An Alternative Poincaré Section or Steady-State Responses and Biurcations o a Duing-Van der Pol Oscillator Jang-Der Jeng, Yuan Kang *, Yeon-Pun Chang Department o Mechanical Engineering, National United University, Taiwan, R.O.C. Department o Mechanical Engineering, Chung Yuan Christian University, Taiwan, R.O.C. *Corresponding author: yk@cycu.edu.tw Abstract: - To apply eective methods to analyze and distinguish all kinds o response patterns is an important issue o nonlinear dynamics. This paper contributes to develop an alternative Poincaré section method to analyze and identiy the whirl responses o a nonlinear oscillator. This integration or analyzing high-order harmonic and chaotic motions is used to integrate the distance between state trajectory and the origin in the phase plane during a speciic period. This response integration process is based on the act that the integration value would be constant i the integration interval is equal to the response period. It provides a quantitative characterization o system responses as the role o the traditional stroboscopic technique (Poincaré section method) to observe biurcations and chaos o the nonlinear oscillators. However, due to the signal response contamination o system, the section points on Poincaré maps might be too close to distinguish, especially or a high-order subharmonic vibration. Combining the capability o precisely identiying period and constructing biurcation diagrams, the advantages o the proposed method are shown by the simulations o a Duing-Van der Pol oscillator. The simulation results show that the high-order subharmonic and chaotic responses and their biurcations can be eectively observed. Key-Words: - Poincaré section method, Chaotic motion, Response integration, Biurcation, Duing-Van der Pol oscillator Introduction In recent twenty years, the problem o identiying and controlling chaos has become a hot topic in the ield o physics, mechanics, engineering and other applied ields [-7]. Because o the sensitivity o chaos to the system parameters and the initial value, identiying and controlling chaos has become important or nonlinear system. A strongly nonlinear system oten exhibits a very complicated phenomenon including not only the periodic motion but also the quasi-periodic and chaotic motions. The identiication o the aorementioned nonlinear behaviors is usually accomplished through the observation on biurcation diagrams in terms o Poincaré section points with assistance o phase plane trajectories. Typical examples o the driven nonlinear oscillators are the Duing oscillator and Van der Pol oscillator. The eorts or searching a better numerical method or nonlinear system have never ceased. The orced Duing and Van der Pol oscillators, which are known to describe many important oscillating phenomena in nonlinear engineering systems, has been investigated extensively in the past decade [8-5]. Much work has already been done to investigate the characteristics o Duing or Van der Pol oscillator [8,, 5]. Stability and Biurcation studies have been carried out, or example, by Duing oscillator [, 5] and Van der Pol oscillator [8, ]. Numerical investigations have provided examples o period doubling biurcations, and chaotic attractors [, 3,, 5]. In applying the incremental harmonic balance method, with the ocus on a Duing oscillator with zero linear stiness, Leung and Fung [] discussed the variation in the biurcation points according to the system s parametric changes and the boundaries o steady solutions o the system in 989. In 987, Parlitz and Lauterborn [8] used winding numbers to describe the dynamic behavior o the Van der Pol oscillator within the system and went on to explain its relationship to the system parameters. In 996, Dooren and Janssen [9] had studied a Duing oscillator under static and large periodic excitation by using the shooting method combined with the Newton method. They ound that there are at least six cascades o period doubling subharmonic solutions in the amplitude o the periodic excitation ixed at speciic values rom biurcation diagram. In 996, Xu and Jiang [] also investigated the global biurcation characteristics o the orced Van der Pol oscillator. ISSN: 9-777 78 Issue 6, Volume 7, June 8
