Research Article Theoretical Analysis and Adaptive ...downloads.hindawi.com/archive/2014/429809.pdfResearch Article Theoretical Analysis and Adaptive Synchronization of a 4D Hyperchaotic

Research ArticleTheoretical Analysis and Adaptive Synchronization of a 4DHyperchaotic Oscillator

T. Fonzin Fozin,1,2 J. Kengne,2,3 and F. B. Pelap1

1 Laboratoire de Mecanique et de Modelisation des Systemes Physiques (L2MSP), Faculty of Science, University of Dschang,P.O. Box 69 Dschang, Cameroon

2 Laboratoire d’Electronique et de Traitement de Signal (LETS), Faculty of Science, University of Dschang,P.O. Box 67 Dschang, Cameroon

3 Laboratoire d’Automatique et Informatique Appliquee (LAIA), IUT-FV Bandjoun, University of Dschang,P.O. Box 67 Dschang, Cameroon

Correspondence should be addressed to F. B. Pelap; [email protected]

Received 31 October 2013; Accepted 14 January 2014; Published 27 February 2014

Academic Editor: Qingdu Li

Copyright © 2014 T. Fonzin Fozin et al.This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We propose a new mathematical model of the TNC oscillator and study its impact on the dynamical properties of the oscillatorsubjected to an exponential nonlinearity. We establish the existence of hyperchaotic behavior in the system through theoreticalanalysis and by exploiting electronic circuit experiments. The obtained numerical results are found to be in good agreement withexperimental observations.Moreover, the new technique on adaptive control theory is used on ourmodel and it is rigorously proventhat the adaptive synchronization can be achieved for hyperchaotic systems with uncertain parameters.

1. Introduction

Over the last four decades, an increasing interest has beenshown on chaotic systems with higher dimensional attractorsknown as hyperchaotic.The interest for hyperchaotic dynam-ics is justified by the rapid development of new techniquesin various areas of physics such as nonlinear circuits [1–6], complex system studies [7, 8], laser dynamics [9–11],secure communication [12, 13], and synchronization [14–21].Hyperchaotic systems are usually classified as chaotic systemswith more than one positive Lyapunov exponent, indicatingthat the chaotic dynamics of the systems are expanded insome directions but rapidly shrink in other directions, whichsignificantly increase the system’s orbital degree or disorderand randomness. Those systems are suitable for engineeringapplication such as secure communications [12–14, 19, 20,22]. In fact, in 1995, Perez and Cerdeira [23] have shownthat by masking signals with simple chaos with only oneLyapunov exponent does not provide high level of security.Hence, the use of more complex hyperchaotic signals is astraightforward way to overcome this limitation because of

their increased randomness and higher unpredictability [24].Therefore the dynamical behavior of several hyperchaoticelectronic circuits has been studied [1, 2, 4–6, 10, 11, 25–27]. Within these, TNC oscillator [2] retains our attention.Indeed, it is extremely simple and provides a powerful toolfor understanding the inherent architectures and dynamicalbehavior of hyperchaotic oscillator as well as their useas chaotic carriers in practical secure communication. Itssynchronization and dynamics have been studied using apiecewise linear (PWL)model [4, 28].Moreover, Peng et al. in[14], address the synchronization of hyperchaotic oscillatorsusing a scalar transmitted signal. Furthermore, this scalarsynchronization does not require either the computation ofthe Lyapunov exponent or initial conditions belonging to thesame basin of attraction. For a possible use of hyperchaoticbehavior in secure communication, the synchronization ofhyperchaotic systems via a single dynamical variable isdemonstrated [20, 29, 30]. In 2001, Miller and Grassi [19]described an experimental realization of observer-basedhyperchaos synchronization using the TNC oscillator [31].X. Wang and Y. Wang in [29] deal with the synchronization

Hindawi Publishing CorporationJournal of ChaosVolume 2014, Article ID 429809, 15 pageshttp://dx.doi.org/10.1155/2014/429809

2 Journal of Chaos

Out

R3

R4 R1

U

VC1 C1 L1L2C2

Vd

Id D

I1I2

VC2

R2

+

−

+

−

+

+

−

−

(a)

ExponentialPWL

V (V)

I(A

)(b)

Figure 1: (a) Schematic diagram of the modified TNC hyperchaotic oscillator. The operational amplifier with feedback resistors 𝑅1, 𝑅3, and

𝑅4acts both as the negative impedance converter (NIC) and the output amplifier. (b) Current-voltage characteristic showing the exponential

and the PWL models of the diode.

