Rich Superstable Phenomena in a Piecewise Constant ...
IEICE TRANS. FUNDAMENTALS, VOL.E89–A, NO.10 OCTOBER 2006
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PAPER
Special Section on Nonlinear Theory and its Applications
Rich Superstable Phenomena in a Piecewise Constant Nonautonomous Circuit with Impulsive Switching Yusuke MATSUOKA†a) , Student Member and Toshimichi SAITO† , Member
SUMMARY This paper studies rich superstable phenomena in a nonautonomous piecewise constant circuit including one impulsive switch. Since the vector field of circuit equation is piecewise constant, embedded return map is piecewise linear and can be described explicitly in principle. As parameters vary the map can have infinite extrema with one flat segment. Such maps can cause complicated periodic orbits that are superstable for initial state and are sensitive for parameters. Using a simple test circuit typical phenomena are verified experimentally. key words: chaos, bifurcation, piecewise constant systems, superstable periodic orbits
1.
Introduction
Impulsive switching is basic nonlinear operation that can cause interesting phenomena. We have studied impulsive switching of capacitor voltage when some condition of state and/or time is fulfilled. Applying such switching to some circuits, we can observe rich bifurcation phenomena of periodic pulse-train, chaos, hyperchaos and synchronization [1]–[3]. Such switching relates deeply to integrate-and-fire neuron models, pulse-coupled neural networks and their applications including associative memories and image segmentation [3]–[9]. Analysis of such switched dynamical systems is important for both basic theory and engineering applications. In this paper we study a simple nonautonomous piecewise constant (ab. PWC) circuit consisting of two capacitors, two signum voltage-controlled current sources (ab. VCCSs) and one impulsive switch. The impulsive switch depends on time and resets a capacitor voltage periodically to a dc base level. The vector field of circuit equation is PWC and trajectories are piecewise linear (ab. PWL) [10]. The embedded return map is one-dimensional PWL and can be described explicitly in principle: it is well suited for theoretical analysis. If the switch is open all the time, the trajectory is to be stable rect-spiral in phase plane and can reach an equilibrium point within finite time. In usual PWL systems a trajectory can not reach an equilibrium point but converges to it [1]. This reachable property can cause various interesting phenomena. The PWC system can generate chaos and it changes to a basic superstable periodic orbit (ab. SSPO) as parameters vary. Between the chaotic and periodic phases Manuscript received January 20, 2006. Final manuscript received May 16, 2006. † The authors are with the Department of Electronics, Electrical and Computer Engineering, Hosei University, Koganei-shi, 1848584 Japan. a) E-mail: [email protected] DOI: 10.1093/ietfec/e89–a.10.2767
we have confirmed various extremely complicated SSPOs: they are superstable for initial state and are very sensitive for parameters. The corresponding return maps have one flat segment and infinite extrema. It should be noted that the SSPOs have very short transient and can reach the steady state very fast. A simple test circuit is fabricated and typical phenomena are verified experimentally. SSPOs have been observed in various systems, e.g., flat-top tent maps relating to optimal limiter control [11], dc/dc converters in discontinuous conduction mode [12] and pulse-coupled artificial spiking neurons [13]. Comparing with these systems, our PWL map has much more complicated shape and exhibits richer phenomena. Our results may be developed into novel bifurcation theory and its application to pulse-coupled neural networks as suggested in [3], [11], [13]. Preliminary results can be found in [14]. 2.
Basic Dynamics of Circuit Model
Figure 1 shows the objective circuit model. The circuit consists of two capacitors, two voltage-controlled current sources (ab. VCCSs) and an impulsive switch S where r1 and r2 represent parasitic resistors. At every period t = nT , S is closed and the capacitor voltage v1 (t) is reset to the base voltage E < 0 where T is a period of clock signal and n is a non-negative integer. For simplicity we assume that the switching is ideal as is [1]–[3]: the reset is instantaneous without delay and continuity property of v2 (t) is held. We then overview the basic circuit dynamics. 2.1 VCCSs Having Linear Characteristics As a preparation, we consider the case where the VCCSs have linear characteristics. i1 = f1 (v1 , v2 ) = −gv1 , i2 = f2 (v1 , v2 ) = −g(v1 − v2 ),
(1)
where g > 0. For simplicity we ignore r1 and r2 . In this case the circuit dynamics is described by
Fig. 1
Objective circuit model with impulsive switching.
c 2006 The Institute of Electronics, Information and Communication Engineers Copyright
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⎧ dv1 ⎪ ⎪ ⎪ = −g(v1 − v2 ) C1 ⎪ ⎪ ⎪ dt ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ dv2 ⎪ ⎪ ⎪ = −gv1 ⎩ C2 dt
S =off,
for
(2)
(v1 (t+), v2 (t+)) = (E, v2 (t)) at t = nT, where t+ ≡ lim t + . Using following dimensionless vari→0
ables and parameters, Eq. (3) is transformed into normalized Eq. (3). 1 dx g 1 v1 , x˙ ≡ v2 , t, x = τ= , y= C1 |E| dτ a|E| C1 g a= , d= T. C2 C1 ⎧ ⎪ ⎪ ⎨ x˙ = −(x − ay) for S =off, ⎪ ⎪ ⎩ y˙ = −(x) (3) (x(τ+), y(τ+)) = (−1, y(τ)) at τ = nd. This system has two positive parameters a and d. For simplicity we assume a > 1 and consider the case of S = open all the time. Let us consider a trajectory starting from (0, Y0 ) at τ = 0 (Fig. 2(a)) and let (0, Yn ) be n-th intersection of the trajectory and y-axis where n is a positive integer. Let the intersection be represented by its y component. The inter1 sections are given by Yn = √ −e− 2 τn Yn−1 and the time from π Yn−1 to Yn is ω where ω ≡ 4a − 1. It is well known that the trajectory converges to the origin as τ goes to ∞. When the switching is applied, the circuit exhibits periodic orbits only as shown in Fig. 2(b). Its theoretical evidence can be found in [1].
2.2 VCCSs Having Signum Characteristics We then consider the case of signum VCCS: ⎧ ⎪ 1 for ⎪ ⎪ i1 = f1 (v1 , v2 ) = I1 sgn(v1 ) ⎨ 0 for sgn(x) = ⎪ ⎪ i2 = f2 (v1 , v2 ) = I2 sgn(v1 −v2 ) ⎪ ⎩ −1 for
x>0 x = 0 (4) x < 0,
where I1 < 0 and I2 < 0. For simplicity we assume that resistors r1 and r2 are large enough and open them. The circuit dynamics is described by ⎧ dv1 ⎪ ⎪ ⎪ = I2 sgn(v1 − v2 ) C ⎪ ⎪ ⎨ 1 dt ⎪ ⎪ ⎪ dv ⎪ ⎪ ⎩ C2 2 = I1 sgn(v1 ) dt
for
S =off, (5)
(v1 (t+), v2 (t+)) = (E, v2 (t)) at t = nT, where n is a non-negative integer. Using following dimensionless variables and parameters, Eq. (5) is transformed into normalized Eq. (6). 1 1 dx |I2 | t, x = v1 , x˙ ≡ , y= v2 , τ= C1 |E| |E| dτ a|E| C1 |I1 | |I2 | a= , d= T. C2 |I2 | C1 |E| x˙ = −sgn(x − ay) for S =off, y˙ = −sgn(x) (6) (x(τ+), y(τ+)) = (−1, y(τ)) at τ = nd. This system has two positive real parameters a and d. a is proportional to a ratio of capacitance and d is proportional to T . The vector field of circuit equation is PWC and trajectories are PWL. As a preparation, we consider the case where S is open all the time (d = ∞). For simplicity we assume a > 1.