In, Kim et al. [7] experimentally investigated the dynamic stabilization in a double-well Duing oscillator. They ound that as the amplitude o driving orce increases through a threshold value, the unstable symmetric orbit was ound to become stable via a reverse supercritical pitchork biurcation by absorbing a pair o stable symmetric orbits. In, a chaotic crisis in orced Duing oscillators had been investigated by Hong and Xu [] using the generalized cell mapping digraph method analyzed nonlinear system. They examined that the chaotic attractor together with its basin o attraction is suddenly destroyed as the parameter passes through the critical value, and the chaotic saddle also undergoes an abrupt enlargement in its size. However, such a detailed model is not always the most suitable approach i the main aim is to obtain eicient predictions o overall system dynamics. The intensive studies by many researchers have revealed many characteristics o the oscillators. However, it is still a diicult task to identiy quantitatively or the high-order o harmonic responses due to the signal contamination o system response. However, an eective method or analyzing the high-order harmonic and chaotic motions in a Duing-Van der Pol system has not been suiciently investigated. For this purpose we propose a response integration method to analyze high-order harmonic and chaotic motions in the nonlinear models o periodically orced Duing- Van der Pol oscillator. This study contributes to the development o eicient integration method to analyze and identiy the whirl responses o a nonlinear system. This integration method provides a quantitative characterization o periodic motion. It is a useul tool to observe the periodic motion which generally cannot be easily distinguished rom either the orbit plot or the Poincaré map. It is shown that this method provides valuable inormation in biurcation diagram such that the parameter range leading to chaos can be easily decided and the number o distinguishable time-domain responses can be determined. Since this method can ilter combination and distinguish response pattern. Some examples are given to illustrate its eectiveness and convenience. Method o response integration algorithm The nonlinear, non-autonomous systems can be expressed in a nondimensional orm o the irstorder dierential equation as ollows: d ~ x = ( ~ x, t), () dt n where ~ x R is the n-dimensional vector o nondimensional variables ; t R is a nondimensional time. An integration algorithm denoted by P nt integration which was deined by integrating integrand unction and expressed by tc nt P ( t ) = + ( x, x& d t () nt c ) tc where the integrand unction can be presented by ( x, x& ) = x( t), x& (t) or x + x&. The integrand represents the distance between trajectories and origin or trajectories and one o the coordinate axes in phase plane. x (t) and x& (t) are time histories o nondimensional displacement and velocity, respectively. t is nondimensional time and, is an arbitrary time or nondimensional parameters, taken at the time o steady-state. It is a changing index which is helpul to distinguish the system s response period. The T is the period o single excitation or the smallesommon multiple period o multiple excitations. The integration interval nt is set to be the predicted oscillating period. Constant P T can be obtained rom time history, when response is P- motion. Thereore, when the integration interval is set at nt ( n is an integer larger than zero), ian be judged that the system periodic response is P-n motion, that is, the n-th is a subharmonic response. Moreover, i the integration interval is set as the excitation period, i.e., T, the results can be used to draw the biurcation diagram o the rotor system. The extraction period is the same as Poincaré section method. When the period o n-th order subharmonic motion is nt, the steady state response has ( x, x& ) = ( x( t + nt ), x& ( t + nt )) (3) Thus, integration P nt o ( x, x& ) dierentiating to time gives dpnt ( tc ) = ( x( tc + nt ), x& ( tc + nt )) dtc ( x( tc ), x& ( tc )) = () Thereore, PnT ( tc ) = constant (5) or Δ PnT = PnT ( tc + ) PnT ( tc ) = (6) The method to dierentiate the system responses with the response integration is based on the relations among extraction time and integration interval nt. The analytical process o response integration is as the ollowing: ISSN: 9-777 79 Issue 6, Volume 7, June 8