of uncertain hyperchaotic and chaotic systems by adaptivecontrol using the TNC hyperchaotic oscillator as the masterand theRossler oscillator as the slave.However, the aforemen-tioned analysis is performed based on the piecewise linearmodel of the system. Furthermore, to our knowledge, nodesign tool exist that may be exploited for design purposesof a Tamasevicius oscillator built by Grassi and Mascolo[4]. In fact, the PWL model represents only a first-orderapproximation of the reality and therefore may give rise todifferent types of bifurcation compared to those exhibits bythe real oscillator. Moreover, Kengne et al. [32, 33] derivesmooth mathematical model instead of PWL to perform theanalysis of the two-stage Colpitts oscillator and the improvedColpitts oscillator.They also synchronize those coupled oscil-lators (with smooth mathematical model) using nonlinearstate observers. Using the same concept, we propose in thispaper a new smooth (exponential) mathematical model toinvestigate the nonlinear dynamics of the TNC oscillator.We address two key issues. The first is to point out boththeoretical and experimental tools, which may be exploitedfor design and control purposes. The second is to completethe results obtained so far by deriving a mathematical modelfrom which a systematic and methodological analysis of thenonlinear dynamics of the oscillator is carried out helping topoint out some unknown and striking behavior of the TNCoscillator.

The organization of the paper is as follows. Section 2deals with the modeling process. The electronic structure ofthe oscillator is presented and the appropriate mathematicalmodel is derived to describe the dynamic behavior of theoscillator. The stability analysis of the equilibrium point and

local bifurcation is presented in Section 3. The evolutionprocess analysis by means of Lyapunov exponent spectrum,bifurcation diagram, and phase portraits is investigated inSection 4, where numerical simulations studies are given.In Section 5, an experimental study is carried out. Thecorresponding electronic circuit is implemented, exhibitingexperimental chaotic attractors in accord with numericalsimulations which serves to validate the model derived inthis work. Section 6 considers the synchronization of suchtype of oscillators. By exploiting recent results on adaptivecontrol theory, a controller is designed that combines thesynchronization of two one-way coupled systems and theestimation of unknown parameters of the drive system.Finally the summary of the results is given in Section 7.

2. Circuit Description andMathematical Model

2.1. Circuit Description. The model under consideration isthe electronic oscillator of the Tamasevicius, Namajuas,and Cenys (TNC) given by Figure 1(a). It consists of acombined parallel-series LC circuit (namely, 𝐿

1𝐶1-𝐿2𝐶2), a

single operational amplifier (op amp) with feedback resistors𝑅1, 𝑅3, 𝑅4implementing both the negative impedance (NIC)

and the output amplifier for the oscillator with a diode. Thisdiode is the nonlinear device responsible for the hyperchaoticbehavior of the oscillator. One should note that as the opamp operates in the linear region for 𝑅

3= 𝑅4, the input

impedance of the NIC satisfies 𝑅 = −𝑅1and the output obeys

𝑉out = (1 + 𝑅4/𝑅3)𝑉𝐶1

.

Journal of Chaos 3

2.2.MathematicalModel. This subsection introduces the newmathematical model of the TNC oscillator by considering thefollowing hypotheses. Firstly, we assume that all capacitors,inductors, and resistors of the oscillator network are linear.Secondly, the current-voltage (I-V) characteristic of the diode𝐷 is modeled with an exponential function (Figure 1(b)) [34]as yields in

𝐼𝑑= 𝑓 (𝑉

𝑑) = 𝐼𝑠[exp(

𝑉𝑑

𝜂𝑉𝑇

) − 1] (1)

in which𝑉𝑑is the voltage drop across the diode, 𝐼

𝑠represents

the saturation current of the junction, 𝜂 stands for the idealityfactor (1 < 𝜂 < 2), and 𝑉

𝑇= 𝑘𝑏𝑇/𝑞 is the thermal

voltage wherein 𝑘𝑏is the Boltzmann constant, 𝑇 the absolute

temperature expressed in Kelvin, and 𝑞 the electron charge.Let us note that at room temperature (300K), we have 𝑉

𝑇≈

26mV. If we denote by 𝐼𝑛(𝑛 = 1, 2) the current flowing

through the inductor 𝐿𝑛and 𝑉

𝐶𝑚

(𝑚 = 1, 2) the voltageacross capacitor 𝐶

𝑚, the Kirchhoff ’s electric circuit laws can

be applied on Figure 1(a) to obtain the upcoming set ofautonomous differential equations describing the dynamicsof the TNC hyperchaotic oscillator:

𝑑𝐼1

𝑑𝑡=

𝑉𝐶1

𝐿1

, (2a)

𝑑𝐼2

𝑑𝑡=

1

𝐿2

(𝑉𝐶1

− 𝑉𝑑− 𝑅2𝐼𝑑) , (2b)

𝑑𝑉𝐶1

𝑑𝑡=

1

𝐶1

(

𝑉𝐶1

𝑅− 𝐼1− 𝐼𝑑) , (2c)

