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Research ArticleComplex Dynamical Behavior of a Two-Stage Colpitts Oscillatorwith Magnetically Coupled Inductors

V. Kamdoum Tamba,1,2 H. B. Fotsin,1 J. Kengne,3 F. Kapche Tagne,2 and P. K. Talla4

1 Department of Physics, Laboratory of Electronics and Signal Processing (LETS), Faculty of Science, University of Dschang,P.O. Box 67, Dschang, Cameroon

2Department of Telecommunication and Network Engineering, IUT-Fotso Victor of Bandjoun, University of Dschang,P.O. Box 134, Bandjoun, Cameroon

3Department of Electrical Engineering, Laboratory of Automation and Applied Computer (LAIA),IUT-Fotso Victor of Bandjoun, University of Dschang, P.O. Box 134, Bandjoun, Cameroon

4Department of Physics, Laboratory of Mechanics and Modelling of Physical Systems (L2MPS), Faculty of Science,University of Dschang, P.O. Box 67, Dschang, Cameroon

Correspondence should be addressed to H. B. Fotsin; [email protected]

Received 26 May 2014; Accepted 11 September 2014; Published 21 October 2014

Academic Editor: Uchechukwu E. Vincent

Copyright © 2014 V. Kamdoum Tamba et al. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

A five-dimensional (5D) controlled two-stage Colpitts oscillator is introduced and analyzed. This new electronic oscillator isconstructed by considering the well-known two-stage Colpitts oscillator with two further elements (coupled inductors and variableresistor). In contrast to current approaches based on piecewise linear (PWL) model, we propose a smooth mathematical model(with exponential nonlinearity) to investigate the dynamics of the oscillator. Several issues, such as the basic dynamical behaviour,bifurcation diagrams, Lyapunov exponents, and frequency spectra of the oscillator, are investigated theoretically and numericallyby varying a single control resistor. It is found that the oscillator moves from the state of fixed point motion to chaos via theusual paths of period-doubling and interior crisis routes as the single control resistor is monitored. Furthermore, an experimentalstudy of controlled Colpitts oscillator is carried out. An appropriate electronic circuit is proposed for the investigations of thecomplex dynamics behaviour of the system. A very good qualitative agreement is obtained between the theoretical/numerical andexperimental results.

1. Introduction

During the last three decades, a tremendous attention hasbeen devoted to design chaotic electronic oscillators. Thefocus on this interesting research field comes mainly fromtwo facts: first, one can observe chaos and can also control thedynamics of the oscillator by simply changing the physicallyaccessible parameters of the oscillator, for example, linearresistor, linear capacitor, voltage levels, coupled inductors,and so forth; second, there are a multitude of applicationsof chaotic electronic oscillators starting from chaotic elec-tronic secure communication to cryptography [1]. In thisregard, the classical Colpitts oscillator with single transistorwas investigated at 1 KHz frequency [2], high (3–300MHz)

frequencies [3], and ultrahigh (300–1000MHz) frequencies[4] using both numerical and experimental methods. Theinterest devoted to this oscillator is motivated by its simplephysical realization and low power requirement. Neverthe-less, the main limitation of the classical Colpitts oscillator isits incapacity to exhibit higher fundamental frequencies inchaotic regime [5]. In order to solve this problem, alternativesto this standard version of the Colpitts oscillator, namely, thetwo-stage and improved version, were reported in [6, 7]. Incomparison to a single stage Colpitts oscillator, the two-stageColpitts oscillator presents better spectral properties whichare suitable for communication application. Some interestingworks [8, 9] have been reported concerning the dynamics andthe control of chaos in two-stage Colpitts oscillator using the

Hindawi Publishing CorporationJournal of ChaosVolume 2014, Article ID 945658, 11 pageshttp://dx.doi.org/10.1155/2014/945658