. First, an initial value n is given and the integration interval is set as nt where n is an integer and T is the excitation period o system. Furthermore, will be varied to assist in the determination o the response period.. The Δ PnT integration versus t are calculated, c and the tc ΔPnT curve is drawn. I the integration Δ PnT is not zero, then the index n should be changed, and the above steps are repeated. 3. When the integration Δ PnT is zero, the integration interval nt is deined as n times o excitation period; the system response is P-n periodic motion, i.e., n -order sub-harmonic vibration. I the index n is changed constantly, no ixed integration value can be obtained, and the system response might be chaotic motion. As proo o the above or periodic motion, see Eq. (5) with period nt the integral P nt is constant against the starting time o integration. Thus, the period nt o a steady state response can be identiied due to the existence o constant P nt. Moreover, the Poincaré section method requires more sampling numbers to eliminate the measurement noise. Based on the deinition described as above, the response period which determined to apply the response integration method needs only to be within the time range. The extraction time range can be an arbitrarily small number. Thereore, when limited experimental measurement data are analyzed, it is more advantageous to use the response integration method. 3 Simulation results and analysis 3. Duing-Van der Pol oscillator The representative nonlinear oscillator subjected to driving periodic orce, evolved rom mechanical system illustrated in Fig., are considered as examples herein to demonstrate the usage and eectiveness o the proposed response integration method. The nonlinear oscillator used in this paper is rom the Duing-Van der Pol equation. The oscillator can be used to model rotor subjected to nonlinear damping and stiness with external excitation. The Duing-Van der Pol oscillator is one o the mosommon types o nonlinear oscillators and has many typical nonlinear phenomena [3, 8]. The governing dierential equation o the nonlinear system can be written as 3 5 m& x + c( c c3x ) x& + kx + kx + k3x = cos( ω t) (7) where m denotes the mass o the system; c, c and c 3 denote damping coeicients respectively; k, k and k 3 denote the stiness coeicients respectively due to material nonlinearity; denotes the amplitude o the external harmonic orce; ω denotes the whirling requency. These parameters also control the nonlinearity o the oscillator. The dot denotes a dierentiation with respect to time t. It is assumed that the above system parameters are dimensionless in the Duing-Van der Pol equation. For m = and c 3 =, Eq. (7) is a amous Duing equation. For m =, c =, k = k3 =, Eq. (7) is a amous Van der Pol equation. For perorming numerical integration, the system equations are transormed into irst-order orm by introducing two variables x and x. x & = x, 3 5 x& = { c( c c3x ) x kx kx k3x m + cos( ω t + φ)} (8) It is diicult to obtain the exact solutions directly. For this reason, the numerical simulations are perormed by employing a ourth-order Runge- Kutta algorithm and a smaller integration step are chosen to be smaller than Δ t = π / to ensure a steady-state solution. x m cosω t γ (x) k(x) γ (x) k(x) γ ( x) = c( c c3x x k + ( x) = k + kx k3x Fig.. Duing-Van der Pol oscillator model or rotor excitation. 3. P T integration biurcation diagram analysis The biurcation diagram is a very eective mean to relect the motion change; the results obtained rom the system exhibiting o non-linear behaviors may be presented. As mentioned earlier, an integration x ) ISSN: 9-777 75 Issue 6, Volume 7, June 8
algorithm denoted by P nt integration which was deined by integrating integrand unction. The integrand unction can be presented by ( x, x& ) = x( t), x& (t) or x + x&. To examine the system responses o the Duing-Van der Pol oscillator compared with dierent integrand unctions have been devised. The biurcation diagrams with dierent integrand unctions are shown in Figs., and. The biurcation responses are calculated an increment Δ =.86 as the variation o the control parameter. The orcing amplitude o the periodic excitation ranges rom =. 95 to =. 