𝑑𝑉𝑑

𝑑𝑡=

(𝑉𝐶1

− 𝐼1− 𝐼𝑑) /𝐶1− (𝐼𝑑− 𝐼2) /𝐶2

1 + 𝑅2𝑑𝐼𝑑/𝑑𝑉𝑑

, (2d)

where the diode current 𝐼𝑑is defined by (1). In contrast to

previous literature [2, 4, 31] based on PWL models of theTNC oscillator, the exponential model is considered in thispaper. Indeed, the PWL model is only its first-order approx-imation and can approximately match only over a limitedregion (Figure 1(b)) and, thus, may exhibit different types ofbifurcation compared to the exponential model as previouslymentioned. Furthermore, owing to the fact that a smoothmathematical model is more tractable both analytically andnumerically, it may be exploited to obtained exact bifurcationstructures occurring in the real TNC hyperchaotic oscillatorcircuit. It is worth mentioning that the bifurcation diagramsare very useful for design purpose as they allow the selectionof the right parameters settings giving rise to a regularor erratic behavior (hyperchaotic mode of operation). Forcomputer simulation, we normalize the state equations (2a)–(2d) by considering the following rescaled variables andparameters:

𝑥 =𝜌𝐼1

𝜂𝑉𝑇

; 𝑦 =𝜌𝐼2

𝜂𝑉𝑇

; 𝑧 =

𝑉𝐶1

𝜂𝑉𝑇

;

𝑤 =𝑉𝑑

𝜂𝑉𝑇

; 𝜌 = √𝐿1

𝐶1

; 𝜏 =𝑡

√𝐿1𝐶1

;

𝜀1=

𝐿1

𝐿2

; 𝛾 =𝑅2𝐼𝑠

𝜂𝑉𝑇

; 𝜎 =𝜌

𝑅1

; 𝛼 =𝜌𝐼𝑠

𝜂𝑉𝑇

;

𝜀2=

𝐶1

𝐶2

.

(3)

Therefore, we obtain the upcoming smooth normalizedfourth-order differential equations:

�� = 𝑧, (4a)

𝑦 = 𝜀1(𝑧 − 𝑤 − 𝛾𝜑 (𝑤)) , (4b)

�� = 𝜎𝑧 − 𝑥 − 𝛼𝜑 (𝑤) , (4c)

�� =𝜎𝑧 − 𝑥 − 𝛼𝜑 (𝑤) − 𝜀

2(𝛼𝜑 (𝑤) − 𝑦)

1 + 𝛾 (𝜑 (𝑤) + 1)(4d)

with

𝜑 (𝑤) = exp (𝑤) − 1, (4e)

where the dots denote differentiation with respect to 𝜏

(renamed as t in the new scale without loss of generality).System (4a)–(4d) present only one nonlinear term and onestate variable (namely,𝑤) is involved in exponential function.The presence of such exponential nonlinearity implies thenonsymmetry of our model. Therefore, the system cannotsupport symmetric orbits.

3. Stationary Point and Its Nature

The notion of fixed points in the state space plays a crucialrole in understanding the dynamics of nonlinear systems.Theequilibria of the system (4a)–(4d) can be calculated by solvingthe following algebraic equations simultaneously:

0 = 𝑧, (5a)

0 = 𝜀1(𝑧 − 𝑤 − 𝛾𝜑 (𝑤)) , (5b)

0 = 𝜎𝑧 − 𝑥 − 𝛼𝜑 (𝑤) , (5c)

0 =𝜎𝑧 − 𝑥 − 𝛼𝜑 (𝑤) − 𝜀

2(𝛼𝜑 (𝑤) − 𝑦)

1 + 𝛾 (𝜑 (𝑤) + 1). (5d)

Computation of (5a)–(5d) shows that the origin𝑂(0, 0, 0, 0)

𝑇 is the only equilibrium point of the system.By linearizing the system at that equilibrium point, thefollowing 4 × 4 Jacobean matrix is obtained:

𝑀𝐽=

[[[[[

[

0 0 1 0

0 0 𝜀1

−𝜀1𝐴4

−1 0 𝜎 −𝛼

−1

𝐴4

𝜀2

𝐴4

𝜎

𝐴4

−𝛼 (1 + 𝜀2)

𝐴4

]]]]]

]

. (6)

4 Journal of Chaos

Then, the eigenvalues associated with this matrix areobtained by solving the following characteristic equation:

𝐴4𝜆4+ 𝐴3𝜆3+ 𝐴2𝜆2+ 𝐴1𝜆 + 𝐴

0= 0 (7)

with

𝐴4= 1 + 𝛾, 𝐴

3= 𝛼 (𝜀

2+ 1) − 𝜎 (1 + 𝛾) ,

𝐴2= (1 + 𝛾) (𝜀

1𝜀2+ 1) − 𝛼𝜀

2𝜎,

𝐴1= 𝛼𝜀2(1 + 𝜀

1) − 𝜎𝜀

1𝜀2(1 + 𝛾) ,

𝐴0= 𝜀1𝜀2(1 + 𝛾) .

