Chaos-low periodic orbits transition in a synchronous switched circuit

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1 Chaos-low periodic orbits transition in a synchronous switched circuit N. JABLI 1, H. KHAMMARI 2 and M.F. MIMOUNI 1 Electrical Department (1) National Engineering School o Monastir Rue Ibn aljazzar, 5019 Monastir (2) Higher Institute o Applied Sciences and Technology o Kairouan Avenue Beit El Hekma 3100 Kairouan TUNISIA nj razin@yahoo.r ; gham.hedi@hotmail.com ; maouzi.mimouni@enim.rnu.tn Abstract: In this paper we consider a modiied model o an Alpazur oscillator with a periodically switching operation. Complex oscillatory phenomena are observed due to both switching operation and the appearance o an additional nonlinear component in circuit model. Indeed, subharmonic oscillations and chaotic behavior are illustrated. In addition, time series, phase trajectories and power spectrum was examined in the analysis o the system dynamics. The transition to chaotic states through doubling period biurcation cascade and the direct transition rom undamental harmonic or odd subharmonic orbits and transition rom a special biurcation paths to chaotic states are discussed in the state space. The role o higher harmonic spectral lines in period-1 responses under parameter variation have seemingly similarities with the results obtained in previous works or non switched dynamical system. Chaotic states resulting rom dierent transition modes are also presented in this paper. For the numerical simulations, techniques or deriving time and composite Poincaré method are applied. Key Words: Switching operation,transition modes, higher harmonic predominance, doubling period cascade. 1 Introduction The electrical circuits including switching actions are discussed as a recent interesting subject o search in much o previous works [1],[10],[12],[13],[16],[21]. Much o such circuits are encountered in Power Electronics ield which is a discipline spawned by real lie applications in industrial, commercial, residential and aerospace environments [2],[3],[14],[19]. Several chaotic behaviors in nonlinear dynamic systems were investigated and classiied in many ormer studies. In a erroresonant circuit, seemingly random oscillations were observed in steady state, the impact o source voltage and initial conditions on the so called chaotic erroresonance was highlighted in [17]. There are two types o chaotic systems, autonomous and non autonomous [18]. In the last case the systems are submitted to an external time varying source. A well known example studied in [6] dealt with the Duing type equation describing an electrical circuit, the amplitude and the requency o the sinusoidal signal both contribute to the chaotic dynamics o such circuit. The Duing-Van der Pol oscillator o [4] shows a broad spectrum o dynamic behaviors, both chaotic as well as periodic. Studies o routes to chaos exhibit several conigurations o biurcation paths leading to chaotic orbits.the period doubling succession plays an important role in the occurrence o chaotic attractors, it generally constitutes a veritable route to chaos. Other biurcation paths leading to chaotic states are available in literature. Among theses cases one can itemize the transition rom period 3 orbits to chaos wherein the oscillator s responses enter abruptly chaotic regions under a parameter variation. In certain cases, pursuing to vary the same parameter, the system dynamics exit chaotic regions and turn again to period 3 orbit. In undamental harmonic regime taken as the standard situation, the system responses are synchronous with period one orbit. Nevertheless, the nature, the period and the stability o the responses can undergo an unexpected change under a parameter variation. Theses sudden changes are called biurcation phenomena. In non linear autonomous systems some biurcation scenarios involve transition to chaotic states via stable equilibrium, limit cycles and quasi-periodic orbits [20]. Traditional biurcation appear when one o the eigenvalues reaches critical values (eigenvalues o the Jacobian Matrix) leading to a change o stability or order or ISSN: Issue 12, Volume 7, December 2008

