of this lemma is presented in Appendix A. Therefore, a 1D Poincaré return map f is defined on L 1 as follows: where y 0 is the y-coordinate of the initial point on L 1 , and y 1 is the y-coordinate at a point on L that the solution leaving L 1 returns to. The representation of f is given in Appendix B.
PTEP 2016, 053A01 T. Tsubone et al. An example of the return map is shown in Fig. 11. Suppose that f takes a maximum value at We consider the case f 2 (T h 0 ) > T h 1 . In this case, the Poincaré return map f is a two-segment piecewise-linear on J .

Lemma 2. If the parameters satisfy
a > 3, and The proof of this lemma can be found in Appendix C. We obtain the following theorem.  Hence, f 2 has an invariant measure that is absolutely continuous [26], ergodic [27], and has a positive Lyapunov exponent, which is strong evidence of chaos. Figure 12 represents a chaosgenerating region in the sense of Theorem 1. The proof of this theorem can be found in Appendix D.

Conclusion
In this study, we propose a simple chaos-generating circuit utilizing a diode, for which the governing equation is a third-order autonomous piecewise-constant constrained differential equation. First, we attempted to explain the fundamental idea of piecewise-constant dynamics by demonstrating a piecewise-constantly expanding oscillation in two dimensions. The expressions of the explicit solution, variation, and the boundary conditions were naturally extended to a higher-dimensional piecewise-constant oscillator. Second, we constructed a Poincaré return map explicitly, and analytically proved the generation of chaos with a positive Lyapunov exponent. Finally, we observed chaos generation in experimental measurements. Piecewise-constant dynamics is one of the simplest examples of piecewise-smooth dynamics and could contribute to the analysis of nonsmooth bifurcations.