(8)

For the chosen parameters as 𝜎 = 0.65, 𝜀1

= 3, 𝜀2

=

3.226, 𝛼 = 5.249 ⋅ 10−5,and 𝛾 = 5.249 ⋅ 10

−6 the eigenvaluessolutions of the characteristic equations (7) computed usingMATHEMATICA software package are 𝜆

1,2= 0.3249 ±

0.9457𝑖 and 𝜆3,4

= −8.7569 ⋅ 10−5

± 3.1109𝑖. Since theeigenvalues are complex conjugate with some positive realparts, the equilibrium point 𝑂(0, 0, 0, 0)

𝑇 is an unstablesaddle focus. Physically, this result supports the fact that theoscillator can oscillate chaotically and excludes the existenceof stable fixed point motion in the system.

4. Bifurcation Analysis

In order to define different routes/cascades to chaos in ourmodel, system (4a)–(4d) is solved numerically through thefourth-order Runge-Kutta integration algorithm. For eachset of parameters used in this work, the time step is alwaysΔ𝑡 ≤ 0.005 and the simulations are done with variables andconstant parameters in extended mode. For each case, thesystem (4a)–(4e) is integrated for a sufficiently long time andthe transient is discarded. Here, two indicators are combinedto identify the type of transition which leads to chaos. Thefirst one deals with the bifurcation diagram and the secondis related to the Lyapunov exponent’s spectra. Accordingto Wolf et al. [35] method, the dynamics of the nonlinearsystem (4a)–(4e) can be classified in terms of the Lyapunovexponents 𝜆

𝑖(𝑖 = 1, 2, 3, 4) as

(1) for an equilibrium point, 𝜆1,2,3,4

< 0;(2) for a limit cycle (periodic orbits), 𝜆

1= 0, 𝜆

2,3,4< 0;

(3) for 2-torus (quasiperiodic orbits), 𝜆1,2

= 0, 𝜆3,4

< 0;(4) for 3-torus, 𝜆

1,2,3= 0, 𝜆

4< 0;

(5) for a chaotic orbits, 𝜆1> 0, 𝜆

2= 0, and 𝜆

3,4< 0;

(6) for hyperchaotic orbits, 𝜆1,2

> 0, 𝜆3= 0, and 𝜆

4< 0;

In the next two subsections, the evolution process of thesystem is analyzed precisely by the means of the Lyapunovexponent’s spectrum, bifurcation diagrams, and phase por-traits when varying the parameters 𝜎 and 𝜀

1in system (4a)–

(4e).

4.1. Influence of 𝜎 with Fix 𝜀1. When the control parameter 𝜎

varies in the range [0.25–0.68], the graph of the bifurcation

diagrams and its corresponding Lyapunov exponent spec-trum are depicted, respectively, in Figures 2 and 3. Each bifur-cation diagram is obtained by plotting the localmaxima of therelated state variable in terms of the control state parameter.In the light of Figures 2 and 3, the following scenariosemerge when monitoring the control parameter 𝜎: periodic→ quasiperiodic → chaos → hyperchaos → chaos →

periodic → hyperchaos → periodic → hyperchaos. Inother word, they aremany windows of hyperchaotic behaviorwith respect to the control parameter 𝜎. Also, in contrastwith the traditional approach which consists of plotting thebifurcation diagram of a single state variable, we have plottedthe bifurcation diagram of all the state variables because ofthe nonuniformity observed between those diagrams. In fact,it can be seen that in the periodic windows, each state variablecycle has a different frequency. However, all state variablesshare common bifurcation point in terms of the controlparameter. In particular, a good coincidence is observedbetween band of chaos and windows of regular behaviorwithin those diagrams.

Moreover, for the set of parameters of Figure 2, variousnumerical computations of the phase space trajectories ofthe system (4a)–(4e) are obtained (Figure 4) confirmingtransitions to hyperchaos. Those curves represent the phaseportraits and their corresponding frequency spectrum com-puted for some discrete values of 𝜎: periodic oscillations →

torus state → periodic oscillations → chaotic oscillations→ hyperchaotic dynamics. This scenario is in conformitywith the bifurcation diagram of Figure 2.

One should note that some new periodic (𝜎 =

[0.38–0.42]) windows and some routes to chaos were onlyobserved by using the exponential model of the diode insteadof PWL model which is only its first approximation. Thesimulation results of the model proposed in [2] for exponen-tial and piecewise-linear models are depicted in bifurcationdiagram of Figure 5. The observe differences are mainly dueto the poor PWL approximation of the exponential character-istic (Figure 1(b)). On the other hand, the nonsmooth natureof the piecewise-linear model of the diode can be responsiblefor the different bifurcation structure exhibited by our modeland the one proposed in [2] as their I-V characteristics are notmatching perfectly.

