On the modeling and nonlinear dynamics of autonomous Silva-Young type chaotic oscillators with flat power spectrum.

On the modeling and nonlinear dynamics of autonomous Silva-Young type chaotic oscillators with flat power spectrum Jacques Kengne and Fabien Kenmogne Citation: Chaos: An Interdisciplinary Journal of Nonlinear Science 24, 043134 (2014); doi: 10.1063/1.4903313 View online: http://dx.doi.org/10.1063/1.4903313 View Table of Contents: http://scitation.aip.org/content/aip/journal/chaos/24/4?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Experimental dynamical characterization of five autonomous chaotic oscillators with tunable series resistance Chaos 24, 033110 (2014); 10.1063/1.4890530 Implementation of an integrated op-amp based chaotic neuron model and observation of its chaotic dynamics Chaos 21, 013105 (2011); 10.1063/1.3548064 Experimental investigation of partial synchronization in coupled chaotic oscillators Chaos 13, 185 (2003); 10.1063/1.1505811 Predicting Phase Synchronization from Nonsynchronized Chaotic Data AIP Conf. Proc. 622, 184 (2002); 10.1063/1.1487533 State space parsimonious reconstruction of attractor produced by an electronic oscillator AIP Conf. Proc. 502, 649 (2000); 10.1063/1.1302447

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CHAOS 24, 043134 (2014)

On the modeling and nonlinear dynamics of autonomous Silva-Young type chaotic oscillators with flat power spectrum Jacques Kengne1 and Fabien Kenmogne2 1

Laboratoire d’Automatique et Informatique Apliqu ee (LAIA), Department of Electrical Engineering, IUT-FV Bandjoun, University of Dschang, Bandjoun (Cameroon) 2 Laboratory of Modeling and Simulation in Engineering, Biomimetics and Prototype, University of Yaound e 1, Yaound e (Cameroon)

(Received 26 September 2014; accepted 20 November 2014; published online 4 December 2014) The nonlinear dynamics of fourth-order Silva-Young type chaotic oscillators with flat power spectrum recently introduced by Tamaseviciute and collaborators is considered. In this type of oscillators, a pair of semiconductor diodes in an anti-parallel connection acts as the nonlinear component necessary for generating chaotic oscillations. Based on the Shockley diode equation and an appropriate selection of the state variables, a smooth mathematical model (involving hyperbolic sine and cosine functions) is derived for a better description of both the regular and chaotic dynamics of the system. The complex behavior of the oscillator is characterized in terms of its parameters by using time series, bifurcation diagrams, Lyapunov exponents’ plots, Poincare sections, and frequency spectra. It is shown that the onset of chaos is achieved via the classical period-doubling and symmetry restoring crisis scenarios. Some PSPICE simulations of the nonlinear dynamics of the oscillator are presented in order to confirm the ability of the proposed mathematical model to accurately describe/predict both the regular and chaotic behaviors C 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4903313] of the oscillator. V Chaotic oscillations with flat power spectrum are desirable for engineering applications such as chaos secure communications, sonar sensors, and random bit generation. The present paper studies the dynamics of a recently introduced fourth order autonomous Silva-Young type chaotic oscillator with flat power spectrum. In contrast to common approaches based on piecewise linear (PWL) models, a smooth mathematical model of the oscillator is obtained by adopting a judicious selection of state variables and exploiting the Shockley diode equation. The mechanism of chaos generation is analyzed in terms of the model parameters. Bifurcation diagrams suggest the occurrence of period-doubling bifurcation to chaos and symmetry restoring crises phenomenon. We numerically demonstrate that chaos occurs for a wide range of parameters. The results of present investigations may serve for rigorous designs of such types of oscillators in relevant engineering applications.

I. INTRODUCTION

Since the discovery of deterministic chaos by the Edward Lorenz1 in the mid 1960, followed by the pioneering work of Pecora and Caroll2 on the synchronization of chaotic systems, the field of nonlinear dynamical systems has drained the attention of researchers worldwide. Such phenomenon characterized by the system’s high sensitivity on initial condition has been deeply investigated in various physical, biological, economical, and engineering systems.3–5 Concerning the practical utility of chaos in electronic engineering, examples include random bit generation; spread spectrum communication, information 1054-1500/2014/24(4)/043134/11/$30.00

