Targeting Periodic Solutions of Chaotic Systems

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1 ISSN (print), (online) International Journal of Nonlinear Science Vol.26(218) No.1,pp Targeting Periodic Solutions of Chaotic Sstems Vaibhav Varshne a, Pooja Rani sharma b, Manish Dev Shrimali b Bibhu Biswal c, Awadhesh Prasad a a Department of Phsics and Astrophsics, Delhi Universit, Delhi 11 7, India b Department of Phsics, Central Universit of Rajasthan, Ajmer , Rajasthan, India c Cluster Innovation Center, Delhi Universit, Delhi 11 7, India (Received 31 Januar 217, accepted 17 April 218) Abstract: A linear augmentation method for stabilizing chaotic sstems around unstable fied points [Sharma et al, Phs. Rev. E., 83, 6721 (211)] is etended to stabilize chaotic oscillators in a desired periodic state around a chosen target point. The chaotic sstem attains a periodic orbit with desired frequenc around a chosen center of rotation that can be controlled b variation of the coupling parameters of the augmented sstems. The dnamical properties of the controlled sstems, verified using Lapunov eponent, show transition from chaotic to periodic state through a boundar crisis route, and near the transition points the sstems posses mutistabilit. Kewords: Linear augmentation; Control; Boundar crisis; Multi stabilit 1 Introduction The framework of coupled nonlinear oscillators has been ver useful to model several tpes of phsical, biological and chemical phenomena [1, 2]. It offers a variet of dnamical behaviors such as limit ccle, multistabilit and chaos. Although chaos is a ver interesting nonlinear effect and has been detected in a large number of different dnamical sstems [3, 4], in practice, however, it is often not desirable [5]. It restricts the operating range of man engineering sstems and renders their long-term prediction impossible. In man real world situations, a regular output, either in the form of a stead state or a periodic orbit, is required in spite of the underling chaotic dnamics of the sstem [6 8]. Hence, the scope of controlling chaos in sstems with comple dnamics is of great interest [9, 1] and during the past few decades several techniques have been proposed. The seminal work of Ott, Grebogi and Yorke [11] known as the OGY method stabilizes the chaotic sstem around one of the man unstable periodic orbits embedded in the chaotic attractor through small time dependent perturbations in an accessible sstem parameter. The OGY method has been successfull applied to man phsical, chemical and engineering sstems [12, 13], but was not found to be useful in sstems with fast dnamics [11]. Pragas proposed a time dela feedback control method [14], a proven and powerful method for stabilizing periodic orbit and controlling chaotic motion. This method continuousl injects control perturbations based on the difference between the current state of the sstem and its period one state of the target orbit in the past. Thus, the intensit of perturbations practicall vanishes once the sstem evolves close to the desired orbit. Several modifications and etensions of this method have also been developed to speed up the stabilizing process while dealing with high periodic orbits [15 18]. Liz has proposed a wa to control chaotic behavior with proportional feedback [19]. Besides the feedback control methods, there also eist open-loop control sstems [2] in which the applied perturbation is independent of the state of the sstem, i.e., there is no feedback loop. In 199, a simple and effective method of chaos control was proposed b Sinha et al [21] that utilizes an error signal proportional to the difference between the goal output and actual output of the sstem, and control is achieved b the error signal reducing to zero adaptivel [22]. A ver different approach to control chaos is also the application of suitable weak periodic forcing that has been studied in various sstems, both theoreticall [23 27] and eperimentall [28]. The influence of periodic forcing on a dripping faucet has been investigated eperimentall b Shoji [29] showing that a Corresponding author. address: awadhesh@phsics.du.ac.in Copright c World Academic Press, World Academic Union IJNS /18