4.2. Influence of 𝜀1with Fix 𝜎. As can be seen from the

bifurcation diagram (Figure 6) and the Lyapunov exponentsspectrum (Figure 7), the system (4a)–(4e) can be hyper-chaotic, chaotic, and periodic when varying 𝜀

1in the range

[3–8.5]. By observing Figures 3 and 7, it appears that thehyperchaotic range is much wider for 𝜀

1variation than when

𝜎 varies.A better understanding of the transition from regular

oscillations to hyperchaos could be used to achieve moreeffective control of chaos and also provide some novel waysof the design using such oscillators.

Although the bifurcation analysis was restricted in thiswork on the effects of parameters 𝜎 and 𝜀

1, it is worth

mentioning that any other parameters which appear in the

Journal of Chaos 5

0.3 0.4 0.5 0.6

𝜎

0.3 0.4 0.5 0.6

𝜎

0.3 0.4 0.5 0.6

𝜎

0.3 0.4 0.5 0.6

𝜎

−60

−40

−20

0

20

x

0

50

100

150

200

250

y

0

20

40

60

80

100

z

−30

−20

−10

0

10

w

Figure 2: Bifurcation diagrams showing the coordinates 𝑥, 𝑦, 𝑧, and 𝑤 in terms of the control parameter 𝜎. The others system parametersare 𝜀1= 3.0, 𝜀

2= 3.226, 𝛾 = 5.429 ⋅ 10

−6, and 𝛼 = 5.429 ⋅ 10−5.

characteristic equation could serve as a bifurcation controlparameter for the TNC system.

5. Experimental Study

Due to the real-time character of its results, compared tothe problems of transient phase duration and numericalinstabilities or time steps inherent to numerical simulations,experimental study appears as a complementary technique tothe methods mentioned above. Moreover, it is of interest toevaluate the influences of the simplifying process on the realbehavior of the oscillator.

5.1. Design of the Experimental Circuit. The circuit diagramof the TNC oscillator depicted in Figure 8 consists of an op.amp (TL082) powered by a symmetric ±12V DC voltagesupply, inductors (𝐿

1∼ 𝐿2) implemented using gyrators [36],

capacitors (𝐶1∼ 𝐶2), resistors (𝑅

1, 𝑅3, and 𝑅

4), and a diode

𝐷 (1𝑁4148) (𝑉𝐷= 0.65V and 𝑅

𝑑= 100Ω at 5mA) in series

with its resistor 𝑅𝑑. In the diagram of Figure 8, the inductors

are expressed in terms of the circuit components as follows:

𝐿1=

𝑅5𝑅7𝑅8𝐶3

𝑅6

; 𝐿2=

𝑅9𝑅11𝑅12𝐶4

𝑅10

. (9)

For the following set of the parameters 𝑅5

= 300Ω,𝑅6= 7.5KΩ, 𝑅

7= 7.5KΩ, 𝑅

8= 5KΩ, and𝐶

3= 10 nF

and 𝑅9= 300Ω, 𝑅

10= 7.5KΩ, 𝑅

11= 7.5KΩ, 𝑅

12= 5KΩ,

and 𝐶4= 10 nF, we determine the corresponding values of

the inductances: 𝐿1= 15mH and 𝐿

2= 5mH. It is worth

mentioning that the use of gyrators instead of real inductors isadvantageous since it becomes possible to monitor the valuesof inductances over a wide range by simply adjust a resistor.Therefore, inductors (𝐿

1and/or 𝐿

2) could also be used as a

bifurcation parameters (by varying 𝜀1and/or 𝜀

2). The other

circuit components are taken as 𝐶1= 18.4 nF, 𝐶

2= 5.7 nF,

𝑅3= 𝑅4= 2.2KΩ, and 𝑅

2= 100Ω.

5.2. Experimental Results. Now, we wish to analyze theeffects of the damping resistor 𝑅

1(that deals with the

6 Journal of Chaos

0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65

𝜎

0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65

𝜎

−0.4

−0.3

−0.2

−0.1

0

0.1

−25

−20

−15

−10

−5

𝜆1𝜆2𝜆3

𝜆4

Figure 3: Lyapunov exponent spectrum versus the control parameter 𝜎 for the system parameters of Figure 2.

control parameter 𝜎) on the dynamics of the TNC oscillator.The experimental results are obtained by slowly adjusting𝑅1and plotted phase space trajectories (𝑉

𝐶1

, 𝑉𝐶2

) using adual trace analog oscilloscope (HAMEG HM-203

−5) in the

𝑋𝑌mode. When monitoring the resistor 𝑅1, it is found that

the experimental oscillator circuit exhibits various types ofbifurcation. Indeed, for 𝑅

1= 4KΩ, a limit cycle behavior

is observed. But when 𝑅1decreases gradually, the following

transition is obtained: periodic oscillations→ torus state →

chaotic oscillations→ hyperchaotic oscillations → periodicoscillations→ hyperchaotic oscillations. The state 𝑤𝑤 is thevoltage across the capacitor 𝐶

2as in practice it is difficult to

measure through an oscilloscope the current. Some periodicwindows appearing between the chaotic and hyperchaoticregimes are also noted and were only figured out numerically(𝜎 = 0.4) using the exponential model of the diode. Figure 9presents the result of the experimental real investigationof the TNC oscillator. From those plots, one should notethat the real circuit exhibits the same bifurcation scenariosobserved using theoretical methods. These results also give agood agreement between numerical and experimental resultsand, thereby, can be considered to validate the mathemat-ical model (exponential model) proposed in this paper todescribe the dynamic behavior of the TNC hyperchaoticoscillator.