2 Journal of Chaos

ideal current source (i.e., power supply) as a control parame-ter which is not easily accessible in practical situations. Thus,it is difficult to control the dynamics of the two-stage Colpittsoscillator using one direct accessible electrical component.To overcome this problem, the present work proposes acontrolled version of a two-stage Colpitts oscillator usinga pair of coupled inductors and a linear resistor which iseasy to use as a control parameter. The control schemereported in [10] exploits the coupling between the passivenetwork and the chaotic Colpitts oscillator through mutualinductors. This technique requires a simpler control circuit,which has the advantage of direct and easy implementation.The single resistor is used to control chaos in the Colpittsoscillator. This control technique is exploited in this paper toinvestigate the rich dynamics of two-stage Colpitts oscillator.However, it could also be used as a mean for parametermodulation strategy for secure communication applications.On the other hand, previous works [11–14] related to theinvestigation of the dynamics of this type of oscillators arebased on piecewise linear models (PWL) though restrictingto the first-order description of the system’s dynamics [8,15]. In fact, the PWL model represents only a first-orderapproximation of the reality; therefore it may give rise todifferent types of bifurcations compared to those exhibitedby the real oscillator. In contrast to previous mathematicalanalysis, the present work proposes a smooth (exponentialmodel) mathematical model to investigate the nonlineardynamics and chaos in the controlled two-stage Colpittsoscillator. Various bifurcation diagrams and correspondinggraphs of Lyapunov exponents are provided to characterizethe dynamics of the system in terms of the control parameter(single variable resistor). Finally, the effects of transistorgain on the dynamics of the controlled two-stage Colpittsoscillator are analyzed.

The layout of this paper is as follows. Section 2 is devotedto the circuit description and mathematical model of thecontrolled two-stage Colpitts oscillator. The circuit diagramof the oscillator is addressed and the corresponding mathe-matical model is derived. Section 3 deals with the dynamicalproperties of controlled two-stageColpitts oscillator. Dissipa-tion and existence of attractors, bifurcation analysis, chaoticbehavior, and interior crisis are investigated. In Section 4,experimental investigations of the dynamical behaviour ofthe system are described. Finally, some concluding remarksare given in Section 5.

2. Circuit Description andMathematical Model

2.1. Circuit Description. The simplest circuit diagram of thecontrolled two-stage Colpitts oscillator is depicted in Fig-ure 1(a). The circuit contains two bipolar junction transistors(BJT) 𝑄

1and 𝑄

2used in the common base configuration as

the nonlinear gain element. The resonant network consistsof coupled inductors 𝐿

1and 𝐿

2, mounted in series with

the biasing and damping resistor 𝑅, and three capacitors(𝐶1, 𝐶2, and 𝐶

3). 𝑅𝐿is a variable resistor which is used

as a control parameter of the oscillator. The ideal current

generator 𝐼0is used to maintain constant biasing emitter

current. It is important to note that, in the circuit diagram,the only nonlinear devices are the bipolar junction transistors(𝑄1and 𝑄

2), which are responsible for the striking complex

behavior exhibited by the oscillator. The BJT model consistsof a nonlinear voltage-controlled resistance 𝑅

𝐸and a linear

current-controlled current source 𝐼𝐸as shown in Figure 1(b).

The main difference between the classical circuit diagram ofthe two-stage Colpitts oscillator and the controlled version isthe presence of coupled inductors and variable resistor in thecollector node of the transistor 𝑄

1.

2.2. Mathematical Model. To model the circuit, someassumptions are considered. Firstly, we assume that thecapacitors, the inductor, and the resistor of the resonantnetwork are linear. Secondly, we neglect parasitic capacitors𝐶𝑏𝑒and 𝐶

𝑏𝑐. Thirdly, the transistors (𝑄

1and𝑄

2) are modeled

as in [15]. The V-I characteristic of the nonlinear resistor 𝑅𝐸

is defined as usual by

𝐼𝐸= 𝑓 (𝑉BE) = 𝐼𝑆 (exp(

𝑉BE𝑉𝑇

) − 1) , (1)

where 𝐼𝐸is the emitter current, 𝑉BE is the voltage across

the B-E junction, 𝐼𝑆is the saturation current of the B-E

junction, 𝑉𝑇= 𝐾𝑏𝑇/𝑒 is a thermal voltage with 𝐾

𝑏the

Boltzmann constant, 𝑇 is the absolute temperature, and𝑒 is the elementary charge. At room temperature, 𝑉

𝑇is

approximately equal to 26mV. Taking into account the BJTmodel of Figure 1(b) and denoting 𝐼