(a)
(b)
(c) Fig. 2 Typical dynamics of PWL circuit for a > 1. (a) Basic trajectory without switching, (b) periodic attractor for a = 50, d = 1.7, (c) wave form where CLK is a clock signal.
(7)
In this case trajectories are stable rectangular-spiral and rotate convergently to the origin [14] as shown in the Fig. 3. In order to clarify the characteristics of trajectory, we define some key points. First, let us consider a trajectory starting form (−1, y0 ) at τ = 0 and let (0, Yn ) be n-th intersection of trajectory and y-axis (see Fig. 3(b)). The first intersection is given by ⎧ 1 ⎪ ⎪ ⎪ for y0 ≥ − , y0 + 1 ⎪ ⎪ ⎨ a Y1 = ⎪ (8) ⎪ ⎪ 1 ⎪ ⎪ ⎩ −α(y0 − 1) for y0 < − , a where α ≡ a−1 a+1 and 0 < α < 1. Second, let τn be the time at which the trajectory reaches Yn (i.e., y(τn ) = Yn ). Then Yn and τn are given by
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(a) (a)
(b)
(c)
(d)
(e)
(f)
(b) Fig. 3 Behavior of trajectory without switching in PWC circuit for a > 1. (a) Time domain, (b) phase plane.
(a)
Fig. 5 Typical phenomena in PWC circuit for a = 4.7. (a) Trajectories diverge for d = 1.6, (b) chaotic attractor for d = 2.25, (c) chaotic attractor for d = 3.0, (d) complicated SSPOs for d = 4.8, (e) complicated SSPOs for d = 5.4, (f) basic SSPO for d = 6.0.
⎧ ⎪ ⎪ ⎪ a(y0 + 1) + 1 ⎪ ⎪ ⎨ τ∞ = ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ −a(y0 − 1) − 1 (b) Fig. 4 Basic dynamics of PWC circuit with switching for a > 1. (a) Time domain where CLK is a clock signal, (b) phase plane.
⎧ ⎪ ⎪ ⎪ (−α)n−1 (y0 + 1) ⎪ ⎪ ⎨ Yn = (−α)Yn−1 = ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ (−α)n (y0 − 1) n τn+1 = τ1 + |Yl+1 − Yl |
1 a 1 for y0 < − , a
for y0 ≥ −
(9)
l=1
⎧ ⎪ ⎪ ⎪ (1 − αn )a(y0 + 1) + 1 ⎪ ⎪ ⎨ =⎪ ⎪ ⎪ ⎪ ⎪ ⎩ −(1 − αn+1 )a(y0 − 1) − 1
1 a 1 for y0 < − . a
for y0 ≥ −
It should be noted that the signum VCCSs cause interesting property τn+1 < τn . It is a great difference from circuit with linear VCCSs [1]. We then have
1 a 1 for y0 < − . a
for y0 ≥ −
(10)
where τ∞ is the time at which the trajectory reaches the origin: (y(τ∞ ) = 0). Eq. (10) means that the trajectory can reach the origin within finite time provided a is finite. Such reachability can cause very interesting phenomena shown in Sects. 3 and 4. In fact, when the switching is applied, the PWC circuit exhibits various interesting phenomena as illustrated in Fig. 4 to Fig. 6. As parameter d increases, chaotic attractor (Figs. 5(b) and (c)) is changed into the basic SSPO with period d (Figs. 5(f) and Fig. 6(c)). Formal definition of SSPO can be found in Sect. 3. Between the chaos and basic SSPO, the system generates a variety of complicated phenomena. For example Figs. 5(d) and (e) seem to be chaotic, however, it is not chaos but complicated SSPOs. 3.
Piecewise Linear Return Maps with a Flat Segment
In order to derive the return map, let LD = {(x, y, τ) | x = −1and τ = nd} and let a point on LD be represented by its y coordinate as shown in Fig. 7. We consider a trajectory
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(a)
(b)
(c) Fig. 6 Typical wave forms in PWC circuit for a = 4.7 where CLK is a clock signal. (a) Chaotic behavior for d = 2.25, (b) complicated SSPOs for d = 5.4, (c) basic SSPO for d = 6.0. (a) to (c) correspond to Figs. 5(b), (e) and (f), respectively.
Fig. 8 Typical shape of return maps for a = 4.7. (a) Trajectories diverge for d = 1.6, (b) chaos for d = 2.25, (c) chaos for d = 3.0, (d) complicated SSPOs for d = 4.8, (e) complicated SSPOs for d = 5.4, (f) basic SSPO for d = 6.0. SFP is superstable fixed point and J is an invariant interval.
point y f is said to be superstable for initial state if Eq. (12) is satisfied. F k (y s f ) = y s f , F l (y s f ) y s f ,
Fig. 7
Definition of the return map in phase plane.
starting from LD just after the switching at τ = 0. The n-th switching occurs at τ = nd and yn ∈ LD is satisfied where yn ≡ y(nd). Since yn determines yn+1 , we can define onedimensional return map F from LD to itself: F : LD −→ LD ,
yn+1 = F(yn ).
(11)
Since the trajectories are PWL, the map can be expressed in principle. Figure 8 shows return maps corresponding to trajectories in Fig. 5. The calculation method is shown in Appendix. In order to consider periodic behavior we give some definitions. A point y f ∈ LD is said to be a period-k point if F k (y f ) = y f and F l (y f ) y f for 1 ≤ l < k where F k denotes the k-fold composition of F and k is a positive integer. We refer to y f with k = 1 as a fixed point. A periodic point y f is d k F (y f )| < 1. A periodic said to be stable for initial state if | dy
d k F (y s f ) = 0. dy
(12)
A periodic orbit consisting of such superstable periodic points, {F(y s f ), F 2 (y s f ), · · · , F k (y s f )} is referred to as a superstable periodic orbit and abbreviated by SSPO. Superstable fixed point is abbreviated by SFP. This SFP corresponds to basic SSPO in Fig. 5(f). We then summarize shape of return maps as the following. (1) The maps are PWL and continuous. The number of segments in the map increases as d increases. d F(y)| ≥ 1 is satisfied for all y as (2) In range of d < a, | dy shown in Figs. 8(a) to (c). (3) When d > a, the maps have one flat segment and infinite d F(y) = 0 is satisfied on the flat segment (see extrema. dy Figs. 8(d) to (f)). (4) If d > a + 1, the map has SFP as shown in Fig. 8(f). Referring to these and map formula in Appendix we can say the following properties. Property1: If d < 2, yn < yn+1 is satisfied for all y as shown in Fig. 8(a). In this case trajectory is divergence. Property2: If 2 < d < a, the return maps have an invariant d F(y)| ≥ 1 are satisfied interval J such that J ⊆ F(J) and | dy
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Fig. 9
Basic existence regions of each phenomena.
on J. In this case the maps generate chaotic behavior on J [15] (see Figs. 8(b) and (c)). Property3: If d > a + 1, the maps have SFP and the circuit exhibits basic SSPO as shown in Fig. 8(f) and Fig. 5(f). Basic SSPO starting from (−1, 0) at τ = 0 has reached the origin at τ = a + 1. It stays on the origin for a + 1 < τ < d and returns (jumps) to (−1, 0) at τ = d. The system repeats in this manner and exhibits basic SSPO in the steady state. Existence regions of these phenomena are shown in Fig. 9: trajectories diverge for d < 2, the circuit exhibits chaotic behavior for 2 < d < a and exhibits basic SSPO for d > a + 1. For a < d < a + 1 the maps have the flat segment, however, SFP does not exist (see Figs. 8(d) and (e)). In this case the circuit can generate various complicated SSPOs. 4.
Fig. 10 Bifurcation diagrams for a = 4.7. (b) is an enlargement of a part of (a). Between d = 4.7 and d = 5.7, complicated SSPOs are generated.