6. Synchronization Study

In this section, we investigate the chaos synchronizationof the model (4a)–(4e) with the realistic assumption ofparameters uncertainty. In fact, the complex behaviors ofhyperchaotic systems are believed to have much wider appli-cations. However, they are few discussions in the literature

when the parameters of the system are unknown. Moreover,in practice, some or all the parameters of the system cannotbe exactly known a priori (due to temperature, externalelectric and magnetic fields, etc.) or may be time varyingand/or inaccessible to direct measurements. Therefore, thesynchronization of two uncertain chaotic systems is essential[37–42].

6.1. Design of the Slave System. For our analysis, it is assumedthat the state variables of the drive system are all accessible tomeasurements and the system parameters are unknown.

Theorem 1. Let the system (4a)–(4e) be written in the follow-ing form:

��𝑚= 𝐹0(𝑋𝑚) + 𝐹1(𝑋𝑚) 𝜃 (10a)

in which we have

𝜃 = (0, 𝜀1, 𝜎, 𝜀2)𝑇

;

𝐹0(𝑋𝑚) = (𝑧

𝑚, 0, −𝑥

𝑚− 𝛼𝜑 (𝑤

𝑚) ,

(−𝑥𝑚− 𝛼𝜑 (𝑤

𝑚)) /𝐵1(𝑤𝑚))𝑇

;

𝐹1(𝑋𝑚)

=

[[[[[

[

0 0 0 0

0 𝑧𝑚− 𝑤𝑚− 𝛾𝜑 (𝑤

𝑚) 0 0

0 0 𝑧𝑚

0

0 0𝑧𝑚

𝐵1(𝑤𝑚)

(−𝛼𝜑 (𝑤𝑚) + 𝑦𝑚)

𝐵1(𝑤𝑚)

]]]]]

]

(10b)

Journal of Chaos 7

0 2 4 6 8 10Freq.

−20

−15

−10

−5

0

5

10

15

−10 −5 0 5 10

y

w

PS(y)

102

100

10−2

10−4

10−6

(a)

0 2 4 6 8 10Freq.

102

100

10−2

10−4

10−6

10−8−30

−25

−20

−15

−10

−5

0

5

10

15

w

−20 −10 0 10 20 30

y

PS(y)

(b)

0 2 4 6 8 10Freq.

102

100

10−2

10−4

10−6

10−8

−20 −10 0 10 20 30

y

−35

−30

−25

−20

−15

−10

−5

0

5

10

15

w

PS(y)

(c)

Figure 4: Continued.

8 Journal of Chaos

0 2 4 6 8 10Freq.

104

102

100

10−2

10−4

10−6−80

−70

−60

−50

−40

−30

−20

−10

0

10

20w

−50 0 50 100

y

PS(y)

(d)

0 2 4 6 8 10Freq.

104

102

100

10−2

10−4

10−6−100

−80

−60

−40

−20

0

20

w

−100 −50 0 50 100 150

y

PS(y)

(e)

Figure 4: Numerical phase portraits of the system obtained for various values of 𝜎: (a) periodic attractor (𝜎 = 0.25); (b) 2D torus state(𝜎 = 0.32); (c) periodic attractor (𝜎 = 0.4); (d) chaos 𝜎 = 0.47; (e) hyperchaos state (𝜎 = 0.65).

with 𝐵1(𝑤𝑚) = 1 + 𝛾(𝜑(𝑤

𝑚) + 1). Then the corresponding

response system which is given by

��𝑠= 𝐹0(𝑋𝑠) + 𝐹1(𝑋𝑠) 𝜃 + 𝑢 (11)

can synchronize with the master system (10a) and (10b), withthe controller 𝑢 defined as

𝑢 = − {𝑒 + 𝐹0(𝑋𝑠) − 𝐹0(𝑋𝑚) + [𝐹

1(𝑋𝑠) − 𝐹1(𝑋𝑚)] 𝜃 (𝑡)} ,

(12a)

𝜃 (𝑡) = −𝐹

𝑇

1(𝑋𝑚) 𝑒, (12b)

𝑒 = 𝑋𝑚− 𝑋𝑠; 𝜃 = (0, 𝜀

1, ��, 𝜀2) , (12c)

where 𝜀1, ��, and 𝜀

2are the estimation of the corresponding

parameters of the drive system (10a) and (10b).