1and 𝐼2as the current

flowing through the inductors 𝐿1and 𝐿

2, respectively, and

𝑉𝐶𝑖(𝑖 = 1, 2, 3) as the voltage across capacitor 𝑐

𝑖(𝑖 = 1, 2, 3),

the state equations of the circuit of Figure 1(a) exploiting theKirchhoff electric circuit law are the following:

𝑐1

𝑑𝑉𝑐1

𝑑𝑡= 𝐼1− 𝛼𝐹1𝑓 (𝑉BE1) , (2a)

𝑐2

𝑑𝑉𝑐2

𝑑𝑡= 𝐼1− 𝐼0+ (1 − 𝛼

𝐹1) 𝑓 (𝑉BE1)

+ (1 − 𝛼𝐹2) 𝑓 (𝑉BE2) ,

(2b)

𝑐3

𝑑𝑉𝑐3

𝑑𝑡= 𝐼𝐿+ (1 − 𝛼

𝐹1) 𝑓 (𝑉BE1) − 𝛼𝐹2𝑓 (𝑉BE2) ,

(2c)

𝐿1𝐿2−𝑀2

𝐿2

𝑑𝐼1

𝑑𝑡= 𝑉0− 𝑉𝐶1− 𝑉𝐶2− 𝑉𝐶3− 𝑅𝐼1−𝑀

𝐿2

𝑅𝐿𝐼2,

(2d)

𝐿1𝐿2−𝑀2

𝑀

𝑑𝐼2

𝑑𝑡= 𝑉0− 𝑉𝐶1− 𝑉𝐶2− 𝑉𝐶3− 𝑅𝐼1−𝐿

𝑀𝑅𝐿𝐼2,

(2e)

where 𝛼𝐹𝑖(𝑖 = 1, 2) is common base forward short-circuit

current gain of the transistors,𝑀 = 𝑘√𝐿1𝐿2represents the

mutual inductance between 𝐿1and 𝐿

2, given the coupling

factor 𝑘, and 𝑉BE1 = 𝑉1 − 𝑉𝐶2 − 𝑉𝐶3 and 𝑉BE2 = −𝑉𝐶2 are,respectively, the base-emitter voltage of transistors 𝑄

1and

Journal of Chaos 3

R

V1

Q1

Q2

I0

RL

I2

I1

V0

VC3

VC1

VC2

+

−

+

−

+

−

L1L2

C1

C3

C2

(a)

C

B

IE IC

RE

E

IB

+

+

−

−

VBE

VCE

𝛼FIE

(b)

Figure 1: (a) Physical realization of the controlled two-stage Colpitts oscillator: the circuit parameters are 𝐶1= 8 nF; 𝐶

2= 𝐶3= 10 nF;

𝐿1= 16 uH; 𝐿

2= 23 uH; 𝑅 = 35Ω; 𝑉

0= 12V; 𝑉

1= 6V; 𝐼

0= 1.750mA; and 𝑅 = 60Ω potentiometer. The BJT model (b) consisting of a

current-controlled current source and a single diode.

𝑄2. For simplicity, we first assume that the common base

forward short-circuit gain of the transistor 𝛼𝐹𝑖= 1 (𝑖 = 1, 2);

that is, we neglect the base current. Equation (2a)–(2e) hasa single equilibrium point (𝑉0

𝐶1, 𝑉0

𝐶2, 𝑉0

𝐶3, 𝐼0

1, 𝐼0

2)𝑇 which can

be obtained by setting the right-hand side of (2a), (2b), (2c),(2d), and (2e) to zero.The following expressions are obtained:

𝑉0

𝐶1= 𝑉0− 𝑉1− 𝑅𝐼0+ 𝑉𝑇ln(1 +

𝐼0

𝐼𝑠

) , (3a)

𝑉0

𝐶2= − 𝑉

𝑇ln(1 +

𝐼0

𝐼𝑠

) , (3b)

𝑉0

𝐶3= 𝑉1, (3c)

𝐼0

1= 𝐼0, (3d)

𝐼0

2= 0. (3e)

For convenient numerical analyses, let us introduce thefollowing set of dimensionless state variables and parameters:

𝑥𝑖𝑉𝑇= 𝑉𝐶𝑖− 𝑉0

𝐶𝑖(𝑖 = 1, 2, 3) ,

𝑥𝑗𝑉𝑇= 𝜌 (𝐼

𝑗− 𝐼0

𝑗) (𝑗 = 4, 5) ,

𝑡 = 𝜏√𝐿𝐶2, 𝜌 = √

𝐿1

𝐶2

,

𝜎1=𝐶2

𝐶1

, 𝜎2=𝐶2

𝐶3

,

𝜎3=

𝐿1𝐿2

(𝐿1𝐿2−𝑀2)

, 𝜎4=

𝐿1𝑀

(𝐿1𝐿2−𝑀2)

,

𝛾 =𝜌𝐼0

𝑉𝑇

, 𝜀 =𝑅

𝜌,

𝜇 =𝐿1𝐿2

𝑀2, 𝛼 =

𝑅𝐿𝑀

𝐿2𝜌.

(4)

The physical parameters of controlled two-stage Colpittsoscillator are 𝐶

1= 8 nF; 𝐶

2= 𝐶3= 10 nF; 𝐿

1= 16 uH;

𝐿2= 23 uH; 𝑅 = 35Ω; 𝑉

0= 12V; 𝑉

1= 6V; 𝐼

0= 1.750mA;

and 𝑅𝐿= 60Ω potentiometer.

Therefore, the above equations (2a)–(2e), according to(4), are rewritten in the dimensionless form as

��1= 𝜎1(𝑥4− 𝛾𝜙 (𝑥

2+ 𝑥3)) , (5a)

4 Journal of Chaos

��2= 𝑥4, (5b)

��3= 𝜎2(𝑥4− 𝛾𝜙 (𝑥

2)) , (5c)

��4= 𝜎3(−𝑥1− 𝑥2− 𝑥3− 𝜀𝑥4− 𝛼𝑥5) , (5d)

��5= 𝜎4(−𝑥1− 𝑥2− 𝑥3− 𝜀𝑥4− 𝜇𝛼𝑥

5) , (5e)

where the dots denote differentiation with respect to 𝜏 and𝜙(𝑦) = exp(−𝑦) − 1.

Note that ourmodel is nonsymmetric due to the presenceof the exponential nonlinearity in (5a)–(5e). Therefore, thesystem cannot support symmetric orbits. In this mathemati-cal modeling, we consider an exponential model instead of apiecewise linear model (PWL) of the oscillator as previouslyadopted by some authors [11–14]. In fact, the PWL modelmay experience different types of bifurcations compared tothe exponential model which it approximates (first-orderapproximation) [8]. Furthermore, the exponential model ismore tractable both numerically and analytically comparedto PWL model. Then exponential model may be exploitedto derive the exact bifurcation structure occurring in thecontrolled Colpitts oscillators. Extended discussions can befound in [15].

3. Dynamical Properties

3.1. Dissipative and Existence of Attractors. The state space ofsystem (5a), (5b), (5c), (5d), and (5e) is five-dimensional.Thevector field on the right-hand sides of system (6) is defined as

V (𝑥) =[[[[[

[

V1(𝑥)

V2(𝑥)

V3(𝑥)

V4(𝑥)

V5(𝑥)

]]]]]

]

=

[[[[[

[

𝜎1(𝑥4− 𝛾𝜙 (𝑥

2+ 𝑥3))

𝑥4

𝜎2(𝑥4− 𝛾𝜙 (𝑥

2))

𝜎3(−𝑥1− 𝑥2− 𝑥3− 𝜀𝑥4− 𝛼𝑥5)

𝜎4(−𝑥1− 𝑥2− 𝑥3− 𝜀𝑥4− 𝜇𝛼𝑥

5)

]]]]]

]

.

(6)

The divergence of the vector field V is evaluated as

∇V (𝑥) =𝜕V1

𝜕𝑥1

+𝜕V2

𝜕𝑥2

+𝜕V3

𝜕𝑥3

+𝜕V4

𝜕𝑥4

+𝜕V5

𝜕𝑥5

= − (𝜎3𝜀 + 𝜎4𝜇𝛼) .