Various Phenomena Including Complicated Superstable Periodic Orbits
Here we focus on the following parameter range: a < d < a + 1.
(13)
In this range the circuit can exhibit rich complicated phenomena, however, their theoretical analysis is very hard and we have performed numerical simulations. Figure 10 shows bifurcation diagrams for a = 4.7. In Fig. 10(b), between d = 4.7 and d = 5.7 the orbits seem to be chaotic orbit. But these orbits are complicated SSPOs. We have confirmed that these SSPOs start from the flat segment (a point of (−1, 0)) and return to the flat segment within finite time. It should be noted that the SSPOs are superstable for initial state but can be very sensitive for parameters. Figure 11 illustrates distributions of a period of complicated SSPOs where PCS denotes a period of complicated SSPOs divided by d (or switching frequency per a period of SSPOs). PCS shows huge complicated variation for small perturbation of d. Figures 11(b) to (d) are enlargements of a part of (a) to (c), respectively. Distributions of PCS have self-similarlike structures. It is arisen from a flat segment and infinite extrema of the return maps. A binary expression of distributions of PCS in parameter space are shown in Fig. 12. A black region means PCS ≤ 5 in Fig. 12(b). PCS shows complicated distributions extensively and varies extremely for a slight variation of a and d.
Fig. 11 Distributions of PCS for a = 4.7. (b) to (d) are enlargements of a part of (a) to (c), respectively.
If parameter a varies in the range of a < d < a + 1, various complicated phenomena can exist as shown in Fig. 13. In Figs. 13(a) and (b), the return maps have the flat segment, whereas the orbits are chaotic orbits in an invariant interval J1 after transient process. Whenever the orbit hits the flat segment, it enters into J1 and becomes chaotic orbit. These phenomena correspond to the white regions in Fig. 12(a). We have also confirmed co-existence phenomena of complicated SSPOs and chaos as shown in Figs. 13(c) and (d). In these maps the orbits hit the flat segment and return to the
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Fig. 12 Binary expressions of PCS distribution in parameter range of a < d < a + 1. (a) In the black regions complicated SSPOs exist. In the white regions chaotic orbits exist. (b) The black regions is PCS ≤ 5.
(a)
(b) Fig. 15 A test circuit using OTAs and typical data. E = −1 V, I1 = I2 0.12 mA, C1 47 nF, C2 10 nF (a = 4.7). (a) Chaos for T 0.88 ms (d=2.25), (b) basic SSPO for T 2.4 ms (d=6.0). (a) and (b) correspond to Figs. 5(b) and (f) and Figs. 6(a) and (c), respectively. Horizontal = v1 [1 V/div.], vertical = v2 [5 V/div.] for phase plane (left columns). Horizontal = t [1 ms/div.], vertical = v1 [1 V/div.] for time-domain (right columns). Fig. 13 Complicated phenomena of return maps with a flat segment for a = 2.2. (a) Chaos for d = 2.3, (b) chaos for d = 2.9, (c) co-existence state of complicated SSPOs and chaos for d = 3.1, (d) co-existence state of complicated SSPOs and chaos for d = 3.14. J1 and J2 are invariant intervals, respectively.
flat segment after several switchings. In this case the orbits are complicated SSPOs. On the other hand invariant interval J2 can exist and the chaotic orbit appears in J2 . Figures 10 to 14 suggest huge variety of SSPOs. 5.
Laboratory Experiments
In order to observe typical phenomena we have fabricated a simple test circuit as shown in Fig. 15. The VCCSs are realized by operational transconductance amplifiers OTAs (LM13600). We assume that OTAs have signum function as Eq. (4). An impulsive switch is implemented using an analog switch S (4066), a dc voltage source E and a clock signal with period T . When clock is applied to the PWC circuit, S is closed and capacitor voltage v1 is reset to the base voltage E. Figures 15(a) and (b) show typical data corresponding to Figs. 5(b) and (f) and Figs. 6(a) and (c), respectively. As a period of clock d increases, it is confirmed that chaotic behavior is changed into basic SSPO. 6.
Fig. 14 Bifurcation diagrams of complicated phenomena for a = 2.2. (a) d increases from d = 2.2, (b) d decreases from d = 3.2.
Conclusions
We have considered a simple nonautonomous PWC circuit with impulsive switching. The vector field is PWC, trajectories are PWL and embedded return map is PWL. The map can have one flat segment and infinite extrema: it causes
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chaos and rich SSPOs. Using the map we have investigated the rich phenomena. Using a simple test circuit, typical phenomena are verified experimentally. Future problems include analysis of bifurcation of SSPOs, design of practical circuits and consideration of engineering applications. Acknowledgments The authors wish to thank Dr. H. Torikai for his helpful comments. This work is supported in part by JSPS KAKENHI under Grant 17560269. References
(a) (b) Key objects in phase plane. (a) Definition of τ˜ n , (b) definition
τ˜ 1 = 1, τ˜ n+1 = (a + αn (1 − a)),
[1] K. Miyachi, H. Nakano, and T. Saito, “Response of a simple dependent switched capacitor circuit to a pulse-train input,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol.50, no.9, pp.1180–1187, 2003. [2] Y. Takahashi, H. Nakano, and T. Saito, “A simple hyperchaos generator based on impulsive switching,” IEEE Trans. Circuits Syst. II, Analog Digit. Signal Process., vol.51, no.9, pp.468–472, 2004. [3] H. Nakano and T. Saito, “Grouping synchronization in a pulsecoupled network of chaotic spiking oscillators,” IEEE Trans. Neural Netw., vol.15, no.5, pp.1018–1026, 2004. [4] J.P. Keener, F.C. Hoppensteadt, and J. Rinzel, “Integrate-and-fire models of nerve membrane response to oscillatory input,” SIAM J. Appl. Math., vol.41, pp.503–517, 1981. [5] R.E. Mirollo and S.H. Strogatz, “Synchronization of pulse-coupled biological oscillators,” SIAM J. Appl. Math., vol.50, pp.1645–1662, 1990. [6] E.M. Izhikevich, “Resonate-and-fire neurons,” Neural Netw., vol.14, pp.883–894, 2001. [7] G. Lee and N.H. Farhat, “The bifurcating neuron network 1,” Neural Netw., vol.14, pp.115–131, 2001. [8] S.R. Campbell, D. Wang, and C. Jayaprakash, “Synchrony and desynchrony in integrate-and-fire oscillators,” Neural Computation, vol.11, pp.1595–1619, 1999. [9] E.M. Izhikevich, “Weakly Pulse-coupled oscillators, FM Interactions, Synchronization, and oscillatory associative memory,” IEEE Trans. Neural Netw., vol.10, no.3, pp.508–526, 1999. [10] T. Tsubone and T. Saito, “Manifold piecewise constant systems and chaos,” IEICE Trans. Fundamentals, vol.E82-A, no.8, pp.1619– 1626, Aug. 1999. [11] C. Wagner and R. Stoop, “Renormalization approach to optimal limiter control in 1-D chaotic systems,” J. Statistical Physics, vol.106, pp.97–107, 2002. [12] T. Kabe, T. Saito, and H. Torikai, “Analysis of piecewise constant models of power converters,” Proc. NOLTA, pp.71–74, 2004. [13] Y. Kon’no, T. Saito, and H. Torikai, “Rich dynamics of pulsecoupled spiking neurons with a triangular base signal,” Neural Netw., vol.18, pp.523–531, 2005. [14] Y. Matsuoka, T. Saito, and H. Torikai, “A piecewise constant switched chaotic circuit with rect-rippling return maps,” Proc. ISCAS, pp.3411–3414, 2005. [15] A. Lasota and M.C. Mackey, Chaos, Fractals, and Noise, second ed., Springer-Verlag, 1994.