scrambling for secure communication, sonar sensors, as well as radar systems.5–10 Owing to the central role played by the chaos generator in such types of applications, the design and implementation of chaotic oscillators11–26 have remained for decades one of the most followed research avenues. An undesirable feature observed in most chaotic oscillators is the noticeable unevenness of the spectral density. This characteristic is inherent to non-autonomous second order oscillators as well as to third order and higher dimensional chaotic/hyperchaotic oscillators. As an attempt to tackle this problem, Silva and Young invented a nonautonomous electronic circuit generating chaotic oscillations with broadband noise-like spectrum.22,23 Later Kandangath and collaborators24 described low-frequency version of the oscillator. The main shortcoming of the nonautonomous chaotic oscillator in many practical applications is that they have sharp 20–30 dB height peaks at the drive frequency and its higher harmonics. To get around this problem, Tamasevicius and coworkers25 suggested the third order autonomous Duffing-Holmes (or Silva-Young) chaotic oscillator. Though sharp peaks are avoided in the power spectrum, the spectral density still exhibits large undesirable unevenness of about 10–15 dB. Recently, Tamaseviciute et al.26 proposed a novel Silva-Young type four order chaotic oscillator characterized by the spectral unevenness of less than 10 dB. Moreover, a very simple second order external resonant circuit inserted in the output follower allows decreasing this value to less than 3 dB. The authors proposed a piecewise-linear model to study the nonlinear behavior of the oscillator which unfortunately, allows only a rough description27,28 of systems dynamics. In, addition, the mechanism of chaos generation in this particular oscillator is still not elucidated. Also, with the motivation

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C 2014 AIP Publishing LLC V


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J. Kengne and F. Kenmogne

to shed more light on the dynamics of this type of oscillator, we propose in the present work, a smooth (i.e., infinitely differentiable) model (obtained from the Shockley diode equation) to explain the rich and complex behavior exhibited by the real oscillator. It should be emphasized that this type of model is perfectly suited for the analysis of semiconductor diode-based nonlinear circuits as recently shown by the work of Kengne and co-workers.29 Moreover, one of the main objectives in this work is to provide some tools (for instance, bifurcation diagrams) that can be exploited for rigorous design of this type of oscillators. The rest of the paper is arranged as follows. Section II is devoted to the modeling process. The electronic structure of the oscillator is presented and a suitable mathematical model is derived to describe the dynamical behavior of the system. In Sec. III, some basic properties of the model are underlined. Section IV focuses on the numerical analysis. Various phase portraits and bifurcation diagrams combined with their corresponding graphs of numerically computed Lyapunov exponents are plotted to reveal different scenarios leading to chaos. In Sec. V, some PSPICE simulations are carried out in order to confirm the ability of the proposed mathematical model to accurately describe both the regular and chaotic dynamics of the system. The results obtained are compared with theoretical ones and a very good agreement is observed. Finally, we end with some concluding remarks in Sec. VI. II. CIRCUIT DESCRIPTION AND STATE EQUATIONS A. Circuit description

The oscillator consists of an op. amplifier (OA1 ) based nonlinear resonant stage with the pair of diodes (D1 , D2 ), the L1 C1 tank, and resistor R4 in the first positive feedback loop, in addition to a second op. amplifier (OA2 ) based linear resonant stage with the L2 C2 tank and the load resistor

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R5 in the second positive feedback loop. The OA3 based stage with spectral correction circuit L3 R8 C3 R7 implements the external output follower. Assuming ideal op. amplifiers (operating in their linear regions), we would like to mention that the pair of semiconductor diodes are the only nonlinear elements responsible of the chaotic behavior of the complete electronic circuit. A symmetrical characteristic30 is obtained by connecting two diodes in antiparallel, i.e., with the two terminals shortened but with opposite polarities. In such type of configuration, the voltage across each diode is equal to the voltage of the resulting two-terminal device, while the current is the sum of the current flowing through each diode. This symmetrical nonlinearity is necessary for the occurrence of symmetric (by inversion with respect to the origin) attractors.30 It should be noted that the oscillator shown in Fig. 1 can be considered as an extended version of the previously reported third-order chaotic oscillator25 in the sense that its order is increased from three to four. From a structural point of view, this oscillator is similar to the second order Silva-Young circuit described in Ref. 22. The only difference lies in the fact that the external periodic excitation in the original Silva-Young circuit is replaced with an internal second order resonant circuit L2 R5 C2 inserted in the second positive feedback loop. This modification yields flat power spectrum of the output signal with no peaks at the resonant frequencies.26 B. State equations