(7)

In view of (7), it can easily be shown that system (5a),(5b), (5c), (5d), and (5e) is dissipative with an exponentialrate 𝑑V/𝑑𝑡 = exp(−𝜎

3𝜀 − 𝜎

4𝜇𝛼). Then, in the dynamical

system (5a), (5b), (5c), (5d), and (5e), a volume element 𝑉0

is apparently contracted by the flow into a volume element𝑉0exp(−𝜎

3𝜀 − 𝜎4𝜇𝛼)𝑡 in time 𝑡. It means that each volume

containing the trajectories of this dynamical system shrinksto zero as 𝑡 → ∞ as an exponential rate. Therefore, all thesedynamical system orbits are eventually confined to a specificsubset that has zero volume; namely, the asymptotic motionsettles onto an attractor of this system [16].

3.2. Chaotic Behavior. Generally, two indicators are used toidentify the type of transition leading to chaos. The firstindicator is the bifurcation diagram which summarizes thedifferent dynamical behaviours exhibited by the oscillator

and the second is the largest one-dimensional (1D) numericalLyapunov exponent defined by

𝜆max = lim𝑡→∞

[(1

𝑡) ln (𝑑 (𝑡))] , (8)

where

𝑑 (𝑡) = √𝛿21+ 𝛿22+ 𝛿23+ 𝛿24+ 𝛿25, (9)

and computed from the following variational equationsobtained by perturbing the solutions of (5a), (5b), (5c), (5d),and (5e) as follows: 𝑥

1→ 𝑥

1+ 𝛿1, 𝑥2→ 𝑥

2+ 𝛿2,

𝑥3→ 𝑥

3+ 𝛿3, 𝑥4→ 𝑥

4+ 𝛿4, and 𝑥

5→ 𝑥

5+ 𝛿5.

𝑑(𝑡) is the distance between neighbouring trajectories in thephase space. Asymptotically 𝑑(𝑡) = exp(𝜆max𝑡). Thus, if𝜆max > 0, neighbouring trajectories diverge and the stateof the oscillator is chaotic. For 𝜆max < 0, these trajectoriesconverge and the state of the oscillator is nonchaotic.The case𝜆max = 0 corresponds to the torus state of the oscillator.

In order to analyse the influence of the control parameteron the dynamics of the controlled two-stage Colpitts oscilla-tor, we set 𝜎

1= 1.250, 𝜎

2= 1.000, 𝜎

3= 2.770, 𝜎

4= 1.853,

𝜀 = 0.875, 𝜇 = 1.562, and 𝛾 = 2.692 and only vary 𝛼 inthe domain 1 ≤ 𝛼 ≤ 2. The system (5a), (5b), (5c), (5d),and (5e) is solved numerically to define routes to chaos inour model using the fourth-order Runge-Kutta algorithm.The initial values of the system (5a), (5b), (5c), (5d), and (5e)are arbitrarily taken as [0.001 0.001 0.001 0.001 0.001].Bifurcation diagram and corresponding graph of 1D largestLyapunov exponent are obtained. Sample results are providedin Figure 2 where a bifurcation diagram associated withthe corresponding graph of largest 1D largest numericalLyapunov exponent summarizes various scenarios leadingto chaos in the controlled two-stage Colpitts oscillator. Thisbifurcation diagram is obtained by plotting the coordinate𝑥5in terms of the control parameter 𝛼. The positive value of

𝜆max is the signature of chaotic oscillations of the oscillator.Note that the various transitions/routes to chaos observedin the controlled two-stage Colpitts oscillator are commonlyobserved in various nonlinear systems [15, 17, 18] includingthe universal Chua circuit, phase-locked loops, and thetransformer-coupled oscillators, just to name a few. Thisserves to justify the richness of the bifurcations in thecontrolled two-stage Colpitts oscillator and also the strikingphenomena exhibited by such oscillators.