5. As observed rom these igures, the results show that the period doubling biurcations and chaotic output responses in biurcation diagram exhibit the similar orms o vibration. Our simulations in Fig. show that the chaotic output responses have the larger scale in comparison with other two cases. Based on above results, in this paper we choose ( x, x& ) = x + x& as the integrand unction...5. P motion -.5 n= -. n= 5 5 5 3 P motion - n= n= 5 5 5 3 P motion - n= n= 5 5 5 3 Fig. 3 Plots o Δ P nt integration with dierent integrand unctions, ( x, x& ) = x( t), ( x, x& ) = x&, ( x, x& ) = x + x&. m =, c =.7, c =., c 3 =., k =. 7, k = 3. 8, k 3 =., ω =., =. 5. ω Fig. Biurcation diagrams with dierent integrand unctions, ( x, x& ) = x( t), ( x, x& ) = x&, ( x, x& ) = x + x& ; m =, c =. 7, c =., c 3 =., k =. 7, k = 3. 8, k 3 =., ω =.. Fig. Biurcation diagrams with whirling requency, ω as the control parameter, the response integration method; Poincaré section method; with ω ISSN: 9-777 75 Issue 6, Volume 7, June 8
c =.5, c =., c 3 =. 8, k =. 5, k =. 65, k 3 =.6, ω =. 6. Fig. 5 Biurcation diagrams with orcing amplitude, as the control parameter, the response integration method; Poincaré section method. The plots o Δ P nt integration with dierent integrand unctions as shown in Figs. 3, and are also the same results. In order to validate the analysis results rom the Response integration method, comparisons are made with Poincaré section method. Figs. and are the biurcation diagrams drawn with both the response integration method and the Poincaré section method, using the whirling requency as the control parameters. The whirling requency ω ranges rom. to.3 with a step o.5. In these calculations, we choose the parameters as m =, c =. 35, c =., c 3 =., k =. 5, k =. 65, k 3 =.6, = 7. 8. Similarly, the present method is evaluated by a comparison with Poincaré section method to show the excellent agreement. Ian be seen that the motion is synchronous with period-one rom. to., only one point is correspondingly shown in the biurcation diagram or every whirling requency. As the whirling requency continues to increase, the motion suddenly goes into chaotic region rom about ω =.. When ω increases urther, at ω =. 6, the response goes out o chaos and turns to period-three motion. Then, with urther variations o ω, the response again goes to chaos at ω =.98. When ω =. 7, the response exits chaos and turns into period doubling biurcations. Comparisons o the response biurcations drawn with two methods are shown in Figs. 5 and 5. The biurcation responses were calculated as the variation o the control parameter. The orcing amplitude o the periodic excitation ranges rom =. to =. 6. The response integration method is evaluated by a comparison with the Poincaré section method to show the excellent agreement with the same set o parameters as in Figs. 5 and 5. Ian be seen that the biurcations o the Poincaré section and the response integration methods almost show the same topology o the structure or the same system parameters. The biurcation diagram shows that period doubling biurcations initially occurs. For the values o ranging rom =. to =. 39, the period doubling biurcations are ound to proceed in these igures. The chaos occurs ater the period-our, period-eight, suddenly and then, biurcates to period-sixteen. Herein, the period doubling biurcation plays an important role in the occurrence o chaotic attractors. As the continues to increase; the motion gradually enters into the region o chaos which remains rom.39 to.6. Nevertheless, it is sometimes diicult to identiy the period doubling biurcation o system using the Poincaré section method to strobe Poincaré section points in Fig. 5. Conversely, when the response integration method is used, the period doubling biurcation characteristics can clearly be observed in the biurcation diagram shown in Fig. 5. Ian be seen that the system exhibits very rich orms o periodic and chaotic vibrations. 3.3 Comparisons with response integration and the Poincaré section methods The Poincaré section is a stroboscopic picture o a motion and consists o the time series at a constant interval o T, with T being the period o excitation. The corresponding Poincaré map is a combination o those return points and these discrete points are oten called an attractor. Examination o the distribution o return points on the Poincaré map can reveal the response o motion. In case o a periodic motion, the n discrete points on the Poincaré map indicate that the period o motion is the P-n. For a chaotic motion, the return points orm a geometrically ractal structure. Figs. 6- show the plots o Δ PnT integration and the Poincaré map with various orcing amplitudes. In these calculations, we choose the parameters as m =, c =. 5, c =., c 3 =. 8, k =. 5, k =.65, k 3 =. 6, ω =. 6. In the analysis o the Poincaré section method, the extraction time ISSN: 9-777 75 Issue 6, Volume 7, June 8