2 nature. Some analysis tools issued rom biurcation theory will be applied to a two dimensional switching dynamical circuit. Circuits with one or more switches, also called on-o circuits, are generally described by dynamical dierential equations switched in a certain manner typically synchronous or asynchronous modes. In synchronous mode, the switching is done by a periodic external independent state excitation [5],[8]. Whereas in asynchronous mode, toggling is controlled by a depending state excitation [9],[10]. Thus a switching circuit can be described as a piecewise switched circuit which assumes dierent topologies at dierent times. It is worth noting that in recent ew years, it has been gradually recognized that the switched dynamical systems exhibit many interesting phenomena such as periodic, quasi-periodic, subharmonic and chaotic behaviors, border collision biurcation and so orth [15]. As an illustrated example, we investigated the behavior o a modiied Alpazur oscillator which includes two nonlinearities: the nonlinear conductor having a cubic current-voltage characteristic and a nonlinear inductor represented similarly by a third order unction. The operational mode o the circuit is changed by a switch turned on and o periodically resulting in a sequence o nonlinear circuits being toggled in a supposedly orderly manner. Thus, the considered switched circuit involves two dierential equations one in each region so the corresponding map is continuous across and its derivatives become discontinuous. Applying the general methodology, the Poincaré mapping was taken as a composite discrete mapping. Three main transition modes will be outlined in this paper such as doubling period biurcations succession, direct transition rom undamental harmonic or odd subharmonic responses to chaos and a special transition through period-n orbits succession, n = 4, 5, 6, System description and general reminders 2.1 Modiied Alpazur oscillator Firstly, we should note that the modiication done on the classical Alpazur oscillator discussed in [8] consists in replacing the linear inductor by a nonlinear one so that the circuit includes two nonlinear elements the conductor G and the inductor L. Such circuit is a combination o a Rayleigh-type oscillator unit and two supplied dc power interchanged by a switch S p which operates at switching edges P 1 and P 2 as shown in Figure 1. For 0 < t τ p1 ; τ p1 = ρ.t, S p is Figure 1: Modiied Alpazur Oscillator circuit In what ollows we will outline the plethora o dierent dynamical behaviors exhibited by the considered oscillator. Section 2 is mainly intended to describe the modiied Alpazur oscillator including an external orce depending switch. Section 3 is devoted to present three dierent ways leading to chaotic behaviors. In section 4, we have attempted to characterize intermittent phenomenon by subharmonic predominance via spectral analysis accordingly to ormer studies [6]. Section 5 is reserved to present chaotic regions resulting rom two dierent transition modes. Figure 2: A chronogram o the switch excitation thrown into contact with position P 1 so the circuit dynamics is{ governed by two dimensional autonomous dφ system: dt = v C dv dt = i G(v) + E 1 v r+r 1 0 < t ρ.t (1) ISSN: Issue 12, Volume 7, December 2008

3 Then it is switched { to position P 2 or so the governing dφ equations are: dt = v C dv dt = i G(v) + E 2 v r+r 2 ρ.t < t T (2) Where the capacitor voltage v(t) and the magnetic lux φ(t) in the inductor are state variables o the system. The nonlinear current-voltage characteristic o the conductor G is a smooth cubic unction G(v) = v v3 and the nonlinear current-lux characteristic o the inductor L is supposed to have the orm i = αφ + βφ 3, α > 0, β > 0. Ater normalization o the state variables and the parameters, the dierential systems (1)and (2) are respectively rewritten as: { d dt = d dt = α + βx (1 ε 1) 1 3 x3 2 + H 1 0 < t ρ.t { (3) d dt = d dt = α + βx (1 ε 2) 1 3 x3 2 + H 2 ρ.t < t T (4) where ε 1 = 1 r+r 1, ε 2 = 1 r+r 2, H 1 = ε 1 E 1, H 2 = ε 2 E 2, C = 1.0, τ p1 = ρ.t and τ p2 = (1 ρ).t (i.e. τ p1 + τ p2 = T ), and by considering the ollowing transormation o the state variables φ = and v =, The switching sequence is given in Figure 2. to see that the switching system possesses a composite solution, we then deine the ollowing two maps: M p1 : IR 2 IR 2 x 0 = ϕ p1 (ρ.t, x 0, λ c, λ 1 ) (7) M p2 : IR 2 