Appendix:
Fig. A· 1 of Dn+1 .
Formulation of Return Map
First let us consider a trajectory starting from (−1, − a1 ) at τ = 0 shown in Fig. A· 1(a). Let the trajectory reach the origin as discussed in Sect. 2.2 and let τ˜ n be the n-th intersection time. Noting Eq. (9) τ˜ n is given by
τ˜ ∞ = (a + α∞ (1 − a)) = a.
(A· 1)
where τ˜ ∞ that is the limit of τ˜ n at which the trajectory reaches the origin. Next, we consider trajectories starting from Ld as shown in Fig. A· 1(b). Let Yn be n-th intersection and τn be n-th intersection time. We then consider the following parameter condition. τ˜ m ≤ d < τ˜ m+1 < τ˜ ∞ ,
(A· 2)
where m is a finite positive integer. Condition (A· 2) guarantees that Yn and τn exist for 1 ≤ n ≤ m and the intersection is at most m-times. We then define some key points Dn+1 and Dˆ n on LD . Let Dn+1 > − a1 be a point such that the trajectory starting from Dn+1 reaches Yn+1 at time τn+1 = d and jumps onto LD instantaneously: F(Dn+1 ) = Yn+1 . It is given d−1 1 ˆ by Dn+1 = a(1−α n ) − 1. Let Dn < − a be a point such that the trajectory starting from Dˆ n reaches Yn at time τn = d and jumps onto LD instantaneously: F(Dˆ n ) = Yn . It is given by d+1 Dˆ n = − a(1−α n+1 ) + 1. Then we have Theorem1: For Condition (A· 2), the return map is given by yn+1 = Fm (yn ) where ⎧ ⎪ for D2 ≤ y f1 (y) ⎪ ⎪ ⎪ ⎪ ⎪ fn+1 (y) for Dn+2 ≤ y < Dn+1 ⎪ ⎪ ⎪ ⎪ ⎪ for − 1a ≤ y < Dm ⎨ fm (y) (A· 3) Fm (y) = ⎪ ⎪ gm (y) for Dˆ m ≤ y < − a1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ gn (y) for Dˆ n ≤ y < Dˆ n+1 ⎪ ⎪ ⎪ ⎩ g (y) for y < Dˆ 1 0 fn (y) ≡ (−1)n−1 (βn (y + 1) + 1 − d), gn (y) ≡ (−1)n (κn (y − 1) + 1 + d). κn ≡ 1 + 2α + · · · + 2αn > 1, βn ≡ κn−1 , κ0 = β1 ≡ 1, n ∈ {1, 2, · · · , m − 1}, Proof : We show only for yn ≥ − 1a . If D2 < yn , the trajectories hit y-axis only once and the first intersection is given by Y1 = yn + 1 is satisfied. Since the trajectories reach Y1 at τ = 1 and y decreases for 1 < τ ≤ d, we obtain yn+1 = Y1 − (d − 1) = yn − (d − 2) ≡ f1 (yn ). If Dn+2 < yn < Dn+1 , the trajectories hit y-axis (n + 1)-times hence Yn+1 exists. Using Eq. (9) we obtain Yn+1 = (−α)n (yn + 1) and τn+1 = (1 − αn )a(yn + 1) + 1. For odd n, y increases after n + 1-st intersection ( τn+1 < τ < d). In this case we have yn+1 = Yn+1 +(d−τn+1 ) = (−1)n (αn +a(1−αn ))(yn +1)−1+d ≡
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fn+1 (y) where αn + a(1 − αn ) ≡ βn+1 . For even n, y decreases for τn+1 < τ < d and we have yn+1 = Yn+1 − (d − τn+1 ) = (αn +a(1−αn ))(yn +1)+1−d ≡ fn+1 (y). If − 1a ≤ yn < Dm , the trajectories hit y-axis m-times hence Ym exists. In a likewise manner as Dn+2 < yn < Dn+1 , we can derive fm . Similar consideration is possible for yn < − 1a . Q. E. D. (Remark) Figures 8(b) and (c) show the case of m = 1 and m = 2, respectively. When m = 1, f1 (y) for − 1a < y and n does not mean. The points Dˆ 1 < · · · < Dˆ m ≤ − 1a ≤ Dm < · · · < D2 are breakpoints. When d approaches to τ˜ ∞ , m increases and Dm and Dˆ m approach to − 1a : lim Dm = m→∞ lim Dˆ m = − 1 . The maps have an invariant interval J as m→∞
a
shown in Figs. 8 (b) and (c) where J ≡ [Y− , Y+ ], Y+ ≡ max(F(Dˆ 1 ), F(D3 )) and Y− ≡ min(F(Dˆ 2 ), F(D2 ), F(− a1 )). Since βn > 1 for n ≥ 2 and κn > 1 for n ≥ 1, the map is expanding on J and the system exhibits chaos [15]. Next we consider the parameter condition: d > τ˜ ∞ = a.
(A· 4)
In this range the trajectory starting from (−1, − a1 ) intersects y-axis infinitely and can reach the origin at τ = τ˜ ∞ . Let D∞ > − a1 (respectively, Dˆ ∞ < − 1a ) be a point on LD such that the trajectory starting from D∞ (respectively, Dˆ ∞ ) reaches the origin at τ = d (F(D∞ ) = F(Dˆ ∞ ) = 0). They are given by D∞ = 1a (d − 1) − 1 and Dˆ ∞ = − a1 (d + 1) + 1. In a likewise manner as proof of Theorem 1, we obtain Theorem2: For d > τ∞ = a, the return map is given by yn+1 = F∞ (yn ) where ⎧ for D2 ≤ y f1 (y) ⎪ ⎪ ⎪ ⎪ ⎪ (y) for Dn+2 ≤ y < Dn+1 f ⎪ n+1 ⎪ ⎪ ⎨ 0 for Dˆ ∞ ≤ y < D∞ (A· 5) F∞ (y) = ⎪ ⎪ ⎪ ⎪ ⎪ gn (y) for Dˆ n ≤ y < Dˆ n+1 ⎪ ⎪ ⎪ ⎩ g (y) for y < Dˆ 1 , 0 where n is a positive integer. If Dˆ ∞ ≤ yn < D∞ , the trajectories can reach the origin at τ = τ∞ < d: yn+1 = F(yn ) = 0. (Remark) Typical return maps are shown in Figs. 8(d) d F∞ (y)) is 0 for Dˆ ∞ ≤ y ≤ D∞ , to (f). Slope of the map ( dy d otherwise | dy F(y)| ≥ 1. The map consists of infinite segments and has infinite extrema. The segments accumulate to Dˆ ∞ or D∞ . For d > a + 1, the map has SFP and the circuit generates basic SSPO. Using the map precise numerical simulation is possible, however, we must approximate the map because it is impossible to calculate and the maps having infinite segments. After trial-and-errors, y is regarded to reach 0 if |y| < 10−7 in the numerical calculation. For example, the map of Eq. (A· 5) can be approximated for n = 85, d = 5.0 and 2.0 < a < 10.0.
Yusuke Matsuoka received the B.E. and M.E. degrees in electrical engineering from Hosei University, Tokyo, Japan, in 2004 and 2006, respectively. He is currently working toward the Ph.D. degree at the Department of Electronics and Electrical Engineering, Hosei University. His research interests are in chaos and bifurcation.
Toshimichi Saito received the B.E., M.E, and Ph.D. degrees in electrical engineering all from Keio University, Yokohama, Japan, in 1980, 1982 and 1985, respectively. He is currently a Professor with the Department of Electronics, Electrical and Computer Engineering, Hosei University, Tokyo. His current research interests include chaos and bifurcation, artificial neural networks, power electronics and signal processing. He is a senior member of the IEEE and a member of the INNS.