equals to the external excitation period o T = π /ω. When =., Fig. 6 shows Δ PnT integration value versus with integration duration. The Δ PnT integration value appears zero as continuously varying at n =. This igure clearly illustrates the system response to be a period-our motion. At this time, the motion is a periodic motion and there is our points in the Poincaré map as shown in Fig. 6. At =. 35, due to a ew Poincaré section points being too close, that is, the contamination o the signal response, thus it is diicult to identiy the order o the subharmonic motion as shown in Fig. 7. But rom Fig. 7, the plot o Δ PnT integration, we can dierentiate the motion clearly to be a subharmonic motion with period-eight. Similarly, at =. 38 it is also diicult to identiy periodic responses; this is due to some Poincaré section points being too close. Details are as shown in Fig. 8. But what we can easily observe is that the motion is a sub-harmonic vibration with period-sixteen. As are increased to.5 and.55, it is ound that the Δ PnT integration values never keep zero as continuously varying even with the integration interval set very large as shown in Figs. 9 and. It indicates that responses are not periodic motions. The results o the Poincaré section method also show the responses to be chaotic motions as shown in Figs. 9 and. Ian be seen that a period doubling route to chaos or a selection o when another system parameters are ixed. The Duing-Van der Pol system also represents the complex dynamic characters at dierent orcing amplitudes obviously.. P motion -.5 - n= n= 5 5 5 3 -. -.5 -. -.5-3..6.8....6.8 Fig. 6 Plots o Δ P integration (let) and Poincaré nt map (right) with =... P-8 motion -.5 - n=6 n=7 n=8 5 5 5 3 -. -.5 -. -.5-3..6.8....6.8 Fig. 7 Plots o Δ P integration (let) and Poincaré nt map (right) with =. 35. - P-6 motion n= n=5 n=6-6 5 5 5 3 - - -3.8....6.8 Fig. 8 Plots o Δ P integration (let) and Poincaré nt map (right) with =. 38. 8 Chaotic motion -8 n= n= - n= 5 5 5 3 - - -3.8....6.8 Fig. 9 Plots o Δ P integration (let) and Poincaré nt map (right) with =. 5. 8-8 - Chaotic motion n= n= 5 5 5 3 - - -3.8....6.8 Fig. Plots o Δ P integration (let) and Poincaré nt map (right) with =. 55. 3. The eect o damping and stiness parameters analysis Damping is also one o the main actors aecting dynamic characteristics o nonlinear systems. Biurcation diagrams shown in Fig. was plotted in the orm o P T integration to be as a unction o the with the variation o damping parameter c. In the range 3. < < 7., the biurcations o the system are illustrated or damping parameters c o.55,.6,.7 and.75. As obvious rom biurcation responses shown in Fig., increasing c can reduce the response amplitude and narrows the chaotic zone. It is urther observed that as the value o c is increased rom.55 to.6, the chaotic zone, where is varied in the range 5.6 < < 6., gradually disappear. For the case o c =.7, the chaotic zone in the hal-let interval, i.e.,.3 < <. 8 was also narrowed in Fig.. For the case o c =. 75 in particular, the chaotic zones entirely disappear in Fig. (d). Ian be seen that the system represents the complex dynamic characters at dierent damping coeicients ISSN: 9-777 753 Issue 6, Volume 7, June 8
obviously. The chaotic zones gradually disappear when the damping parameter is increased. This eature indicates that the damping eect is relatively sensitive to the dynamic responses o a system. The eect o changing stiness parameter is also investigated in this paper. c =.55 occur in the ranges 3.66 < <. 7,.55 < < 5. 76, 6.3 < < 6.38 and 6.6 < < 7.. As obvious rom biurcation responses shown in Fig., increasing k also reduces the response amplitude and narrows the chaotic zone. It is observed that as the value o k is increased rom.8 to.36, the chaotic zones gradually decrease. When k increases urther, at k =.5 as shown in Fig., the chaotic regions are ound to turn into two rom three. Then, with urther variations o k, Fig. (d) shows that the chaotic zones disappear and the level o vibration is lower when the stiness is larger (at k =. 