IR 2 = ϕ p2 ((1 ρ).t,, λ c, λ 2 ) (8) Where ϕ p1 (ρ.t, x 0, λ c, λ 1 ) = ϕ(ρ.t, x 0, λ c, λ 1, λ 2 ) ϕ p2 ((1 ρ).t, x 0, λ c, λ 2 ) = ϕ((1 ρ).t, x 0, λ c, λ 1, λ 2 ) Poincaré section is given by choosing transversal plan deined t = k.t, the resulting composite mapping M c expressed as: M c : IR 2 IR 2 x 0 M c (x 0 ) = M p2 M p1 (x 0 ) (9) For ixed points investigation we merely solve M c (x 0 ) = x 0. Additional condition about the derivative is required or solving such equation: M c x 0 = M p 2 t=t. M p 1 t=(1 ρ).t x 0 t=ρ.t (10) 2.2 Some properties o Poincaré map The dierential systems (3) and (4) can be written in a compact orm: dx dt = p 1 (x, λ, λ 1 ), dx dt = p 2 (x, λ, λ 2 ), 0 < t ρ.t ρ.t < t T (5) The general solution o such kind o dierential equation can be expressed as : x(t) = ϕ(t, x 0, λ, λ 1, λ 2 ) (6) Where t IR is the time, x 0 IR 2 is the state variable, (λ c, λ 1, λ 2 ) IR 3. λ c is a common parameter or the unctions p1 and p2, whereas λ 1 and λ 2 are speciic parameters or each o them respectively. These unctions are assumed to be smooth and dierentiable as many times as necessary in regard to the variables and parameters. Reerring to the illustrative trajectory example shown in Figure 3, and taking into account the border continuity conditions, it is obvious Figure 3: Composite trajectory 2.3 Basic methods o analysis In this section, we will briely introduce the method previously discussed in [12] to investigate the governing equations o system under study, which typically consist o two nonlinear autonomous systems. More generally, we consider the ollowing n-dimensional ISSN: Issue 12, Volume 7, December 2008

4 dierential equations: The solution o these equations can be expressed as: For 0 < t ρ.t, S p is at p 1 : x(t) = ϕ p1 (t, x 0, λ c, λ 1 ) (11) For ρ.t < t T, S p is at p 2 : x(t) = ϕ p2 (t ρ.t, ϕ p1 (t, x 0, λ c, λ 1 ), λ c, λ 2 ) (12) based on the equations, we can easily deine the ollowing two maps: In practice the Poincaré section can be generated by choosing the transversal plan deined at t = ρ.t. Thus, By using the property o the switching actions, Poincaré mapping was constructed as a composite discrete mapping M c o M p1 and M p2 : M c : IR n IR n x 0 M c (x 0 ) = M p2 M p1 (x 0 ) (13) It is obvious to veriy that a ixed point x 0 in the chosen surace section, is a point o the map M c, where M c (x 0 ) = x 0. The knowledge o the singularities o M c enables to study the dynamical behaviors o the initial system. Indeed a a ixed point o M c is associated to a periodical solution having the period o the undamental harmonic (or a period-1 orbit) whereas a k-order cycle o M c is associated to a periodical solution having the period o a k-order subharmonic (or a period-k orbit). To numerically solve the equation mentioned above we can apply the Newton s method. In act, we must calculate the derivative term presented in this method witch is deined as: We note that the eigenvalues o the Jacobian matrix Mc x 0 at a ixed point are called the characteristic multipliers, and they will be noted here by (s i ) i=1, 2,... n, which are the roots o the characteristic equation deined as: χ(µ) = M c x 0 si n = 0 (14) Generally, there are three critical points associated to the value o (s i ), called biurcation points: Fold and Pitchork biurcation (s = 1), Flip or perioddoubling biurcation (s = 1) and Neimark biurcation (complex conjugate multipliers with modulus 1). Consider a period-1 solution o (4) and (5), its Fourier series expansion and the corresponding requency spectrum (made o o lines). Let r be place occupied by an order p higher harmonic spectral line rom an ordering based on the amplitudes o spectral lines in descending order [6]. A given order p higher harmonic is said predominant i it occupies the second place (r = 2) in the above classiication. It is said ully predominant i it occupies the irst place (r = 1). Choosing an arbitrary parameter plane, it can be considered as made up o sheets (oliated representation), each one being associated with a well deined response. For two dimensional nonlinear dynamical system, every period k orbit (associated with order k cycle) possesses two multipliers s 1 and s 2 determined rom the Jacobian matrix, so a old biurcation curve is the junction o two sheets o same period, one