2767
PAPER
Special Section on Nonlinear Theory and its Applications
Rich Superstable Phenomena in a Piecewise Constant Nonautonomous Circuit with Impulsive Switching Yusuke MATSUOKA†a) , Student Member and Toshimichi SAITO† , Member
SUMMARY This paper studies rich superstable phenomena in a nonautonomous piecewise constant circuit including one impulsive switch. Since the vector field of circuit equation is piecewise constant, embedded return map is piecewise linear and can be described explicitly in principle. As parameters vary the map can have infinite extrema with one flat segment. Such maps can cause complicated periodic orbits that are superstable for initial state and are sensitive for parameters. Using a simple test circuit typical phenomena are verified experimentally. key words: chaos, bifurcation, piecewise constant systems, superstable periodic orbits
1.
Introduction
Impulsive switching is basic nonlinear operation that can cause interesting phenomena. We have studied impulsive switching of capacitor voltage when some condition of state and/or time is fulfilled. Applying such switching to some circuits, we can observe rich bifurcation phenomena of periodic pulse-train, chaos, hyperchaos and synchronization [1]–[3]. Such switching relates deeply to integrate-and-fire neuron models, pulse-coupled neural networks and their applications including associative memories and image segmentation [3]–[9]. Analysis of such switched dynamical systems is important for both basic theory and engineering applications. In this paper we study a simple nonautonomous piecewise constant (ab. PWC) circuit consisting of two capacitors, two signum voltage-controlled current sources (ab. VCCSs) and one impulsive switch. The impulsive switch depends on time and resets a capacitor voltage periodically to a dc base level. The vector field of circuit equation is PWC and trajectories are piecewise linear (ab. PWL) [10]. The embedded return map is one-dimensional PWL and can be described explicitly in principle: it is well suited for theoretical analysis. If the switch is open all the time, the trajectory is to be stable rect-spiral in phase plane and can reach an equilibrium point within finite time. In usual PWL systems a trajectory can not reach an equilibrium point but converges to it [1]. This reachable property can cause various interesting phenomena. The PWC system can generate chaos and it changes to a basic superstable periodic orbit (ab. SSPO) as parameters vary. Between the chaotic and periodic phases Manuscript received January 20, 2006. Final manuscript received May 16, 2006. † The authors are with the Department of Electronics, Electrical and Computer Engineering, Hosei University, Koganei-shi, 1848584 Japan. a) E-mail: [email protected] DOI: 10.1093/ietfec/e89–a.10.2767
we have confirmed various extremely complicated SSPOs: they are superstable for initial state and are very sensitive for parameters. The corresponding return maps have one flat segment and infinite extrema. It should be noted that the SSPOs have very short transient and can reach the steady state very fast. A simple test circuit is fabricated and typical phenomena are verified experimentally. SSPOs have been observed in various systems, e.g., flat-top tent maps relating to optimal limiter control [11], dc/dc converters in discontinuous conduction mode [12] and pulse-coupled artificial spiking neurons [13]. Comparing with these systems, our PWL map has much more complicated shape and exhibits richer phenomena. Our results may be developed into novel bifurcation theory and its application to pulse-coupled neural networks as suggested in [3], [11], [13]. Preliminary results can be found in [14]. 2.
Basic Dynamics of Circuit Model
Figure 1 shows the objective circuit model. The circuit consists of two capacitors, two voltage-controlled current sources (ab. VCCSs) and an impulsive switch S where r1 and r2 represent parasitic resistors. At every period t = nT , S is closed and the capacitor voltage v1 (t) is reset to the base voltage E < 0 where T is a period of clock signal and n is a non-negative integer. For simplicity we assume that the switching is ideal as is [1]–[3]: the reset is instantaneous without delay and continuity property of v2 (t) is held. We then overview the basic circuit dynamics. 2.1 VCCSs Having Linear Characteristics As a preparation, we consider the case where the VCCSs have linear characteristics. i1 = f1 (v1 , v2 ) = −gv1 , i2 = f2 (v1 , v2 ) = −g(v1 − v2 ),
(1)
where g > 0. For simplicity we ignore r1 and r2 . In this case the circuit dynamics is described by
Fig. 1
Objective circuit model with impulsive switching.
c 2006 The Institute of Electronics, Information and Communication Engineers Copyright
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⎧ dv1 ⎪ ⎪ ⎪ = −g(v1 − v2 ) C1 ⎪ ⎪ ⎪ dt ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ dv2 ⎪ ⎪ ⎪ = −gv1 ⎩ C2 dt
S =off,
for
(2)
(v1 (t+), v2 (t+)) = (E, v2 (t)) at t = nT, where t+ ≡ lim t + . Using following dimensionless vari→0
ables and parameters, Eq. (3) is transformed into normalized Eq. (3). 1 dx g 1 v1 , x˙ ≡ v2 , t, x = τ= , y= C1 |E| dτ a|E| C1 g a= , d= T. C2 C1 ⎧ ⎪ ⎪ ⎨ x˙ = −(x − ay) for S =off, ⎪ ⎪ ⎩ y˙ = −(x) (3) (x(τ+), y(τ+)) = (−1, y(τ)) at τ = nd. This system has two positive parameters a and d. For simplicity we assume a > 1 and consider the case of S = open all the time. Let us consider a trajectory starting from (0, Y0 ) at τ = 0 (Fig. 2(a)) and let (0, Yn ) be n-th intersection of the trajectory and y-axis where n is a positive integer. Let the intersection be represented by its y component. The inter1 sections are given by Yn = √ −e− 2 τn Yn−1 and the time from π Yn−1 to Yn is ω where ω ≡ 4a − 1. It is well known that the trajectory converges to the origin as τ goes to ∞. When the switching is applied, the circuit exhibits periodic orbits only as shown in Fig. 2(b). Its theoretical evidence can be found in [1].
2.2 VCCSs Having Signum Characteristics We then consider the case of signum VCCS: ⎧ ⎪ 1 for ⎪ ⎪ i1 = f1 (v1 , v2 ) = I1 sgn(v1 ) ⎨ 0 for sgn(x) = ⎪ ⎪ i2 = f2 (v1 , v2 ) = I2 sgn(v1 −v2 ) ⎪ ⎩ −1 for
x>0 x = 0 (4) x < 0,
where I1 < 0 and I2 < 0. For simplicity we assume that resistors r1 and r2 are large enough and open them. The circuit dynamics is described by ⎧ dv1 ⎪ ⎪ ⎪ = I2 sgn(v1 − v2 ) C ⎪ ⎪ ⎨ 1 dt ⎪ ⎪ ⎪ dv ⎪ ⎪ ⎩ C2 2 = I1 sgn(v1 ) dt
for
S =off, (5)
(v1 (t+), v2 (t+)) = (E, v2 (t)) at t = nT, where n is a non-negative integer. Using following dimensionless variables and parameters, Eq. (5) is transformed into normalized Eq. (6). 1 1 dx |I2 | t, x = v1 , x˙ ≡ , y= v2 , τ= C1 |E| |E| dτ a|E| C1 |I1 | |I2 | a= , d= T. C2 |I2 | C1 |E| x˙ = −sgn(x − ay) for S =off, y˙ = −sgn(x) (6) (x(τ+), y(τ+)) = (−1, y(τ)) at τ = nd. This system has two positive real parameters a and d. a is proportional to a ratio of capacitance and d is proportional to T . The vector field of circuit equation is PWC and trajectories are PWL. As a preparation, we consider the case where S is open all the time (d = ∞). For simplicity we assume a > 1.
(a)
(b)
(c) Fig. 2 Typical dynamics of PWL circuit for a > 1. (a) Basic trajectory without switching, (b) periodic attractor for a = 50, d = 1.7, (c) wave form where CLK is a clock signal.