2 14 International Journal of NonlinearScience,Vol.26(218),No.1,pp periodic eternal force can induce a transition from periodic to chaotic motion. Although the aforementioned schemes are well applicable and verified theoreticall as well as eperimentall for several sstems, man sstems are still not amenable to such control because the require the access of a suitable internal parameter of the sstem. Another issue with these schemes is the enhancement of the sstem s dimension. Recentl, linearl augmented sstem without periodic forcing have been studied etensivel and used to stabilize the unstable fied point of the sstem [3], to control bistabilit [31], for controlling dnamical behavior of driven response sstem [32], for controlling multistabilit and dnamics of hidden attractors [33, 34] and for stabilizing the stationar state solutions [35]. Motivated b the fact that changing the internal parameter of a sstem is not alwas feasible, We propose a new control scheme b augmenting the underling sstem with a periodic forcing. We target the sstem to a periodic orbit with desired frequenc around a targeted point in the phase space. We show that this scheme steers the sstem to a desired periodic or stead state. In contrast to the earlier schemes of controlling chaos, this method leaves the sstem s internal parameters unchanged. In this work, apart from the main results of targeting the sstem to a periodic orbit with desired frequenc around a targeted point in the phase space, we also discuss a transition from chaos to periodic behavior through the boundar crises route which is the global bifurcation. We also discuss the presence of bistabilit [36, 37] at the transition point. The paper is organized as follows. The details of the coupling scheme are presented in Sec. 2, followed b the results and discussions in Sec.3, and the summar in Sec Coupling scheme Consider a sstem of general nonlinear oscillators Ẋ = F (X), where X is the m dimensional vector of dnamical variables and F (X) is the vector field. In the previous work [3], the unstable fied point of such kind of nonlinear sstems was stabilized using a specific tpe of coupling with the m dimensional linear dnamical variable (U) as Ẋ = F (X) + ϵu U = ku ϵ(x B), (1) with U = [u,,,...] T and B = [b,,,...] T where T indicates the transpose. Here ϵ is the coupling strength between the oscillator and the linear sstems, k is the deca parameter of the linear sstem, and B is a control parameter of the augmented sstem. In the absence of coupling, the linear sstem approaches zero for finite positive k. Therefore, without coupling, the sstem will maintain its natural dnamics. The oscillator sstem can be stabilized to a desired stead state with the proper choice of parameter B [3, 38, 39]. The above choice of U and B stabilizes the sstem onl around an eisting unstable fied point as well as around newl created fied points. Stabilizing the sstem around a periodic orbit, however, would require at least two linear augmented sstems because the target limit ccle would be two dimensional in the phase space. In this work our aim is to stabilize a chaotic sstem, simultaneousl, in a periodic orbit (instead of fied points) around a specific target point and of a desired frequenc. In order to achieve this, we modif the linear augmentation coupling in Eq.1 as Ẋ = F (X) + ϵu U = K U ϵ(x Γ), (2) with U = [u 1, u 2,..., u r,,..., ] T, K = [k 1, k 2,..., k r,,..., ] T, Γ = [b 1 +α cos ωt, b 2 +α cos ωt,..., b r +α cos ωt,..., ] T, and is the Hadamard or entrwise product. In this paper we present successful control of Lorenz sstem and Rössler sstem for the simple case r = 2, k 1 = k 2 = k and b 1 = b 2 = b. Here Γ i = b i α cos ωt where α cos ωt is the eternal periodic ecitation and α is its amplitude. Note that for α =, i.e. with no eternal periodic forcing, Eq. (1) is recovered. In this work, with fied k =.1 [3], we change the parameters ϵ, b, α and ω to achieve the targeted control. The targeted control can also achieved for other values of b 1, b 2, k 1, k 2. But for simplicit we kept them equal. All the simulations in this work are done using 4th order Runge Kutta method with step size of.1. IJNS for contribution: editor@nonlinearscience.org.uk