Denoting by ILn ðn ¼ 1; 2Þ the current flowing through the inductor Ln , and VCm ðm ¼ 1; 2Þ the voltage across capacitor Cm ; the Kirchhoff’s electric circuit laws can be applied to the schematic diagram of Fig. 1 to obtain the following set of differential equations describing the dynamics of the oscillator:

FIG. 1. Circuit diagram of the fourthorder autonomous Silva-Young type oscillator.26


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8 dIL > > > L1 1 ¼ ðk 1ÞVd R4 Id ðR3 þ r1 ÞIL2 ðk 1ÞR5 IL2 ; > > dt > > > > dV > C2 > > ¼ IL2 ; C2 > > dt < dIL > L2 2 ¼ R6 ðIL1 Id Þ VC2 ðR5 þ r2 ÞIL2 ; > > dt > > > > > dV IL1 Id d > > ¼ C1 ; > > dId dt > > 1 þ R4 : dVd

(1)

where Id the diode current, dId =dVd represents its derivative with respect to the voltage drop Vd and r1 (reps. r2 ) denotes the internal series resistance of inductor L1 (resp. L2 ). The current-voltage (I-V) characteristic of the pair of diodes ðD1 and D2 Þ is obtained from the Shockley diode equation29,31,32 as follows: Id ¼ f ðVd Þ ¼ ID1 ID2 ¼ IS ½exp ðVd =gVT Þ 1 IS ½exp ðVd =gVT Þ 1 ¼ 2IS sinhðVd =gVT Þ;

(2)

where IS is the saturation current of the junction; VT ¼ kb T=q is the thermal voltage with kb is the Boltzmann constant, T is the absolute temperature expressed in Kelvin, q is the electron charge, and g is the ideality factor (1 < g < 2). One should remark that the original state vector can still be obtained provided that VC1 ¼ Vd þ R4 f ðVd Þ . Equations related to the correction circuit L3 R8 C3 R7 are rather trivial and will not be presented here for simplicity. Also, we would like to stress that the choice of the state variables adopted here represents a straightforward way of solving the problem of transcendental equations faced when a resistor is series connected with a diode. Otherwise, a piecewise linear26,27 model will be inevitable for the mathematical modeling of the system. With the following change of variables and parameters: pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi q ¼ L1 =C1 ; t ¼ s L1 C1 ; Vref ¼ gVT ; x1 ¼ Vd =Vref ; x2 ¼ qIL1 =Vref ; x3 ¼ VC2 =Vref ; x4 ¼ qIL2 =Vref ; w ¼ VC1 =Vref ; e ¼ R5 =ðR5 þ R2 Þ; cL ¼ L1 =L2 ; e1 ¼ 2R4 Is =Vref ; e2 ¼ 2qIs =Vref ; cC ¼ C1 =C2 ; a ¼ R6 =q;

b ¼ ðR3 þ r1 Þ=q;

c ¼ R5 =q;

d ¼ ðR5 þ r2 Þ=q;

The non-dimensional circuit equations are defined by the following smooth nonlinear fourth order differential equations: 8 > x2 e2 sinhðx1 Þ > > ; x_ 1 ¼ > > > 1 þ e1 coshðx1 Þ > < x_ 2 ¼ ðk 1Þx1 e1 sinhðx1 Þ bx2 cðk 1Þx4 ; > > > x_ 3 ¼ cC x4 ; > > > > : x_ 4 ¼ cL ax2 þ ae2 sinhðx1 Þ ex1 x3 dx4 ;

(4)

where the dots denote differentiation with respect to the dimensionless time s. It can be seen that only one state variable (namely, x1 ) is involved in the hyperbolic nonlinearities of the model in (4). Also note that all the state variables are real and may be captured in real experiment with a standard oscilloscope. In the mathematical model (4) ten parameters can be identified. Two of them (namely, e1 and e2 ) depend on intrinsic diode parameters and consequently will be kept constant (as well as b, c, d, r, and cL ) during all the numerical experiments. Thus, the bifurcation analysis of the system will be carried out with respect to the control parameters k, a, and cC respectively. For the numerical analysis, the following values of electronic components are employed: L1 ¼ L2 ¼ 20 mH, L3 ¼ 16 mH, C1 ¼ 47 nF, C2 tuneable, C3 ¼ 175 nF, R1 tuneable, R2 ¼ 20 kX, R3 ¼ 60 X, R8 ¼ 62 X, R4 ¼ 68 kX, R5 ¼ 100 X, R6 tuneable, R7 ¼ 1:5 kX, r1 ¼ r2 ¼ 2 X, D ¼ 1N4148

k ¼ 1 þ R1 =ðR2 þ R5 Þ :