(7)
In this case trajectories are stable rectangular-spiral and rotate convergently to the origin [14] as shown in the Fig. 3. In order to clarify the characteristics of trajectory, we define some key points. First, let us consider a trajectory starting form (−1, y0 ) at τ = 0 and let (0, Yn ) be n-th intersection of trajectory and y-axis (see Fig. 3(b)). The first intersection is given by ⎧ 1 ⎪ ⎪ ⎪ for y0 ≥ − , y0 + 1 ⎪ ⎪ ⎨ a Y1 = ⎪ (8) ⎪ ⎪ 1 ⎪ ⎪ ⎩ −α(y0 − 1) for y0 < − , a where α ≡ a−1 a+1 and 0 < α < 1. Second, let τn be the time at which the trajectory reaches Yn (i.e., y(τn ) = Yn ). Then Yn and τn are given by
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(a) (a)
(b)
(c)
(d)
(e)
(f)
(b) Fig. 3 Behavior of trajectory without switching in PWC circuit for a > 1. (a) Time domain, (b) phase plane.
(a)
Fig. 5 Typical phenomena in PWC circuit for a = 4.7. (a) Trajectories diverge for d = 1.6, (b) chaotic attractor for d = 2.25, (c) chaotic attractor for d = 3.0, (d) complicated SSPOs for d = 4.8, (e) complicated SSPOs for d = 5.4, (f) basic SSPO for d = 6.0.
⎧ ⎪ ⎪ ⎪ a(y0 + 1) + 1 ⎪ ⎪ ⎨ τ∞ = ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ −a(y0 − 1) − 1 (b) Fig. 4 Basic dynamics of PWC circuit with switching for a > 1. (a) Time domain where CLK is a clock signal, (b) phase plane.
⎧ ⎪ ⎪ ⎪ (−α)n−1 (y0 + 1) ⎪ ⎪ ⎨ Yn = (−α)Yn−1 = ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ (−α)n (y0 − 1) n τn+1 = τ1 + |Yl+1 − Yl |
1 a 1 for y0 < − , a
for y0 ≥ −
(9)
l=1
⎧ ⎪ ⎪ ⎪ (1 − αn )a(y0 + 1) + 1 ⎪ ⎪ ⎨ =⎪ ⎪ ⎪ ⎪ ⎪ ⎩ −(1 − αn+1 )a(y0 − 1) − 1
1 a 1 for y0 < − . a
for y0 ≥ −
It should be noted that the signum VCCSs cause interesting property τn+1 < τn . It is a great difference from circuit with linear VCCSs [1]. We then have
1 a 1 for y0 < − . a
for y0 ≥ −
(10)
where τ∞ is the time at which the trajectory reaches the origin: (y(τ∞ ) = 0). Eq. (10) means that the trajectory can reach the origin within finite time provided a is finite. Such reachability can cause very interesting phenomena shown in Sects. 3 and 4. In fact, when the switching is applied, the PWC circuit exhibits various interesting phenomena as illustrated in Fig. 4 to Fig. 6. As parameter d increases, chaotic attractor (Figs. 5(b) and (c)) is changed into the basic SSPO with period d (Figs. 5(f) and Fig. 6(c)). Formal definition of SSPO can be found in Sect. 3. Between the chaos and basic SSPO, the system generates a variety of complicated phenomena. For example Figs. 5(d) and (e) seem to be chaotic, however, it is not chaos but complicated SSPOs. 3.
Piecewise Linear Return Maps with a Flat Segment
In order to derive the return map, let LD = {(x, y, τ) | x = −1and τ = nd} and let a point on LD be represented by its y coordinate as shown in Fig. 7. We consider a trajectory
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(a)
(b)
(c) Fig. 6 Typical wave forms in PWC circuit for a = 4.7 where CLK is a clock signal. (a) Chaotic behavior for d = 2.25, (b) complicated SSPOs for d = 5.4, (c) basic SSPO for d = 6.0. (a) to (c) correspond to Figs. 5(b), (e) and (f), respectively.
Fig. 8 Typical shape of return maps for a = 4.7. (a) Trajectories diverge for d = 1.6, (b) chaos for d = 2.25, (c) chaos for d = 3.0, (d) complicated SSPOs for d = 4.8, (e) complicated SSPOs for d = 5.4, (f) basic SSPO for d = 6.0. SFP is superstable fixed point and J is an invariant interval.
point y f is said to be superstable for initial state if Eq. (12) is satisfied. F k (y s f ) = y s f , F l (y s f ) y s f ,
Fig. 7
Definition of the return map in phase plane.
starting from LD just after the switching at τ = 0. The n-th switching occurs at τ = nd and yn ∈ LD is satisfied where yn ≡ y(nd). Since yn determines yn+1 , we can define onedimensional return map F from LD to itself: F : LD −→ LD ,
yn+1 = F(yn ).
(11)
Since the trajectories are PWL, the map can be expressed in principle. Figure 8 shows return maps corresponding to trajectories in Fig. 5. The calculation method is shown in Appendix. In order to consider periodic behavior we give some definitions. A point y f ∈ LD is said to be a period-k point if F k (y f ) = y f and F l (y f ) y f for 1 ≤ l < k where F k denotes the k-fold composition of F and k is a positive integer. We refer to y f with k = 1 as a fixed point. A periodic point y f is d k F (y f )| < 1. A periodic said to be stable for initial state if | dy
d k F (y s f ) = 0. dy
(12)
A periodic orbit consisting of such superstable periodic points, {F(y s f ), F 2 (y s f ), · · · , F k (y s f )} is referred to as a superstable periodic orbit and abbreviated by SSPO. Superstable fixed point is abbreviated by SFP. This SFP corresponds to basic SSPO in Fig. 5(f). We then summarize shape of return maps as the following. (1) The maps are PWL and continuous. The number of segments in the map increases as d increases. d F(y)| ≥ 1 is satisfied for all y as (2) In range of d < a, | dy shown in Figs. 8(a) to (c). (3) When d > a, the maps have one flat segment and infinite d F(y) = 0 is satisfied on the flat segment (see extrema. dy Figs. 8(d) to (f)). (4) If d > a + 1, the map has SFP as shown in Fig. 8(f). Referring to these and map formula in Appendix we can say the following properties. Property1: If d < 2, yn < yn+1 is satisfied for all y as shown in Fig. 8(a). In this case trajectory is divergence. Property2: If 2 < d < a, the return maps have an invariant d F(y)| ≥ 1 are satisfied interval J such that J ⊆ F(J) and | dy
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Fig. 9
Basic existence regions of each phenomena.
on J. In this case the maps generate chaotic behavior on J [15] (see Figs. 8(b) and (c)). Property3: If d > a + 1, the maps have SFP and the circuit exhibits basic SSPO as shown in Fig. 8(f) and Fig. 5(f). Basic SSPO starting from (−1, 0) at τ = 0 has reached the origin at τ = a + 1. It stays on the origin for a + 1 < τ < d and returns (jumps) to (−1, 0) at τ = d. The system repeats in this manner and exhibits basic SSPO in the steady state. Existence regions of these phenomena are shown in Fig. 9: trajectories diverge for d < 2, the circuit exhibits chaotic behavior for 2 < d < a and exhibits basic SSPO for d > a + 1. For a < d < a + 1 the maps have the flat segment, however, SFP does not exist (see Figs. 8(d) and (e)). In this case the circuit can generate various complicated SSPOs. 4.
Fig. 10 Bifurcation diagrams for a = 4.7. (b) is an enlargement of a part of (a). Between d = 4.7 and d = 5.7, complicated SSPOs are generated.