In order to study the effects of the transistor gain 𝛽 onthe dynamics of the oscillator, we reconsidered system (2a),(2b), (2c), (2d), and (2e) with 𝛼

𝐹𝑖= 1 (𝑖 = 1, 2). Then the

normalized circuit equations can be rewritten as follows:

��1= 𝜎1(𝑥4− 𝛼𝐹1𝛾𝜙 (𝑥2+ 𝑥3)) , (10a)

��2= 𝑥4+ (1 − 𝛼

𝐹1) 𝛾𝜙 (𝑥

2+ 𝑥3) + (1 − 𝛼

𝐹2) 𝛾𝜙 (𝑥

2) , (10b)

��3= 𝜎2(𝑥4+ (1 − 𝛼

𝐹1) 𝛾𝜙 (𝑥

2+ 𝑥3) − 𝛼𝐹2𝛾𝜙 (𝑥2)) , (10c)

��4= 𝜎3(−𝑥1− 𝑥2− 𝑥3− 𝜀𝑥4− 𝛼𝑥5) , (10d)

��5= 𝜎4(−𝑥1− 𝑥2− 𝑥3− 𝜀𝑥4− 𝜇𝛼𝑥

5) , (10e)

Journal of Chaos 5

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 20

1

2

3

4

5

𝛼

x5

(a)

−0.05

0

0.05

0.1

0.15

0.2

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2𝛼

𝜆m

ax

(b)

Figure 2: Bifurcation diagram of the controlled two-stage Colpitts oscillator (a) showing the state variable 𝑥5versus control parameter 𝛼 and

corresponding graph of 1D largest Lyapunov exponent (b). The parameters are 𝜎1= 1.250, 𝜎

2= 1.000, 𝜎

3= 2.770, 𝜎

4= 1.853, 𝜀 = 0.875,

𝜇 = 1.562, 𝛾 = 2.692, and 𝛼𝐹= 1.

6

4

2

01 1.2 1.4 1.6 1.8 2

𝛼

x5

(a)

6

4

2

01 1.2 1.4 1.6 1.8 2

𝛼

x5

(b)

4

3

2

1

01 1.2 1.4 1.6 1.8 2

𝛼

x5

(c)

Figure 3: Bifurcation diagrams of the controlled chaotic Colpitts oscillator showing the effects of the transistor gain (𝛽 = 𝛼𝐹/1 − 𝛼

𝐹) on the

dynamics of the oscillator plotted for (a) 𝛼𝐹= 1, (b) 𝛼

𝐹= 0.9803, and (c) 𝛼

𝐹= 0.9876. The rest of the system parameters are those in Figure 2.

where the state variables and corresponding normalizationare as defined previously. The relationship between thetransistor gain (𝛽) and common base forward short-circuitgain of the transistor (𝛼

𝐹) is defined as

𝛽 =𝛼𝐹

1 − 𝛼𝐹

. (11)

We assume that the two transistors are equivalent; that is,(𝛼𝐹1= 𝛼𝐹2). According to (11), the numerical value of 𝛼

𝐹is

0.9803 for 𝛽 = 50 and 0.987 for 𝛽 = 80 are used in thesimulations.The effect of the transistor gain on the dynamicsof the oscillator is illustrated by the bifurcation diagram ofFigure 3. We observe that this diagram is similar to the onepreviously obtained with the ideal transistor model for 𝛼

𝐹𝑖=

1 (𝑖 = 1, 2) (see Figure 3(a)). Nevertheless, it is obvious thatthe value of (𝛼

𝐹𝑖= 1 (𝑖 = 1, 2)) induces a horizontal stretching

of the control parameter 𝛼 (see Figures 3(b) and 3(c)).

Using the same parameters settings of Figure 2 and circuitparameters given in Figure 1, various numerical phase por-traits and their corresponding power spectra were obtainedconfirming transitions/routes to chaos depicted previously(see Figure 4). The broadband noise-like power spectrum isthe signature of a chaotic behaviour of the oscillator.

A blow-up of the bifurcation diagram of Figure 2 in theregion of period 3 showing saddle-node bifurcation (SN),period-doubling (PD), and interior crisis (IC) is depicted inFigure 5.