Proof. The error of the dynamical system can be rewritten as

𝑒 = ��𝑠− ��𝑚

= 𝐹0(𝑋𝑠) + 𝐹1(𝑋𝑠) 𝜃 (𝑡) − 𝑒 − 𝐹

0(𝑋𝑠)

+ 𝐹0(𝑋𝑚) − [𝐹

1(𝑋𝑠) − 𝐹1(𝑋𝑚)] 𝜃 (𝑡) − 𝐹

0(𝑋𝑚)

− 𝐹1(𝑋𝑚) 𝜃 = −𝑒 + 𝐹

1(𝑋𝑚) [𝜃 (𝑡) − 𝜃] .

(13)

Journal of Chaos 9

0.3 0.4 0.5 0.6

𝜎

−20

0

20

40

60

80

z

(a)

0.3 0.4 0.5 0.6

𝜎

z

−2

0

2

4

6

(b)

Figure 5: Comparison of the transitions exhibited by the different diode models presented in [2]. (a) Bifurcation diagram for the exponentialmodel. (b) Bifurcation diagram for piecewise-linear model.

3 4 5 6 7 8

𝜀1

3 4 5 6 7 8

𝜀1

3 4 5 6 7 8

𝜀1

3 4 5 6 7 8

𝜀1

−60

−40

−20

0

20

40

x

0

50

100

150

200

y

−20

0

20

40

60

80

z

−40

−30

−20

−10

0

10

20

w

Figure 6: Bifurcation diagrams showing the coordinates 𝑥, 𝑦, 𝑧, and 𝑤 in terms of the control parameter 𝜀1using the system parameters of

Figure 2 with 𝜎 = 0.65.

10 Journal of Chaos

3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5

𝜀1

3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5

𝜀1

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

−20

−25

−15

−10

−5

𝜆1𝜆2𝜆3

𝜆4

Figure 7: Lyapunov exponent spectrum versus the control parameter 𝜀1.

−

−−

−−

+

++

++

11

1111

1111

4

44

44

2

22

22

3

33

33

1

11

11

U4

U5U2

U1U3

R9

R10

R11

R12R8

R1

R4

R3

R7

R6

R5

C4

C2C

D

1

C3

300300

100

7.5 k

2.2 k

2.2 k

1.7 k

5 k

7.5 k7.5 k

7.5 k

10nF

5nF

10nF

1.66k

18.13 nF

Rd

Figure 8: Experimental setup formeasurements on the hyperchaotic oscillator. 𝐿1(resp., 𝐿

2) is implemented by the op amps𝑈

1and𝑈

2(resp.,

𝑈4and 𝑈

5) based network following the approach presented in [32].

At this level, we consider the following Lyapunov functioncandidate:

𝑉(𝑒, 𝜃 (𝑡)) =1

2𝑒𝑇𝑒 +

1

2(𝜃 (𝑡) − 𝜃)

𝑇

(𝜃 (𝑡) − 𝜃) (14)

whose time derivative along the trajectory is given by

�� =1

2( 𝑒𝑇𝑒 + 𝑒𝑇

𝑒)

+1

2{(

𝜃 (𝑡))

𝑇

(𝜃 (𝑡) − 𝜃) + (𝜃 (𝑡) − 𝜃)𝑇 𝜃 (𝑡)}

=1

2(−𝑒𝑇𝑒 + (𝜃 (𝑡) − 𝜃)

𝑇

𝐹𝑇

1(𝑋𝑚) 𝑒

− 𝑒𝑇𝑒 + 𝑒𝑇𝐹1(𝑋𝑚) (𝜃 (𝑡) − 𝜃))

+1

2((

𝜃 (𝑡))

𝑇

(𝜃 (𝑡) − 𝜃) + (𝜃 (𝑡) − 𝜃)𝑇 𝜃 (𝑡))

= − 𝑒𝑇𝑒 ≤ 0.

(15)

Journal of Chaos 11

0.99 0.995 1 1.005 1.01ww

−15

−10

−5

0

5

10

15

20

z

(a)

0.98 0.99 1 1.01 1.02ww

−20

−15

−10

−5

0

5

10

15

20

z

(b)

0.98 0.99 1 1.01 1.02ww

−20

−10

0

10

20

z

(c)

0.95 1 1.05ww

−40

−20

0

20

40

z

(d)

Figure 9: Continued.

12 Journal of Chaos

0.9 1 1.1ww

−60

−40

−20

0

20

40

60

z

(e)

Figure 9: Experimental phase portraits (left hand side) obtained from the circuit using scope in Lissajou mode and their numericalcorrespondences (right hand side). Experimental measurements for the control parameters are (a) 𝑅

1= 4 kΩ, (b) 𝑅

1= 2.82 kΩ, (c)

𝑅1= 2.25 kΩ, (d) 𝑅

1= 1.92 kΩ, and (e) 𝑅

1= 1.38 kΩ.