(3)

(g ¼ 1:9, VT ¼ 26 mV; and IS ¼ 2:682 nA). With this set of electronic components values, the system model parameters values are fixed to: e1 ¼ 7:3836 103 , e2 ¼ 7:08317 105 , b ¼ 0:09811, c ¼ 0:15329, d ¼ 0:15636, r ¼ 4:97512 103 , and cL ¼ 1:00. III. BASIC PROPERTIES OF THE MODEL A. Symmetry

It is obviously that system (4) is invariant under the transformation: ðx1 ðsÞ; x2 ðsÞ; x3 ðsÞ; x4 ðsÞÞ () ðx1 ðsÞ; x2 ðsÞ; x3 ðsÞ; x4 ðsÞÞ. Therefore, if ðx1 ðsÞ; x2 ðsÞ; x3 ðsÞ; x4 ðsÞÞ is a solution of system (4) for a specific set of parameters, then ðx1 ðsÞ; x2 ðsÞ; x3 ðsÞ; x4 ðsÞÞ is also a solution for the same parameters set. The origin of the new system coordinate Oð0; 0; 0; 0Þ is a trivial symmetric static solution. As a consequence, attractors in state space have to be symmetric by inversion with respect to the origin; otherwise they must appear in pairs, to restore the exact symmetry of the model equations. This exact symmetry could serve to explain the occurrence of several co-existing attractors in state space. Furthermore, it represents a good way to test the scheme used for numerical integration. B. Fixed point analysis

It is known that the equilibrium points play a crucial role on the dynamics of nonlinear systems.33 The fixed


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points of system (4) can be found by solving the following nonlinear system: 8 > x2 e2 sinhðx1 Þ > > ¼ 0; > > > 1 þ e1 coshðx1 Þ > < ðk 1Þx1 e1 sinhðx1 Þ bx2 cðk 1Þx4 ¼ 0; (5) > > > cC x4 ¼ 0; > > > > : cL ax2 þ ae2 sinhðx1 Þ ex1 x3 dx4 ¼ 0: After some algebraic manipulations, it is shown that the origin Oð0; 0; 0; 0Þ is the only trivial equilibrium point. By linearizing around the origin Oð0; 0; 0; 0Þ, the Jacobian matrix of system (4) is given by 2 3 e2 1 0 0 6 1 þ e1 7 1 þ e1 6 7 7: (6) ð Þ k 1 e b 0 c k 1 MJ=O ¼ 6 1 6 7 4 5 0 0 0 cC cL ðae2 rÞ acL cL dcL With the following specific set of control parameters values (for which the system exhibits a typical double-band chaotic attractor (see Sec. IV)): k ¼ 2:0, a ¼ 0:597858, cc ¼ 0:83928, the eigenvalues of the above Jacobian matrix are: k1 ¼ 0:6290, k2 ¼ 1:6311, and k3;4 ¼ 0:067760:8952i. Since there are eigenvalues with real parts of different signs, the origin is a saddle equilibrium point of system (4). Therefore, it is possible to generate chaos with our model in (4).

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DL ¼ k þ

1

k X

jkkþ1 j

j¼1

kj ;

(7)

where the integer k, which represents the number P of degrees of P freedom, meets the condition kj¼1 kj 0 and kþ1 j¼1 kj < 0, the Lyapunov exponents being ordered such that kj kjþ1 . Briefly recall that the Kaplan-Yorke dimension is an estimate for the information dimension. In other words, it is a measure of the degree of disorder of the points on the attractor or, more precisely, it specifies the amount of information needed to locate the system in the phase space. To gain further insight about the dynamics of the system under investigation, we compute the Poincare sections, the time series of state variables as well as corresponding frequency spectra. B. Routes to chaos

To investigate the sensitivity of the system with respect to a single parameter, we fix a ¼ 0:5979, cC ¼ 0:83928 and vary k in the range 1:0 k 2:0. When monitoring, the control parameter k, various types of bifurcations are