is related to a saddle ixed point (s 1 < 1, 0 < s 2 < 1) the other to a ixed point with s i < 1, i = 1, 2, (stable node, or stable ocus). A lip biurcation curve is the junction o three sheets. one is associated to a stable period k cycle with s i < 1, i = 1, 2, another with a saddle type period k cycle with s 1 < 1, s 2 < 1, the third sheet is associated to a period 2k cycle having s i < 1, i = 1, 2 (stable node, or stable ocus). A pitchork biurcation curve is the junction o our sheets. Three are related to node cycles having the same order and the ourth is linked to a saddle cycle having the same order too. 3 Transition modes to chaotic orbits The main property o a chaotic response is that it is not asymptotically stable and closely correlated initial conditions have trajectories which quickly become unrelated. Among the characteristics o chaos, that can be quantiied we distinguish apparent randomness in the time variation, the broad band components to the power spectrum and sensitive dependence on initial conditions. The Lyapunov characteristic exponent gives the rate o exponential divergence rom perturbed initial conditions, The dominant Lyapunov exponent is one o the most widely used indicators to describe the qualitative behavior in a dynamical system. A system o order n acquire n Lyapunov exponents. In a chaotic behavior, the Lyapunov exponents can measure how two dierent trajectories starting rom dierent initial conditions converge to or diverge rom each other. Under a parameter change, transition to chaotic states can be gradually or abruptly as it will be seen in next sections. Varying system parameters, one can observe three dierent cases leading to chaotic orbits, the classical doubling period biurcation, direct transition to chaotic states rom undamental harmonic or odd subharmonic solutions and a special case raising the succession o subharmonic orbits o orders 4, 5,6 and 7 ended by chaotic orbit. Many interesting eatures investigated in phase plan known as strange attractors are shown in illustrative igures below. ISSN: Issue 12, Volume 7, December 2008

5 3.1 Transition to chaos via doubling period succession The period doubling biurcation plays an important role in the occurrence o chaotic attractors. In a Duing type equation [6], the biurcation structure shows lip biurcation closed curves surrounding chaotic regions. Thus when a system parameter is varied, the responses gradually enter into chaotic regions then its exit these domains in order to turn to lower order cycles. This sequence is repeatedly reproduced as many times as the lip biurcation curves number is. Aiming to maintain an acceptable stable operating point one has to investigate in the system parameters in order to narrow the chaotic zone or make it entirely disappear. It is worth noting, that coming close to the chaotic regions, the lip biurcations become increasingly closer to each other. Crossing a small range o T values, one can meet our dierent solutions schemed in phase plane as shown in Figure 4, it is obvious to remark that when T increases rom T = 0.98 to T = the system responses undergo a doubling period cascade o biurcations. The red points plotted in all phase portraits belong to the discrete trajectory generated by the composite Poincaré map. (a1) T = (a2) T = (a3) T = (a4) T = Figure 5: phase portraits or dierent values o T and ρ = 0.58,(a1) chaotic orbit, (a2) period-12 orbit, (a3) period-6 orbit, (a4) period-3 orbit variables (, ) or dierent values o the parameter T in order to give an illustrative example o intermittent chaotic states resulting rom doubling period cascades.it is shown that the system intermittently biurcates rom an initial regular chaotic state or T = to lower subharmonic orbits or increasing values o T and then turns again to another chaotic state or T = through the same biurcation type but in the reverse path. 3.2 Direct transition rom harmonic or subharmonic orbits to chaos Figure 4: phase portraits or dierent values o T and ρ = 0.58,(a1) period-4 orbit, (a2) period-8 orbit, (a3) period-16 orbit, (a4) chaotic state Figure 5 exhibits a doubling period sequence o a period-3 orbit under variation o T. chaotic state is obtained here or decreasing values o T rom to In a doubling period cascade o biurcation, the biurcation points become closer and closer to each other, so that at a certain order n it is diicult to distinguish period-2 n orbits. In Figure 6, we plot phase portraits o the state From [11] it was stated that it is suicient or one dimensional case to have a period 3 orbit to deduce the existence o a chaotic behavior. But later, it was proved that actually many other biurcations types can lead to chaotic orbits such as the period doubling biurcation succession o either a ixed point giving rise to 2,4,8,16,...