4. Experimental Study

In this section, theoretical results obtained in the previouspart are verified by carrying out an experimental study.The complex behaviour of the oscillator is investigated bymonitoring a single variable resistor 𝑅

𝐿, while keeping the

rest of electronic components values constant. This section is

6 Journal of Chaos

−10 −5 0 5−2

−1

0

1

2

3

4

5

6

0 0.5 1 1.5 2Frequency

PSD

xj(𝜏)

xk(𝜏)

10−4

10−3

10−2

10−1

101

100

(a)

−10 −5 0 5−4

−2

0

2

4

8

6

0 0.5 1Frequency

PSD

xj(𝜏)

xk(𝜏)

10−4

10−3

10−2

10−1

101

100

(b)

−10−15 −5 0 5−4

−2

0

2

4

6

8

10

0 0.2 0.4 0.6 0.8Frequency

PSD

xj(𝜏)

xk(𝜏)

10−4

10−3

10−2

10−1

101

100

(c)

−15 −10 −5 0 5

−4

−2

0

2

4

6

8

12

10

0 0.2 0.4 0.6 0.8Frequency

PSD

xj(𝜏)

xk(𝜏)

10−5

10−4

10−3

10−2

10−1

101

100

(d)

−15 −10 −5 0 5

−4

−2

0

2

4

6

8

12

10

0 0.2 0.4 0.6 0.8Frequency

PSD

xj(𝜏)

xk(𝜏)

10−3

10−2

10−1

101

100

(e)

−20 −10 0 10−4

−2

0

2

4

6

8

12

14

10

0 1 2 3 4Frequency

PSD

xj(𝜏)

xk(𝜏)

10−5

10−4

10−3

10−2

10−1

101

100

(f)

Figure 4: Computer generated phase portraits (left) of the system projected onto the plane (𝑥𝐾− 𝑥𝑗) showing routes to chaos in terms of

control parameter 𝛼 and corresponding power spectra (right): (a) period 1 for 𝛼 = 0.6, (b) period 2 for 𝛼 = 1.3, (c) period 4 for 𝛼 = 1.4, (d)chaos for 𝛼 = 1.5, (e) period 3 for 𝛼 = 1.68, and (f) chaos for 𝛼 = 1.8. The parameters are defined in text. Note that 𝑥

𝑘= 𝑥1+ 𝑥2+ 𝑥3and

𝑥𝑗= 𝑥2+ 𝑥3.

Journal of Chaos 7

1.67 1.68 1.69 1.7 1.71 1.72 1.730

0.5

1

1.5

2

2.5

3

3.5

4

PDICSN

x5

𝛼

Figure 5: Blow-up of the bifurcation diagram of Figure 2 in the region of period 3 window showing saddle-node bifurcation (SN), period-doubling (PD), and interior crisis (IC). The rest of the parameters are those in Figure 4.

R6

R7

R1

R2

R3 R4

R

3560

8n

10n

10n

0

R5

100 k

100 k

56 k

6 V

12 V

12 V

16 𝜇H 23 𝜇H

TX

11 k 5.6 k

560

14 k

TL082TL082

Q2N2222

Q2N2222

U6AU5A

42

3

0

0

1

8 1

24

3 8

−

−

+

+

−

+

−

+

−

+

V0

V2

V3

Q2

Q3

C1

C2

C3

L1 L2 RL

V+

V−V+

V−

Figure 6: Experimental setup for measurements on the controlled two-stage Colpitts oscillator. The biasing current is constructed by the op.amplifiers (U1A and U2A) based network. Both 𝑄

1and 𝑄

2are 2N2222 bipolar junction transistor type.

Figure 7: Photograph of the experimental controlled two-stage Colpitts oscillator circuit.

8 Journal of Chaos

−10 −5 0 5−2

−1

0

1

2

3

4

5

6xj(𝜏)

xk(𝜏)

(a)

−10 −5 0 5−4

−2

0

2

4

6

8

xj(𝜏)

xk(𝜏)

(b)

xj(𝜏)

xk(𝜏)

−15 −10 −5 0 5−4

−2

0

2

4

6

8

10

(c)

Figure 8: Continued.