IV. NUMERICAL STUDY A. Numerical methods

In order to investigate various transitions to chaos in the 4D autonomous Silva-Young oscillator, system (4) is integrated numerically using the classical fourth-order RungeKutta integration scheme. Throughout this work, the time grid is always kept Dt ¼ 0:005 and the computations are performed out using variables and constants parameters in extended mode. For each set of parameters, system (4) is integrated for a sufficiently long time and the transient is discarded. Various bifurcation diagrams, combined with the corresponding graphs of three largest Lyapunov exponents are plotted to define the type of transition leading to chaos in the system. The bifurcation diagrams are obtained by plotting the local maxima of state variables in terms of the bifurcation control parameter that is varied in tiny steps, whereas the Lyapunov exponents are computed numerically with the help of the reliable algorithm of Wolf and collaborators.34 Unlike some other methods, which only compute the largest Lyapunov exponent, the algorithm of Wolf et al. calculates the full spectrum of the Lyapunov exponents and thus helps to distinguish between chaotic attractors characterized by only one positive exponent, and hyperchaotic attractors marked by more than one positive exponent. Furthermore, the Lyapunov dimension of the attractors is computed following the definition of Kaplan and Yorke:

FIG. 2. Bifurcation diagram (a) showing local maxima of the coordinate x4 ðsÞ of the attractor and corresponding graph of three largest Lyapunov exponents (b) versus parameter k. The positive value of k1 is the signature of chaotic motion.


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observed. In particular, for values of k < kth ¼ 1:305, there is no oscillation in the system. When increasing k beyond the threshold value kth , the system undergoes a Hopf bifurcation giving rise to a stable period-1 limit cycle. Further increasing k, this period-1 limit cycle converts to a chaotic single band (i.e., asymmetric) attractor via a period doubling bifurcation. Past the critical value kc ¼ 1:727, both asymmetric chaotic attractors merges to form a double-band symmetric orbit via the well known symmetry restoring crisis phenomenon. This bifurcation sequences are well illustrated in Fig. 2(a). Where, we have plotted the maxima of the coordinate x4 ðsÞ in terms of the control parameter k. Correspondingly, the graphs of three largest Lyapunov exponents are depicted in Fig. 2(b). A very good coincidence is observed between the two diagrams. In particular, bands of chaos characterized by positive values of k1 can easily be identified. It is interesting to note that although the Sylva-Young oscillator considered here is a 4D system, there is always one positive Lyapunov

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exponent meaning that the system is simply chaotic (but not hyperchaotic). In particular, the spectrum of the four Lyapunov exponents computed for k ¼ 2:0 is: k1 ¼ 0:1078, k2 ¼ 0:00, k3 ¼ 0:1283, and k4 ¼ 0:2426. To confirm the scenario to chaos shown in Fig. 2, sample phase portraits along with corresponding time traces computed for some discrete values of the control parameter k. Asymmetric attractors pairs are observed in Figs. 3(a-i)–3(d-i), while a double-band strange attractor is depicted in Fig. 3(e-i). To better illustrate the complexity of the attractor depicted in Fig. 3(e-i), various 2D dimensional projections are presented in Fig. 4. In the same line, the Poincare section of the attractor and the power spectrum of the coordinate wðsÞ are shown in Figs. 5(a) and 5(b), respectively. The broadband nature of the power spectrum is a characteristic feature of chaos. Furthermore, we provide in Fig. 6 the time traces of the coordinate wðsÞ to confirm the crisis induced intermittency occurring in the system.

FIG. 3. Three dimensional views of the attractor projected onto the ðw; x2 ; x3 Þ space (left) of the system showing routes to chaos (in terms of the control parameter k) and corresponding time traces (right): (a) Period-1 for k ¼ 1:50, (b) Period-2 for k ¼ 1:70, (c) Period-4 for k ¼ 1:718, (d) single band chaos for k ¼ 1:723, (e) double band chaotic attractor for k ¼ 2:0.


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FIG. 3. (Continued).

FIG. 4. Two dimensional projections of the double-band chaotic attractor shown in Fig. 3(e) highlighting the complexity of the system. However, this attractor is simply chaotic (but not hyperchaotic) despite the fact of being four dimensional.