-order cycles or o k-order cycle leading to 2k,4k,8k,16k,...order cycles. From Figure 7, the graph clearly shows the abrupt transition rom subharmonic response o order 3 to chaotic state, under a parameter variation rom ρ = to ρ = Another remarkable observation is that the direct transition happens rom odd subharmonic orbit to a chaotic state, this can be illustrated by a second example o transition rom period-5 orbit to chaos or a parameter change rom ρ = ρ = see Figure 9. In Figure 11, it is shown the transition rom undamental harmonic orbit to chaotic state by varying H 2 rom to ISSN: Issue 12, Volume 7, December 2008

6 WSEAS TRANSACTIONS on CIRCUITS and SYSTEMS (a1) T = (a5) T = Figure 7: (a) period-3 orbit, (b) chaotic orbit (a2) T = (a6) T = (a1) T = (a2) T = Figure 8: (a) period-3 orbit, (b) chaotic orbit (a3) T = (a7) T = (a4) T = 5.08 (a8) T = Figure 9: (a) period-5 orbit, (b) chaotic orbit Figure 6: intermittency o chaotic states Figures 8 and 10 show the direct transition rom period-3 orbits to chaos or relatively small and large values o parameter T. Note that the obtained results are derived rom observed examples. Hence the statement that there is a direct transition rom an odd subharmonic to chaotic state is a subject or urther investigations and development. 3.3 (a1) T = (a2) T = Figure 10: (a) period-3 orbit, (b) chaotic orbit or ε2 = To our knowledge this succession o appearance o subharmonics with incremental order leading to a chaotic behavior, dierent rom doubling period cascade and rom direct transition rom odd period-n orbit to chaotic orbit, is a special interesting case. Thus urther investigation is necessary to characterize this behavior in dynamical switching systems. Transition to chaos via a special period-n orbit succession Varying the parameter ε2 over a small range o magnitudes, our investigation indicates that the traces in Figure 12 change ollowing a certain order o merging o period-n, n = 4, 5, 6, 7 orbits ended by a chaotic state ISSN: Issue 12, Volume 7, December 2008

7 (a) H 2 = (b) H 2 =0.836 Figure 11: (a) period-1 orbit, (b) chaotic orbit 4 Higher harmonic predominance 4.1 Spectral analysis o period-1 orbits Generally, in series resonance circuits periodic responses, terms o harmonics higher than the third are certain to be present but are ignored to this order. Nevertheless in certain cases namely in nonlinear systems described by a Duing type equation, the periodic responses spectra can include higher harmonics with amplitudes greater than the third and even the undamental harmonic. Spectral analysis o periodic responses in such systems led to ind the role linking between higher harmonics amplitudes and a particular old biurcation structure namely isoordinal cascade o lips[6]. This structure contains a inite set o pairs o Fold biurcation curves (lips) associated to well deined domains or which the amplitude o a rank-m higher harmonic line, m = 1, 2, 3,..., has the place r (r = 1, 2) rom an ordering based on higher harmonics amplitudes in descending order. Such domains deined or ixed points biurcation structure are called predominance domains o the rank-m harmonics r = 2, and ull predominance domains when r = 1. For a period-1 response, the corresponding spectral ordering contains a supplementary useul inormation about the response that should be considered among other characteristics such as order, stability and the nature o the period-1 orbit. Figure 12: phase portraits or dierent values o ε 2,(a1) period-4 orbit, (a2) period-5 orbit, (a3) period-6 orbit, (a4) period-7 orbit, (a5) chaotic state The transition rom period-4 orbit to period-5 orbit, then rom period-5 orbit to period-6 orbit and so on may be related to a certain type o biurcation that is not revealed here. But one can say that it is not a old or pitchork biurcation, because the order o periodic orbits which undergo such type o biurcation does not change. Furthermore, the hypothetical biurcation is not a lip a period doubling biurcation. Border collision which plays an important role in switched dynamical systems was said in recent studied that it can be connected to saddle-node biurcation. The biurcation succession identiied can have a subtle link with border collision biurcation. 