Various Phenomena Including Complicated Superstable Periodic Orbits
Here we focus on the following parameter range: a < d < a + 1.
(13)
In this range the circuit can exhibit rich complicated phenomena, however, their theoretical analysis is very hard and we have performed numerical simulations. Figure 10 shows bifurcation diagrams for a = 4.7. In Fig. 10(b), between d = 4.7 and d = 5.7 the orbits seem to be chaotic orbit. But these orbits are complicated SSPOs. We have confirmed that these SSPOs start from the flat segment (a point of (−1, 0)) and return to the flat segment within finite time. It should be noted that the SSPOs are superstable for initial state but can be very sensitive for parameters. Figure 11 illustrates distributions of a period of complicated SSPOs where PCS denotes a period of complicated SSPOs divided by d (or switching frequency per a period of SSPOs). PCS shows huge complicated variation for small perturbation of d. Figures 11(b) to (d) are enlargements of a part of (a) to (c), respectively. Distributions of PCS have self-similarlike structures. It is arisen from a flat segment and infinite extrema of the return maps. A binary expression of distributions of PCS in parameter space are shown in Fig. 12. A black region means PCS ≤ 5 in Fig. 12(b). PCS shows complicated distributions extensively and varies extremely for a slight variation of a and d.
Fig. 11 Distributions of PCS for a = 4.7. (b) to (d) are enlargements of a part of (a) to (c), respectively.
If parameter a varies in the range of a < d < a + 1, various complicated phenomena can exist as shown in Fig. 13. In Figs. 13(a) and (b), the return maps have the flat segment, whereas the orbits are chaotic orbits in an invariant interval J1 after transient process. Whenever the orbit hits the flat segment, it enters into J1 and becomes chaotic orbit. These phenomena correspond to the white regions in Fig. 12(a). We have also confirmed co-existence phenomena of complicated SSPOs and chaos as shown in Figs. 13(c) and (d). In these maps the orbits hit the flat segment and return to the
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Fig. 12 Binary expressions of PCS distribution in parameter range of a < d < a + 1. (a) In the black regions complicated SSPOs exist. In the white regions chaotic orbits exist. (b) The black regions is PCS ≤ 5.
(a)
(b) Fig. 15 A test circuit using OTAs and typical data. E = −1 V, I1 = I2 0.12 mA, C1 47 nF, C2 10 nF (a = 4.7). (a) Chaos for T 0.88 ms (d=2.25), (b) basic SSPO for T 2.4 ms (d=6.0). (a) and (b) correspond to Figs. 5(b) and (f) and Figs. 6(a) and (c), respectively. Horizontal = v1 [1 V/div.], vertical = v2 [5 V/div.] for phase plane (left columns). Horizontal = t [1 ms/div.], vertical = v1 [1 V/div.] for time-domain (right columns). Fig. 13 Complicated phenomena of return maps with a flat segment for a = 2.2. (a) Chaos for d = 2.3, (b) chaos for d = 2.9, (c) co-existence state of complicated SSPOs and chaos for d = 3.1, (d) co-existence state of complicated SSPOs and chaos for d = 3.14. J1 and J2 are invariant intervals, respectively.
flat segment after several switchings. In this case the orbits are complicated SSPOs. On the other hand invariant interval J2 can exist and the chaotic orbit appears in J2 . Figures 10 to 14 suggest huge variety of SSPOs. 5.
Laboratory Experiments
In order to observe typical phenomena we have fabricated a simple test circuit as shown in Fig. 15. The VCCSs are realized by operational transconductance amplifiers OTAs (LM13600). We assume that OTAs have signum function as Eq. (4). An impulsive switch is implemented using an analog switch S (4066), a dc voltage source E and a clock signal with period T . When clock is applied to the PWC circuit, S is closed and capacitor voltage v1 is reset to the base voltage E. Figures 15(a) and (b) show typical data corresponding to Figs. 5(b) and (f) and Figs. 6(a) and (c), respectively. As a period of clock d increases, it is confirmed that chaotic behavior is changed into basic SSPO. 6.
Fig. 14 Bifurcation diagrams of complicated phenomena for a = 2.2. (a) d increases from d = 2.2, (b) d decreases from d = 3.2.
Conclusions
We have considered a simple nonautonomous PWC circuit with impulsive switching. The vector field is PWC, trajectories are PWL and embedded return map is PWL. The map can have one flat segment and infinite extrema: it causes
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chaos and rich SSPOs. Using the map we have investigated the rich phenomena. Using a simple test circuit, typical phenomena are verified experimentally. Future problems include analysis of bifurcation of SSPOs, design of practical circuits and consideration of engineering applications. Acknowledgments The authors wish to thank Dr. H. Torikai for his helpful comments. This work is supported in part by JSPS KAKENHI under Grant 17560269. References
(a) (b) Key objects in phase plane. (a) Definition of τ˜ n , (b) definition
τ˜ 1 = 1, τ˜ n+1 = (a + αn (1 − a)),
[1] K. Miyachi, H. Nakano, and T. Saito, “Response of a simple dependent switched capacitor circuit to a pulse-train input,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol.50, no.9, pp.1180–1187, 2003. [2] Y. Takahashi, H. Nakano, and T. Saito, “A simple hyperchaos generator based on impulsive switching,” IEEE Trans. Circuits Syst. II, Analog Digit. Signal Process., vol.51, no.9, pp.468–472, 2004. [3] H. Nakano and T. Saito, “Grouping synchronization in a pulsecoupled network of chaotic spiking oscillators,” IEEE Trans. Neural Netw., vol.15, no.5, pp.1018–1026, 2004. [4] J.P. Keener, F.C. Hoppensteadt, and J. Rinzel, “Integrate-and-fire models of nerve membrane response to oscillatory input,” SIAM J. Appl. Math., vol.41, pp.503–517, 1981. [5] R.E. Mirollo and S.H. Strogatz, “Synchronization of pulse-coupled biological oscillators,” SIAM J. Appl. Math., vol.50, pp.1645–1662, 1990. [6] E.M. Izhikevich, “Resonate-and-fire neurons,” Neural Netw., vol.14, pp.883–894, 2001. [7] G. Lee and N.H. Farhat, “The bifurcating neuron network 1,” Neural Netw., vol.14, pp.115–131, 2001. [8] S.R. Campbell, D. Wang, and C. Jayaprakash, “Synchrony and desynchrony in integrate-and-fire oscillators,” Neural Computation, vol.11, pp.1595–1619, 1999. [9] E.M. Izhikevich, “Weakly Pulse-coupled oscillators, FM Interactions, Synchronization, and oscillatory associative memory,” IEEE Trans. Neural Netw., vol.10, no.3, pp.508–526, 1999. [10] T. Tsubone and T. Saito, “Manifold piecewise constant systems and chaos,” IEICE Trans. Fundamentals, vol.E82-A, no.8, pp.1619– 1626, Aug. 1999. [11] C. Wagner and R. Stoop, “Renormalization approach to optimal limiter control in 1-D chaotic systems,” J. Statistical Physics, vol.106, pp.97–107, 2002. [12] T. Kabe, T. Saito, and H. Torikai, “Analysis of piecewise constant models of power converters,” Proc. NOLTA, pp.71–74, 2004. [13] Y. Kon’no, T. Saito, and H. Torikai, “Rich dynamics of pulsecoupled spiking neurons with a triangular base signal,” Neural Netw., vol.18, pp.523–531, 2005. [14] Y. Matsuoka, T. Saito, and H. Torikai, “A piecewise constant switched chaotic circuit with rect-rippling return maps,” Proc. ISCAS, pp.3411–3414, 2005. [15] A. Lasota and M.C. Mackey, Chaos, Fractals, and Noise, second ed., Springer-Verlag, 1994.