75 ). k =.8 c =.6 k =.36 c =.7 k =.5 (d) c =.75 Fig. Eect o damping coeicient, c on the system responses o biurcation diagrams using the response integration method with c =., c 3 =., k =. 8, k =. 65, k 3 =. 6, ω =.. Biurcation diagrams shown in Fig. was plotted in the orm o P T integration to be as a unction o the with various stiness parameters k. For a lower stiness value o k =. 8, it is clearly seen that there are several chaotic regions (d) k =.75 Fig. Eect o stiness coeicient, k on the system responses o biurcation diagrams using the response integration method with c =. 5, c =., c 3 =., k =. 65, k 3 =. 6, ω =.. ISSN: 9-777 75 Issue 6, Volume 7, June 8
regular and periodic as the phase portrait shown in Fig.. There is our points in the Poincaré map as shown in Fig.. Thereore, the system response is a period-our motion. The result is the same as the plot o Δ P n T integration as shown in Fig.. 8 Chaotic motion n= n= -8 5 5 5 3 Fig. 3 Phase portrait, Poincaré map and integration plot or c =. 55. Δ P nt 8-8 n= n= Chaotic motion 5 5 5 3 Fig. 5 Phase portrait, Poincaré map and integration plot or k =. 8. Δ P nt P motion - n= n= - 5 5 5 3 Fig. Phase portrait, Poincaré map and integration plot or c =. 7. Δ P nt Figs. 3- show the phase portrait, the Poincaré map and the plots o Δ P n T integration with various damping parameters, c =. 55 and c =. 7. In these calculations, we choose the other parameters as the same with Fig.. The chaotic motion. was plotted or c =. 55 and the periodic motion or c =.7. When c =. 55, the phase portrait also shows irregular motion as shown in Fig. 3. There are strange attractors representing chaotic motion in the Poincaré map as shown in Fig. 3. Furthermore, the Δ P n T integration values never keep zero as continuously varying even with the integration interval set very large ( n = 3 ) as shown in Fig. 3. It indicates that responses are not periodic motions. When c =. 7, the motion is Figs. 5-6 show the phase portrait, the Poincaré map and the plots o Δ P n T integration with various damping parameters, k =. 8 and k =. 36. In these calculations, we choose the other parameters as the same with Fig.. When k =. 8, the phase portrait also shows irregular motion as shown in Fig. 5. The chaotic response keeps active. As observed rom Figs. 5, the results o the Poincaré section method also show the responses to be chaotic motion. At this time, it is ound that the Δ P nt integration values never keep zero as continuously varying even with the integration interval set very large ( n = ) as shown in Fig. 5. When k =. 36, Ian be seen rom the Figs. 6 and 6 that the system vibration is periodthree motion. The result is the same as the plot o Δ P nt integration as shown in Fig. 6. This eature indicates that the stiness eect is relatively sensitive to the dynamic responses o a nonlinear system. From above results, it is concluded that the stiness and damping o the system can eectively suppress chaotic vibration and reduce vibration amplitude. ISSN: 9-777 755 Issue 6, Volume 7, June 8
3 P-3 motion - n= - n= -3 5 5 5 3 Fig. 6 Phase portrait, Poincaré map and integration plot or k =. 36. Δ P nt oscillator in some cases. Applying this method, the eects o the change in the stiness and the damping coeicients on the vibration eatures are investigated. With variation o system parameters, such as damping, stiness, whirling speed and external orcing amplitude, chaotic response can be observed, along with other complex dynamics such as period- doubling biurcation in the response. The numerical simulated results show that the response integration technique will be available to observe biurcations and chaos o the nonlinear oscillator eectively and can precisely identiy high-order subharmonic motion. By applying our proposed method and combining with the GA method, the control and synchronization o chaos in nonlinear dynamical systems are also interesting topics or uture studies. Conclusions This study proposes the response integration algorithm to analyze the vibration responses o a Duing-Van der Pol oscillator with periodic excitation. This method numerically integrates the distance between state trajectory and the origin in the phase plane during