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The period doubling route to chaos and the attractor restoring crisis are also observed when using either a or cC as bifurcation control parameter (see Fig. 7). Also, it is important to stress that the scenario to chaos observe in this work is typical of the c order non-autonomous Duffing oscillator from which the circuit of Fig. 1 originates. C. Comparison with the piecewise linear (PWL) model

In the pioneering work of Ref. 26, the authors proposed a PWL model of the 4D autonomous Silva-Young oscillator in the dimensionless form as follows: 8 x_ 1 ¼ x2 ; > > < x_ 2 ¼ kuðx1 Þ x1 bx2 þ cðk 1Þx4 ; (8a) x_ 3 ¼ cC x4 ; > > : x_ 4 ¼ cL ðax2 x3 dx4 Þ; where the parameters (k, a, c, d, cC , and cL ) are same as above while the state variables are now defined as follows: x1 ¼ VC1 =Vref , x2 ¼ qIL1 =Vref , x3 ¼ qIL2 =Vref , x4 ¼ VC2 =Vref (with Vref ¼ 2V0 ). Here, V0 0:5 V stands as the forward voltage drop across the diodes. The nonlinear function uð:Þ involving the current-voltage (I-V) characteristic of the diodes is approximated by a three segments piecewise linear function 8 < 0:5; if x < 0:5 uðxÞ ¼ x; if jxj 0:5 (8b) : 0:5; if x > 0:5;

FIG. 5. Poincare section (in the hyper-plane x2 ¼ 0) of the 4D double-band chaotic attractor projected onto the ðx1 ; x3 Þ plane (a) and the corresponding frequency distribution of the coordinate wðsÞ. Parameters are same as in Fig. 4.

where the dot denotes once more differentiation with respect to the dimensionless time s as previously mentioned. The relative simplicity of the model in (8) is remarkable as only one nonlinear term is involved. However, we would like to stress that in contrast to previous literature26 based on PWL model of the same oscillator, the exponential model is considered throughout this work. Indeed, the PWL model is “nonsmooth” and the corresponding vector field is of C0

FIG. 6. Illustration of the symmetry restoring crisis. For k < kc 1:727, there are two mirror image chaotic attractors, one with wðsÞ < 0 and the one with wðsÞ > 0. The two attractors merge to form unique attractor with mirror symmetry at k ¼ kc 1:727. The figure depicts: (a) the time series of the attractor with wðsÞ > 0 for k ¼ 1:723; (b) and (c) the time series of the attractor for k ¼ 1:729 and k ¼ 2:0 (respectively) past the symmetry restoring crisis.


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FIG. 7. Bifurcation diagrams (a) and (b) showing local maxima of the coordinate x4 ðsÞ of the attractor obtained, respectively, for varying cC and a: (a) a ¼ 0:597858, k ¼ 2:0; (b) cC ¼ 0:83928, k ¼ 2:0.

type, while the vector field associated to the exponential model is of C1 type. Briefly recall that a vector field uðxÞ is said to be of class Ck if u is k-time differentiable with respect to x and the kth derivative uðkÞ is continuous. In this respect, the PWL model may undergo different bifurcation scenarios compared to the smooth/hyperbolic model, which it approximates.27–29 To evaluate the performances of both the PWL and the smooth models of the system, we provide in Fig. 8 the bifurcation diagrams of the state vector coordinate IL2 ðtÞ versus the tuneable parameter k. Fig. 8(a) is computed with the PWL model, while Fig. 8(b) corresponds to the smooth/ hyperbolic model. Chaos is predicted in both PWL and smooth models (Fig. 8(b)) for higher values of k. However, it can be notice that completely different dynamical behaviours are observed for lower values of the control parameter. This clearly illustrates the incapacity of the PWL model to accurately describe the dynamics of the real 4D autonomous Silva-Young chaotic oscillator. In contrast, even tiny windows of regular behavior of the real oscillator (see Sec. V) can be captured with the exponential model proposed in this work. A two parameters diagram (see Fig. 9) computed with

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FIG. 8. Bifurcation diagrams showing local maxima of the current iL2 ðtÞ in terms of parameter k computed numerically using, respectively, the PWL model (a) and the smooth model (b) with the same parameters’ setting of Fig. 2. Both models predict chaos in the system for high values of the control parameter k while, completely different dynamics are observed for lower values of the same parameter.

FIG. 9. Two parameters phase diagram showing the region of chaotic dynamics in the ða; kÞ plane.