Appendix:
Fig. A· 1 of Dn+1 .
Formulation of Return Map
First let us consider a trajectory starting from (−1, − a1 ) at τ = 0 shown in Fig. A· 1(a). Let the trajectory reach the origin as discussed in Sect. 2.2 and let τ˜ n be the n-th intersection time. Noting Eq. (9) τ˜ n is given by
τ˜ ∞ = (a + α∞ (1 − a)) = a.
(A· 1)
where τ˜ ∞ that is the limit of τ˜ n at which the trajectory reaches the origin. Next, we consider trajectories starting from Ld as shown in Fig. A· 1(b). Let Yn be n-th intersection and τn be n-th intersection time. We then consider the following parameter condition. τ˜ m ≤ d < τ˜ m+1 < τ˜ ∞ ,
(A· 2)
where m is a finite positive integer. Condition (A· 2) guarantees that Yn and τn exist for 1 ≤ n ≤ m and the intersection is at most m-times. We then define some key points Dn+1 and Dˆ n on LD . Let Dn+1 > − a1 be a point such that the trajectory starting from Dn+1 reaches Yn+1 at time τn+1 = d and jumps onto LD instantaneously: F(Dn+1 ) = Yn+1 . It is given d−1 1 ˆ by Dn+1 = a(1−α n ) − 1. Let Dn < − a be a point such that the trajectory starting from Dˆ n reaches Yn at time τn = d and jumps onto LD instantaneously: F(Dˆ n ) = Yn . It is given by d+1 Dˆ n = − a(1−α n+1 ) + 1. Then we have Theorem1: For Condition (A· 2), the return map is given by yn+1 = Fm (yn ) where ⎧ ⎪ for D2 ≤ y f1 (y) ⎪ ⎪ ⎪ ⎪ ⎪ fn+1 (y) for Dn+2 ≤ y < Dn+1 ⎪ ⎪ ⎪ ⎪ ⎪ for − 1a ≤ y < Dm ⎨ fm (y) (A· 3) Fm (y) = ⎪ ⎪ gm (y) for Dˆ m ≤ y < − a1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ gn (y) for Dˆ n ≤ y < Dˆ n+1 ⎪ ⎪ ⎪ ⎩ g (y) for y < Dˆ 1 0 fn (y) ≡ (−1)n−1 (βn (y + 1) + 1 − d), gn (y) ≡ (−1)n (κn (y − 1) + 1 + d). κn ≡ 1 + 2α + · · · + 2αn > 1, βn ≡ κn−1 , κ0 = β1 ≡ 1, n ∈ {1, 2, · · · , m − 1}, Proof : We show only for yn ≥ − 1a . If D2 < yn , the trajectories hit y-axis only once and the first intersection is given by Y1 = yn + 1 is satisfied. Since the trajectories reach Y1 at τ = 1 and y decreases for 1 < τ ≤ d, we obtain yn+1 = Y1 − (d − 1) = yn − (d − 2) ≡ f1 (yn ). If Dn+2 < yn < Dn+1 , the trajectories hit y-axis (n + 1)-times hence Yn+1 exists. Using Eq. (9) we obtain Yn+1 = (−α)n (yn + 1) and τn+1 = (1 − αn )a(yn + 1) + 1. For odd n, y increases after n + 1-st intersection ( τn+1 < τ < d). In this case we have yn+1 = Yn+1 +(d−τn+1 ) = (−1)n (αn +a(1−αn ))(yn +1)−1+d ≡
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fn+1 (y) where αn + a(1 − αn ) ≡ βn+1 . For even n, y decreases for τn+1 < τ < d and we have yn+1 = Yn+1 − (d − τn+1 ) = (αn +a(1−αn ))(yn +1)+1−d ≡ fn+1 (y). If − 1a ≤ yn < Dm , the trajectories hit y-axis m-times hence Ym exists. In a likewise manner as Dn+2 < yn < Dn+1 , we can derive fm . Similar consideration is possible for yn < − 1a . Q. E. D. (Remark) Figures 8(b) and (c) show the case of m = 1 and m = 2, respectively. When m = 1, f1 (y) for − 1a < y and n does not mean. The points Dˆ 1 < · · · < Dˆ m ≤ − 1a ≤ Dm < · · · < D2 are breakpoints. When d approaches to τ˜ ∞ , m increases and Dm and Dˆ m approach to − 1a : lim Dm = m→∞ lim Dˆ m = − 1 . The maps have an invariant interval J as m→∞
a
shown in Figs. 8 (b) and (c) where J ≡ [Y− , Y+ ], Y+ ≡ max(F(Dˆ 1 ), F(D3 )) and Y− ≡ min(F(Dˆ 2 ), F(D2 ), F(− a1 )). Since βn > 1 for n ≥ 2 and κn > 1 for n ≥ 1, the map is expanding on J and the system exhibits chaos [15]. Next we consider the parameter condition: d > τ˜ ∞ = a.
(A· 4)
In this range the trajectory starting from (−1, − a1 ) intersects y-axis infinitely and can reach the origin at τ = τ˜ ∞ . Let D∞ > − a1 (respectively, Dˆ ∞ < − 1a ) be a point on LD such that the trajectory starting from D∞ (respectively, Dˆ ∞ ) reaches the origin at τ = d (F(D∞ ) = F(Dˆ ∞ ) = 0). They are given by D∞ = 1a (d − 1) − 1 and Dˆ ∞ = − a1 (d + 1) + 1. In a likewise manner as proof of Theorem 1, we obtain Theorem2: For d > τ∞ = a, the return map is given by yn+1 = F∞ (yn ) where ⎧ for D2 ≤ y f1 (y) ⎪ ⎪ ⎪ ⎪ ⎪ (y) for Dn+2 ≤ y < Dn+1 f ⎪ n+1 ⎪ ⎪ ⎨ 0 for Dˆ ∞ ≤ y < D∞ (A· 5) F∞ (y) = ⎪ ⎪ ⎪ ⎪ ⎪ gn (y) for Dˆ n ≤ y < Dˆ n+1 ⎪ ⎪ ⎪ ⎩ g (y) for y < Dˆ 1 , 0 where n is a positive integer. If Dˆ ∞ ≤ yn < D∞ , the trajectories can reach the origin at τ = τ∞ < d: yn+1 = F(yn ) = 0. (Remark) Typical return maps are shown in Figs. 8(d) d F∞ (y)) is 0 for Dˆ ∞ ≤ y ≤ D∞ , to (f). Slope of the map ( dy d otherwise | dy F(y)| ≥ 1. The map consists of infinite segments and has infinite extrema. The segments accumulate to Dˆ ∞ or D∞ . For d > a + 1, the map has SFP and the circuit generates basic SSPO. Using the map precise numerical simulation is possible, however, we must approximate the map because it is impossible to calculate and the maps having infinite segments. After trial-and-errors, y is regarded to reach 0 if |y| < 10−7 in the numerical calculation. For example, the map of Eq. (A· 5) can be approximated for n = 85, d = 5.0 and 2.0 < a < 10.0.
Yusuke Matsuoka received the B.E. and M.E. degrees in electrical engineering from Hosei University, Tokyo, Japan, in 2004 and 2006, respectively. He is currently working toward the Ph.D. degree at the Department of Electronics and Electrical Engineering, Hosei University. His research interests are in chaos and bifurcation.
Toshimichi Saito received the B.E., M.E, and Ph.D. degrees in electrical engineering all from Keio University, Yokohama, Japan, in 1980, 1982 and 1985, respectively. He is currently a Professor with the Department of Electronics, Electrical and Computer Engineering, Hosei University, Tokyo. His current research interests include chaos and bifurcation, artificial neural networks, power electronics and signal processing. He is a senior member of the IEEE and a member of the INNS.