0 20 40 60

t

0 20 40 60

t

0 20 40 60

t

0 20 40 60

t

−2

−1.5

−1

−0.5

0

e x(t)

−1.5

−1

−0.5

0

0.5

e y(t)

−1.5

−1

−0.5

0

0.5

e z(t)

e w(t)

−0.6

−0.4

−0.2

0

0.2

0.4

Figure 10: Time traces of error signals showing transition to synchronized states of the system and its observer. Initial conditions are𝑋𝑚(0) =

(2, 1.2, 1, 0)𝑇 and𝑋

𝑠(0) = (0.1, 0.2, −0.1, −0.4)

𝑇. We choose the parameters of the drive system as 𝜀1= 3, 𝜀

2= 3.226, and 𝜎 = 0.65 to ensure a

hyperchaotic behavior.

Journal of Chaos 13

0 20 40 601.5

2

2.5

3

3.5

t

�� 1(t)

(a)

t

0 20 40 600.2

0.65

1

𝜎(t)

(b)

t

0 20 40 60

2.813

3.226

3.4

2.4

�� 2(t)

(c)

t

0 20 40 600

1

1.5

0.5

2

2.5

MSE

(d)

Figure 11: Time evolution of the estimated parameters results (a)–(c) and the mean squared error (d). Details of the captures are (a) 𝜀1(𝑡), (b)

��(𝑡), and (c) 𝜀2(𝑡).

Owing to the result (15), we conclude that 𝑉(𝑒, 𝜃(𝑡)) is aLyapunov function of the error system and that (13) is stableat the origin, traducing the fact that 𝑒, 𝜃 are bounded.

6.2. Numerical Verification. Referring to Theorem 1 above,the response system is described by the following sets of first-order differential equations:

𝜀1= − (𝑧

𝑚− 𝑤𝑚− 𝛾𝜑 (𝑤

𝑚)) 𝑒𝑦, (16a)

𝜎 = − (𝑧𝑚𝑒𝑧+ (𝑧𝑚/𝐵1(𝑤𝑚)) 𝑒𝑤) , (16b)

𝜀2= (

(𝛼𝜑 (𝑤𝑚) − 𝑦𝑚)

𝐵1(𝑤𝑚)

) 𝑒𝑤, (16c)

��𝑠= 𝑧𝑚− (𝑥𝑠− 𝑥𝑚) , (16d)

𝑦𝑠= 𝜀1(𝑧𝑚− 𝑤𝑚− 𝛾𝜑 (𝑤

𝑚)) − (𝑦

𝑠− 𝑦𝑚) , (16e)

��𝑠= ��𝑧𝑚− 𝑥𝑚− 𝛼𝜑 (𝑤

𝑚) − (𝑧

𝑠− 𝑧𝑚) , (16f)

��𝑠=

(��𝑧𝑚− 𝑥𝑚− 𝛼𝜑 (𝑤

𝑚) − 𝜀2(𝛼𝜑 (𝑤

𝑚) − 𝑦𝑚))

𝐵1(𝑤𝑚) − (𝑤

𝑠− 𝑤𝑚)

. (16g)

The numerical computations of (16a)–(16g) are per-formed using the fourth-order Runge-Kutta integration algo-rithm with a time step of Δ𝑡 = 0.005. The initial conditions

on parameters are taken as 𝜀1(0) = 1.8, ��(0) = 0.4, and

𝜀2(0) = 2.5. We select the parameter of the drive system as

𝜀1= 3, 𝜎 = 0.65, and 𝜀

2= 3.226 to ensure a hyperchaotic

behavior. Numerical results are shown on Figures 10 and 11.From those graphs, it can be noted that the hyperchaoticsynchronization and the unknown parameters identificationare achieved simultaneously after a short transient time.Obviously, this perfect synchronization may be exploitedin secure communication using the technique of parametermodulation/estimation [37–39] as the system can be designedand implemented at any given frequency band depending onthe values of circuit components.

7. Conclusion

In this paper, we have built a new mathematical model basedon the exponential nonlinearity to examine the behavior of asimple 4D hyperchaotic electronic oscillator. This exponen-tial model takes advantage of the PWL approximation as itprovides a close form description of the oscillator’s dynamics.Dynamics of the system subjected to this exponential modelhave been analyzed both through bifurcation diagrams,Lyapunov exponents, phase portraits, and routes to chaos.By monitoring the parameters, the system has exhibitedmany phase portraits with different shapes from periodicto hyperchaotic oscillations. A real physical prototype wasdesigned and implemented on a bread board to study the

14 Journal of Chaos

dynamics of our model. Theoretical and experimental resultsobtained have been compared and a very good agreementwasobserved. In the context of the application to secure commu-nication, the synchronization of coupled hyperchaotic systemof themodel with uncertain parameters has also been studiedusing the recent results on adaptive control theory.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

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