3 V. Varshne et al.: Targeting Periodic Solutions of Chaotic Sstems 15 3 Results and discussion 3.1 Lorenz sstem Control without periodic forcing We first eamine the chaotic Lorenz sstem coupled to two linear augmentation sstems and without periodic forcing (α = ), and are coupled with u 1 and u 2 respectivel as, ẋ = σ( ) + ϵu 1 ẏ = r z + ϵu 2 ż = µz u 1 = k 1 u 1 ϵ( b 1 ) u 2 = k 2 u 2 ϵ( b 2 ). (3) For σ=1, r=28, µ=8/3 the uncoupled sstem (ϵ = ) has three unstable fied points and the sstem dnamics is chaotic. We have used two linear sstems, u 1 and u 2, instead of one linear sstem as considered in fied point targeting [3]. The phase space diagram in parameters space ϵ where = b b 2 2 is shown in Fig. 1(. The dashed line corresponds to the contour where the largest Lapunov eponent changes its sign. The smbols U and S corresponds to unstable and stable fied point regions respectivel. Fig. 1(b) shows the chaotic oscillator dnamics in absence of coupling (ϵ = ). With the presence of coupling (ϵ = 2) and in absence of the periodic forcing term (α = ), the sstem converges to a fied point at the coordinate (b 1,b 2 ) as shown in Fig. 1(c). Results shown are for k 1 = k 2 = k =.1 and b 1 = b 2 = b = 1. This result suggests that with two linear augmentation sstems, this method could be used for stabilizing chaotic sstems around target fied points at different locations in the phase space S U b) c) Figure 1: ( Phase diagram in parameters space ϵ for α = where = b b 2 2 [see Eq. (3)] where (b 1,b 2 ) are the coordinates around which we want oscillation. Here regions marked U and S correspond to the oscillator and stable fied point motions respectivel. The demarcation line dividing the two regions is where the largest Lapunov eponent changes its sign. The trajectories for oscillator and targeted fied point of the sstem, Eq. (3), (b) ϵ = and (c) ϵ = 2 for b 1 = b 2 = b = 1 and k 1 = k 2 = k = Targeting periodic orbit We now eamine the chaotic Lorenz sstem coupled to two linear augmentation sstems and with periodic forcing (finite amplitude α), and are coupled with u 1 and u 2 respectivel as ẋ = σ( ) + ϵu 1 ẏ = r z + ϵu 2 ż = µz u 1 = k 1 u 1 ϵ( b 1 α cos ωt) u 2 = k 2 u 2 ϵ( b 2 α cos ωt). (4) For α = 1 and ω = 1, the phase diagram in parameter space ϵ, where = (b b 2 2 ) and (b 1,b 2 ) is the point around which we want the periodic oscillation, is shown in Fig. 2(. The dashed line and dotted points correspond, IJNS homepage:

4 16 International Journal of NonlinearScience,Vol.26(218),No.1,pp R1 R2 R A b) c) d) 6 e) Figure 2: ( Phase diagram in parameter space ϵ for ω = 1 and α = 1 where = b b 2 2. The dashed line and dotted points correspond to the contour at A =.4 where A = ( 2 + ȳ 2 ). The solid black line is plotted manuall to separate regions R 2 and R 3. (b) The variation of A as a function of at ϵ = 2, and targeted orbits at (c) b = 2, (d) b = 9 and (e) b = 25 in plane, Eq. (4). respectivel, to the contour at A =.4, where A = ( 2 + ȳ 2 ) and and ȳ are the time averages of and. This difference signifies the convergence of the center of rotation to the targeted point. Here the solid black line is drawn manuall as a guide to the ee to separate the regions R 2 and R 3. Both regions R 1 and R 3 correspond to the regions where the periodic orbit targeting does not work. In region R 3 the motion remains chaotic similar to the uncoupled Lorenz attractor and in the region R 1 the motion is periodic but not according to the targeted scheme. In region R 2, for smaller value of, the motion is periodic. In the intermediate values of, there are all kind of attractors (chaotic, quasiperiodic and periodic), and for larger value there is onl period-1 attractor. The details of the transitions across different regions are also shown in Fig. 2 for fied ϵ = 2. The variation of A as a function of shown in Fig. 2(b) indicates that the slope is close to unit, i.e., an increment in will have equal increment in A. To verif periodic oscillation around a targeted point in the phase space, we choose b 1 =b 2 =b, and obtained trajectories for three different values of b, i.e., 2, 9 and 25. These phase space trajectories are shown in Fig. 2(c), (d) and (e) respectivel. Fig. 2(d) shows that the oscillations of quasiperiodic attractor tpe can also be obtained through this scheme. As shown in Fig. 2(c), the oscillations targeted are for small but their range is the same as the range of uncoupled Lorenz attractor. All the three attractors are in the region R 2 showing successful control. The quantitative details of the motion