4.2 Simple predominance case From igure 13, (a i ), i = 1, 6 are the phase trajectories o ixed points corresponding to dierent values o H 2, (b i ) are the corresponding -spectra. For H 2 = 0.01, the higher harmonic classiication shown in Figure 13 (b 1 ) presents the undamental harmonic in the irst rank, and the order-2 higher harmonic in the second rank so this case is regarded as a simple predominance o second higher harmonic. The higher harmonic spectral lines classiication is then {1, 2, 3, 5, 4, 8, 7,...}. For the same ixed parameters a slight variation o H 2 around H 2 = 0.01 make the spectral lines amplitudes vary smoothly so that it exists a predominance domain o order-2 higher harmonic containing at least such point. Computation o predominance domains corresponding to the dierent higher harmonic orders in a given parameter plane provide supplementary characterization o the periodic orbits identiied in these domains. Increasing the value o H 2 to H 2 = 0.24 we have also a ixed point having the phase portrait o Figure 13 (a 2 ) but seemingly with simple predominance o third higher harmonic see Figure 13 (b 2 ). The spectral lines classiication is deduced rom Figure 13(b2) as ISSN: Issue 12, Volume 7, December 2008

8 {1, 3, 5, 4, 2, 8, 7,...}. A irst glimpse to the reorganization o the spectral lines classiication under variation o H 2 rom 0.01 to 0.24 leads to point out several changes. Firstly the order-2 higher harmonic moves rom the 2nd to the 5th rank and we have the order-3 higher harmonic instead o it (i.e in 2nd rank). Secondly the obtained spectrum or H 2 = 0.24 reveals that the amplitudes o order-3 and order-5 harmonics are apparently equal. This kind o situation was called in [6] a permutation point o the third and the ith higher harmonic. Subsequently or H 2 = 0.71 the obtained period-1 orbit spectra is characterized by a ourth order higher harmonic simple predominance see Figure 13 (a 3 ) and (b 3 ). Thus, The spectral lines classiication {1, 4, 2, 5, 3, 8, 7,...} corresponds to a periodic solution which belongs to a ourth order higher harmonic predominance domain and it is close to a permutation point o the second and the ith higher harmonics being in the third rank o the classiication. The studied switched circuit, under a gradual increasing o H 2 values, exhibits an intermittent appearance o order m higher harmonics simple predominance: m=2,3,4.., thereore an isoordinal cascade o lips may exist in the neighborhood o the H 2 values range given above. The identiication o such biurcation structure in a given parameter plane is let to urther researches. Accordingly to their power spectra, the remaining ixed points o Figure 13 ((a 4 ),(a 5 ),(a 6 )) obtained or H 2 = 5.185,H 2 = 8.76 and H 2 = are associated to ull predominance o higher harmonics o orders 2,3 and 4 respectively. For H 2 = 5.185, the corresponding spectral lines classiication is {2, 3, 1, 4, 5, 6, 11,...}, see Figure 13(b4); it is obvious to remark that the undamental harmonic occupies the third rank and the second order higher harmonic has the greatest amplitude in the called spectral lines classiication. For H 2 = 8.76, the spectral lines classiication o a period -1 solution is {3, 4, 1, 2, 6, 5, 10,...},see Figure 13(b5) and the undamental harmonic is similarly in the third rank but the 2nd order higher harmonic moves to the ourth rank o the decreasing amplitudes classiication. A particular case in which the undamental harmonic moves to the ourth rank or H 2 = and the classiication becomes {4, 2, 3, 1, 6, 4, 10, 7,...} see Figure 13(b6). For such values range o H 2 or in other words or large amplitudes o the second DC generator E 2, the higher harmonics play an important role regarding to their increasing amplitudes and become more predominant than the low order ones in the Fourier spectrum o period-1 solutions. The trajectories shown in phase plane present as more undulations as H 2 increases. (a1) H 2 = 0.01 (a2) H 2 = 0.24 (a3) H 2 = 0.71 A A A A (b1) power spectrum o (b2) power spectrum o (b3) power spectrum o 4.3 Full predominance case (a4) H 2 = (a5) H 2 = 8.76 A A (b4) power spectrum o (b5) power spectrum o (a6) H 2 = (b6) power spectrum o Figure 13: Higher harmonic predominance ISSN: Issue 12, Volume 7, December 2008