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the exponential model showing regions of irregular behavior (chaos) in the ða; kÞ plane computed with cC ¼ 3:30. This diagram is of great importance for a practical circuit design of a real 4D autonomous Siva-Young oscillator. V. PSPICE SIMULATIONS

Following the theoretical analysis presented above, it is predicted that the oscillator under investigation can demonstrate very complex and striking dynamic behaviours for some suitable parameters settings. The aim of this section is to implement the circuit diagram of Fig. 1 in PSPICE and carry out subsequent series of simulations in order to validate the mathematical model proposed in this work. Furthermore, it is of interest to evaluate the effects of simplifying assumptions (e.g., ideal diode model and ideal op amplifiers) adopted during the modelling process on the real behavior of the oscillator in PSPICE.35,36 For this end, the dynamics of

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the circuit in Fig. 1 is simulated in PSPICE with the same parameters settings in Sec. IV for C2 ¼ 56 nF, R6 ¼ 390 X with varying R1 in order to confirm the sensitivity of the system with respect to parameter k as in Sec. IV. When monitoring (increasing) the control resistor R1 in tiny steps, a rich variety of bifurcations is observed. For values of R1 < 8540 X, only fixed point motion is observed. When R1 is increased past this critical value, a stable limit cycle of approximately 4:320 kHz takes place. Further increasing R1 , the stable limit cycle born from the Hopf bifurcation undergoes a series of period-doubling bifurcations culminating to an asymmetric (one band) chaotic attractor. As R1 is further increased, the asymmetric chaotic orbit suddenly converts to a double-band strange attractor via the symmetry restoring crisis event. This bifurcation sequence perfectly agrees with the theoretical analysis presented in Sec. IV. Sample results showing typical dynamic states of the system are depicted in Fig. 10. From the graphs in Fig. 10, a very good similarity

FIG. 10. Sample PSPICE simulation results (left) and corresponding theoretical phase space trajectories (right) obtained by a direct numerical integration of the system model (Eq. (2)) confirming the scenario to chaos in the system for varying R1 (i.e., parameter k): (a) Period-1 for R1 ¼ 10050 X, (b) Period-2 for R1 ¼ 14600 X, (c) Period-4 for R1 ¼ 15150 X, (d) single band chaos for R1 ¼ 15300 X, (e) double band chaotic attractor for R1 ¼ 20100 X. Initial conditions are: iL1 ð0Þ ¼ 0 mA, iL2 ð0Þ ¼ 0 mA VC1 ð0Þ ¼ VC2 ð0Þ ¼ 0:1 V. The rest of circuit components’ values are defined in the text whereas the numerical phase portraits are obtained with the parameters in Fig. 3.


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FIG. 10. (Continued).

TABLE I. Comparison between theoretical and PSPICE simulation results in terms of the control parameter R1 (recall that k ¼ 1 þ R1 =ðR2 þ R5 Þ). The rest of circuit parameters are defined in the text. Theoretical values of R1 Transition (period-n ! period-m) Hopf bifurcation Period-1 ! Period-2 Period-2 ! Period-4 Crisis (single-band ! double-band)

SPICE simulations values of R1

PWL model (X)

Smooth model (X)

Ideal op. amplifier (X)

Real op. amplifier (X)

3618 9547.5 10151 10372

6130.5 13829 14372 14613

8160 14280 14840 15060

8510 14565 15115 15345

between theoretical phase portraits and PSPICE simulation results can be observed. However, slight discrepancies that may be attributed to the simplifying assumptions adopted during the modeling process can be noted between the bifurcations points in PSPICE compared to the results from the theoretical analysis (see Table I). VI. CONCLUDING REMARKS

To summarize, this paper has proposed a new mathematical model for a better description of the nonlinear dynamics of 4D autonomous Silva-Young type chaotic oscillators. Using various nonlinear analysis tools including bifurcation diagrams, Lyapunov exponents’ plots, time series, and frequency spectra, the dynamics of the system has been characterized with respect to its numerous parameters. It was found/ revealed that chaos arise in the system through the classical period-doubling and symmetry restoring crises scenarios when monitoring the control parameters in tiny ranges. The smooth mathematical model is advantageous compared to its PWL approximation as it provides a more accurate description of the oscillator and, consequently, it stands as a

promising tool to carry out further studies on the nonlinear and chaotic dynamics of 4D autonomous Silva-Young type oscillators. A close agreement was observed between PSPICE based simulations and the theoretical analysis. Also, we would like to stress that the analysis performed in this work could be extended to the particular cases of 3D SilvaYoung type oscillators as well as to any semiconductor diode-based chaotic or hyperchaotic oscillator, in general. 1

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