a speciic period. It provides a quantitative characterization o system responses and can replace the role o the traditional stroboscopic technique (Poincaré section method) to observe biurcations and chaos o the nonlinear oscillators. Utilizing the P nt - integral quantity analytical method, the periodic response o the system can be clearly and simply drawn. It is a very useul tool to observe vibration responses which generally cannot be easily distinguished rom the time history, the phase portrait and the Poincaré map. Due to the signal response contamination o system, thus it is diicult to identiy the high-order responses o the subharmonic motion because o the sampling points on Poincaré map too near each other. Even the system responses will be made misjudgments. The response integration method is used to analyze nonlinear system. When P nt (or Δ P nt = ) is a ixed constant, the system response is periodic motion according to the deinition o the response integration method. Such characteristics can avoid the disturbances described above. Thereore, it is more advantageous to use this method or analysis regarding limited measuring data or numerical simulation. In this paper we have inspected the Δ PnT integration plot and P T integration biurcation diagram to identiy responses o a nonlinear Acknowledgements This work had supported by the National Science Council o R.O.C. with grant No. NSC-9--E- 33-6 and R&D Center or Membrane Technology in CYCU due to the center-oexcellence program rom the Ministry o Education o R.O.C. Reerences: [] Thompson, J. M. T., Stewart H. B., Nonlinear dynamics and chaos, Chichester: Wiley; 986. [] Leung, A. Y. T., and Fung, T. C., Construction o Chaotic Regions, Journal o Sound and Vibration, Vol. 86, 989, pp. 555. [3] Jackson, E. A., Perspectives o Nonlinear Dynamics, Cambridge University Press, New York, 99. [] Holmes, J., P., Nonlinear Oscillation and Biurcation o Vector Fields, Springer, New York, 993. [5] Lakshmanan, M., and Murali, K., Chaos in Nonlinear Oscillators: Controlling and Synchronization, World Scientiic, Singapore, 996. [6] Lei, Y. M., and Xu, W., Chaos control by harmonic excitation with proper random phase, Chaos, Solitons & Fractals, Vol.,, pp. 75-8. [7] Kyprianidis, I. M., Volos, Ch. K. and Stouboulos, I. N., Suppression o chaos by linear resistive coupling, WSEAS Trans. on Circuits and Systems, Vol., 5, pp. 57-53. [8] Parlitz, U., and Lauterborn, W., Period- Doubling Cascades and Devil s Staircases o ISSN: 9-777 756 Issue 6, Volume 7, June 8
the Driven Van der Pol Oscillator, Physical Review A, Vol. 36, 987, pp. 8-3. [9] Dooren, R. V., and Janssen, H., A Continuation Algorithm or Discovering New Chaotic Motion in Forced Duing Systems, Journal o Computational and Applied Mathematics, Vol. 66, 996, pp. 57-5. [] Xu, J. X., and Jiang, J., The global biurcation characteristics o the orced Van der Pol oscillator, Chaos, Solitons & Fractals, Vol. 7, 996, pp. 3-9. [] Aguirre, J., and Sanjuán, M. A. F., Unpredictable behavior in the Duing oscillator: Wada basins, Physica D, Vol. 7,, pp. 5. [] Acho L., Rolon, J. and Benitez, S. A., Chaotic oscillator using the Van der Pol dynamic immersed into a jerk system. WSEAS Trans. on Circuits and Systems, Vol. 3,, pp. 98 99. [3] Qun He, Wei Xu, Haiwu Rong, Tong Fang, Stochastic biurcation in Duing-Van der Pol oscillators, Physica A, Vol. 338,, pp. 39-33. [] Hong, L. and Xu, J., A Chaotic Crisis between Chaotic Saddle and Attractor in Forced Duing Oscillators, Communication in Nonlinear Science and Numerical Simulation, Vol. 9,, pp. 33-39. [5] Jing, L., and Wang, R., Complex dynamics in Duing system with two external orcings, Chaos, Solitons and Fractals, Vol. 3, 5, pp. 399. [6] Volos, Ch. K., Kyprianidis, I. M. and Stouboulos,, I. N., Synchronization o Two Chaotic Duing-type Electrical Oscillators, Proceeding th WSEAS International Conerence on Circuits and Systems, 6, pp.79-8. [7] Kim, Y., Lee, S. Y., and Kim, S. Y., Experimentally observation o dynamic stabilization in a double-well Duing oscillator, Physics Letters A, Vol. 75,, pp.5-59. [8] Kakmeni, F. M. M., Bowong, S., Tchawoua, C., and Kaptouom, E., Strange attractors and chaos control in a Duing-Van der Pol oscillator with two external periodic orces, Journal o Sound and Vibration, Vol. 77,, pp. 783-799. ISSN: 9-777 757 Issue 6, Volume 7, June 8