9 This raises the ollowing question: do the higher harmonic predominance change under parameter variation shown in the ixed points spectra is connected to an isoordinal cascade o old biurcation similarly to the case o Duing type equation as in [6]? To settle this issue urther investigation to characterize certain biurcation structure in switching dynamical systems through higher harmonic predominance is needed. 5 Chaotic states rom dierent transition modes 5.1 Diagram o a reverse bubble The doubling period biurcation succession leads generally to a chaotic state it is also called Myrberg cascade o biurcations. Two illustrative patterns o such type o biurcation were observed in Chua s circuit namely diagram o a bubble and diagram o a reverse bubble [7]. The reverse bubble reveals the unavoidable reversals o doubling period which is a typical phenomenon o nonlinear circuits. Increasing the values o H 2 the Alpazur circuit responses cross the regions D 1, D 2, D 3, D 4, D 5 displayed in igure 17. Each region is associated to harmonic,subharmonic or chaotic behavior. Figure 14: (a) chaotic orbit, (b)period-2 orbit D 5 D 4 D 3 D 2 D 1 H Figure 17: reverse bubble amid two direct transition modes Figure 15: (a) chaotic orbit, (b)period-2 orbit Figure 16: (a) chaotic orbit, (b)period-1 orbit 5.2 Transition modes between the ive regions The region D 1 corresponds to period 1 orbit, changing the value o H 2 rom to 0.836, the responses suddenly go into chaotic region D 2. The undamental harmonic orbit and the chaotic orbit obtained rom transition rom D 1 to D 2 or vice versa are given in igure 11. The chaotic domain D 2 is located in parameter interval H 2 [0.836; 1.55]. For H 2 = 1.58 a transition rom chaotic region D 2 to low order orbits region D 3 happens resulting in a period 2 orbit see igure 14. Varying H 2 rom H 2 = 1.7 to H 2 = 1.8, the responses go out the region D 3 and turn again to a chaotic region D 4, the transition orbits are given in igure 15. Then, or H 2 = 2.95 the responses exit the region D 4 and enter abruptly the region D 5 giving rise to a period-1 orbit see igure 16.The chaotic domain D 4 is enclosed in parameter interval H 2 [1.8; 2.95]. ISSN: Issue 12, Volume 7, December 2008

10 5.3 Miscellaneous chaotic regions It is worth noting that the chaotic regions D 2 and D 4 do not outcome rom only one transition mode. Either transition rom undamental harmonic orbit or rom doubling period succession can lead to a chaotic state inside such regions. In a parameter plane a closed lip biurcation curve includes period doubling sequences associated with a corresponding reverse period doubling cascade,chaotic regions generally appear amid these two cascades. The chaotic states belonging to such regions which purely result rom doubling period succession are dissimilar rom those o the regions D 2 and D 4. 6 Conclusion In this paper, we have considered a modiied Alpazur oscillator and through numerical investigation a variety o observations over time, Poincaré maps, phase trajectories, and power spectrum is established. Three main remarks are deduced rom this work, even though rigorous proo is needed. The irst point deals with biurcations leading to chaotic states, three dierent ways are described: doubling period cascade o biurcation, direct transition rom undamental harmonic or odd subharmonic orbits to chaos and a special biurcation succession ollowing a certain order ( ) o subharmonic responses merging ended by a chaotic orbit. The second point is the higher harmonic behavior under a parameter variation which encourages to prospect the existence o isoordinal cascade o lips in a proper parameter plane. The third point concerns the existence o chaotic regions resulting rom either doubling period cascade or direct transition rom a undamental harmonic orbit. The results reported in this paper pave the