Journal of Chaos 9

xj(𝜏)

xk(𝜏)−15 −10 −5 0 5

−4

−2

0

2

4

6

8

10

12

(d)

xj(𝜏)

xk(𝜏)−15 −10 −5 0 5

−5

0

5

10

15

(e)

xj(𝜏)

xk(𝜏)−20 −15 −10 −5 0 5 10

−5

0

5

10

15

(f)

Figure 8: Experimental phase portraits (right) obtained from the circuit of Figure 8 using a dual trace oscilloscope in the 𝑋𝑌 mode. Thecollector voltage of BJT 𝑄

1(V𝑐1+ V𝑐2+ V𝑐3) is connected to the 𝑋 input and the 𝑌 channel displays the collector voltage of BJT 𝑄

2(V𝑐2+ V𝑐3):

(a) period 1 for 𝑅𝐿= 60Ω; (b) period 2 for 𝑅

𝐿= 90Ω; (c) period 4 for 𝑅

𝐿= 102Ω; (d) chaos for 𝑅

𝐿= 120Ω; (e) period 3 for 𝑅

𝐿= 148Ω;

and (f) chaos for 𝑅𝐿= 160Ω. Corresponding numerical phase portraits (right).

10 Journal of Chaos

also provided in order to evaluate the effects of the simplifyingassumptions adopted during the modelling process on thereal dynamics of the controlled two-stage Colpitts oscillator.

4.1. Design of the Experimental Setup. Theexperimental setupfor measurements on the controlled two-stage Colpitts oscil-lator is depicted in Figure 6. This figure is carried out usingoperational amplifier (TL082), bipolar junction transistors(2N2222), variable transformer, high precision resistors, andcapacitors with corresponding values fixed as above.The biasis provided by a 12VDC symmetric source. In order to insuregood functioning of the circuit during experimental process,the ideal current source is replaced by the network consistingof the operational amplifier UA1 associated with five resistors𝑅𝑖(𝑖 = 1, 2, . . . , 5). This configuration satisfies the following

condition:

𝑅1

𝑅2

=𝑅3

𝑅4+ 𝑅5

. (12)

The expression of the ideal current source 𝐼0is as follows

[10]:

𝐼0=𝑅2

𝑅1𝑅5

𝑉𝑖, (13)

where 𝑉𝑖is the output voltage of the network (i.e., an invert-

ing amplifier) using the operational amplifierUA2.Accordingto (12) and (13), we derive the following relationship betweenthe control voltage 𝑉

𝑖and the ideal current source 𝐼

0as

𝐼0= 10−3𝑉𝑖. (14)

The photograph of the experimental controlled two-stageColpitts oscillator is presented in Figure 7.

4.2. Experimental Results. In this section, we analyse theeffects of the variable resistor (𝑅

𝐿) on the dynamics of the

controlled two-stage Colpitts oscillator. When monitoringthe single variable resistor (𝑅

𝐿), it is found that the electronic

circuit experiences a rich and striking dynamical behaviourand various types of bifurcation. Some sample phase portraitsobtained experimentally are shown in Figure 8. This figure(right side) presents the real dynamics of controlled two-stage Colpitts oscillator. It is observed in Figure 8 thatthe real circuit presents the same bifurcation scenarios asshown using analytical methods (period 1 → period 2 →period 4 → chaos → period 3 → chaos). A verygood qualitative agreement is obtained between numerical(left side of Figure 8) and experimental results (right side ofFigure 8).

5. Conclusion

This paper has introduced and investigated the dynamicsof the new controlled two-stage Colpitts oscillator. Theproposed oscillator in its regular/periodic state can be usedfor instrumentation in laboratory. In its irregular state, it canalso be exploited as high frequency chaotic signals generators

suitable for chaos based communication. The dynamics ofthe oscillator is easily controlled via a single variable resistor.The circuit diagram of the oscillator was presented andthe modeling process using exponential nonlinearities wasperformed to derive the set of coupled first-order nonlinearordinary differential equations describing the behavior ofthe oscillator. Various bifurcation diagrams associated withtheir graph of 1D largest numerical Lyapunov exponent wereobtained showing transitions/routes to chaos in terms of thecontrol parameter (variable resistor). Theoretical/numericaland experimental results were compared and a very goodagreement was observed. It was found that the oscillatormoves from the state of fixed point motion to chaos viathe usual paths of period-doubling and interior crisis routeswhenmonitoring the control parameter (variable resistor𝑅

𝐿)

in tiny steps.

Conflict of Interests

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

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