way to urther investigations or proper understanding o the biurcation paths leading to chaotic states in switched dynamical systems. These results also call or urther developments o the theory o higher harmonic oscillations in regard to old biurcation cascade (or isoordinal cascade o lips). Reerences: [1] D. Dai, C. K. Tse, and X. Ma Symbolic Analysis o Switching Systems: Application to Biurcation Analysis o DC/DC Switching Converters, IEEE trans. circuits and systems, vol.52,no.8,august [2] H. H. C. Iu and C. K. Tse Biurcation Behavior in Parallel-Connected Buck Converters, IEEE trans. circuits and systems,o solid-state circuits, vol.48,no.2,februry [3] S. Jalali, I. Dobson, R. H. Lasseter and G. Venkataramanan Switching Time Biurcations in a Thyristor Controlled Reactor, IEEE Transactions on Circuits and Systems-I: Fundamental Theory and Applications, Vol. 43,NO.3,March 1996 [4] J.D Jeng, Y. Kang and Y.P. Chang An Alternative Poincar Section or Steady-State Responses and Biurcations o a Duing-Van der Pol Oscillator, WSEAS Transactions on systems Issue 6, Volume 7, June [5] H. Kawakami and R. Lozi Switched dynamical systems-dynamical o a class o circuits with switch. Proceedings o the RIMS conerence on structure and biurcations o dynamical systems, World scientiic: Singapore;1992, [6] H. Khammari,C. Mira and J.P. Carcassès Behavior o harmonics generated by a Duing type equation with a nonlinear damping. Part I. International Journal o Biurcation and Chaos, Vol. 15, No. 10, ,2005. [7] L. Kocarev, K.S. Halle, K. Eckert and L.O. Chua Experimental observation o antimonotonicity in Chua s circuit, Chua s Circuit: a paradigm or chaos,world Scientiic Publishing [8] T. Kousaka,Y. Umakoshi, and H.Kawakami Biurcation o nonlinear circuits with periodically operating switch, Electronic and communications in Japan, Prt 3,vol.84,No.1,2001. [9] T. Kousaka, T. Ueta, Y. Ma and H. Kawakami Biurcation analysis o a piecewise smooth system with non-linear characteristics, International journal o circuit theory and applications, 33: , [10] T. Kousaka, T. Ueta, Y. Ma and H. Kawakami Control o chaos in a piecewise smooth nonlinear system, Chaos, Solutions and Fractals 27 (2006) [11] T. Y. Li and J.A Yorke Period Three Implies Chaos, The American Mathematical Monthly, Vol. 82, No. 10 (December, 1975), pp [12] Y. Ma, H. Kawakami, and C. K. Tse Biurcation Analysis o Switched Dynamical Systems with Periodically Moving Borders,IEEE trans. circuits and systems, vol.51,no.6,june [13] Y. Ma, H. Kawakami,C. K. Tse and T. Kousaka. General Consideration For Modeling And Biurcation Analysis o Switched dynamical systems,international Journal o Biurcation and Chaos, Vol. 16, No. 3, ,2006. [14] Y. Ma, M. Agarwal and S. Banerjee Border Collision Biurcation in a Sot Impact System, Physics Letters A354(2006) ISSN: Issue 12, Volume 7, December 2008

11 [15] Y. Ma, T. Kousaka and H. Kawakami Connecting Border Collision witb Saddle-Node Biurcation in Switched Dynamical Systems, IEEE Transactions on Circuits and Systems-II: Express Bries, ,2005 [16] G. M. Maggio, M. di Bernardo and M. P. Kennedy Nonsmooth Biurcation in a Piecewise-Linear Model o the Colpitts Oscillator, IEEE Transactions on Circuits and Systems-I: Fundamental Theory and Applications, Vol. 47,NO.8,August 2000 [17] K. Milicevic, I. Rutnik and I. Lukacevic Impact o Voltage Source and Initial Conditions on the Initiation o Ferroresonance,WSEAS Transactions on circuits and systems, Issue 8, Volume 7, August [18] M.S. Papadopoulou, I.M.Kyprianidis, and I.N. Stouboulos Complex Chaotic Dynamics o the Double-Bell Attractor, WSEAS Transactions on circuits and systems Issue 1, Volume 7, January [19] T. Saito, T. Kabe, Y. Ishikawa, Y. Matsuoka and H. Torikai Piecewise Constant Switched Dynamical Systems in Power Electronics, International Journal o Biurcation and Chaos, Vol. 17, No. 10, ,2007. [20] C. K. Tse, Y. M. Lai and H. H. C. Iu Hop Biurcation and Chaos in a Free-Running Current-Controlled Cuk Switching Regulator, IEEE Transactions on Circuits and Systems- I: Fundamental Theory and Applications, Vol. 47,No.4,April 2000 [21] Y. Zhou, J.N. Chen, H.H.C. Iu and C.K. Tse Complex Intermittency in Switching Converters,International Journal o Biurcation and Chaos, Vol. 18, No. 1, ,2008. ISSN: Issue 12, Volume 7, December 2008