are shown in Fig. 3. The bifurcation diagram, m (the maima of ) as a function of, plotted in Fig. 3( shows that for intermediate values of the chaotic motion persists and then the sstem becomes periodic as approaches 1. This is further confirmed in Fig. 3(b) with largest Lapunov eponent being positive in this region [4]. From Fig. 3 ( and (b) it can be concluded that as increases there are all tpe of attractors possible at different values of the coupling parameter. One noticeable transition is from quasiperiodic to periodic motion near 9 where two Lapunov eponent are zero which confirms the presence of quasiperiodic attractor. Another important observation is shown in Fig. 3(c) where numericall calculated frequenc Ω of the full sstem is plotted. Here the measured sstem frequenc is equal to the frequenc of the applied periodic forcing, sa around Ω 1 for lower values of. Then it begins to fluctuate because the sstem behavior is different in this region. However, in the region of periodic-1 orbits (in R 2 ), the measured sstem frequenc is again the same as the frequenc of the applied periodic forcing, i.e., Ω 1 = ω. This implies that at higher values of, when there is period-1 orbit we get the targeted frequenc. Therefore, Fig. 3( and Fig. 3(c) suggest that one can get a periodic orbit of targeted frequenc in different locations around (b 1,b 2 ) in phase space using the proposed coupling scheme Transition from chaos to periodic motions We also analze the nature of transitions, from R 1 to R 2 and from R 3 to R 2 see in Fig 3. Here we take the specific case of b = 15 (=15 2), which shows a transition from the region R 1 to R 2, where chaotic attractor is suddenl destroed as the control parameter passes through its critical value. The largest Lapunov eponent and bifurcation diagram for b = 15 are plotted in Fig. 4 ( and (b) respectivel showing the transition from chaos to periodic behavior around ϵ.14. Fig. 4 (c) and (d) are the corresponding chaotic and periodic attractors respectivel across this transition. Here the transition is sudden in the sense that chaotic attractor disappears to periodic attractor as coupling strength is increased. In order to understand this transition we compute the transient time (n t ) for a given initial condition that settles to a periodic orbit. In Fig. 4 (e) the mean transient lifetime n t averaged over 1 initial conditions is plotted as a function of (ϵ ϵ c ), IJNS for contribution: editor@nonlinearscience.org.uk

5 V. Varshne et al.: Targeting Periodic Solutions of Chaotic Sstems 17 m 4 2 b) λ ma c) Ω Figure 3: Characteristics of the oscillator motion depicted in Fig. 2 for ϵ = 1. ( bifurcation diagram, m (b) largest Lapunov eponent, λ ma and (c) resulting frequenc, Ω as a function of parameter. where ϵ c is the critical parameter value, ϵ c =.14. This shows a power law behavior as n t (ϵ ϵ c ) β, where β is the critical eponent. Fitting the data with this scaling form gives the eponent β =.48. This tpe of scaling behavior has been studied in Ref. [41] for the mass spring model. The have shown that this model can reproduce the dnamical characteristics of a periodicall forced dripping faucet observed eperimentall b Shoji s [41], where it is claimed that such transitions are due to boundar crises. Similarit of the numerical results of our stud confirms that the transition from chaos to period-1 oscillation is via boundar crises. Similar results obtained (results are not shown here) for other values of can be interpreted as transitions for all values of are through a boundar crises route. 3.2 Rössler sstem In order to generalize the proposed scheme to other sstems, we consider the Rössler oscillator: ẋ = ( z) + ϵu 1 ẏ = ϵu 2 ż =.2 + z ( 4) u 1 = k 1 u 1 ϵ( b 1 α cos ωt) u 2 = k 2 u 2 ϵ( b 2 α cos ωt). (5) This sstem also have three unstable fied points and the sstem dnamics is chaotic Targeting periodic orbit The phase diagram in parameter space ϵ for ω = 1 and α = 1 is shown in Fig. 5 (. The dashed line corresponds to the contour at A =.4. Region R 1 is uncontrolled one where motion is either chaotic or periodic, similar to the uncoupled Rössler attractors [Fig. 5 (d)]. In region R 2 we get the periodic oscillations around the desired point. The variation of A as a function of in Fig. 5(b) shows that slope is close to unit, i.e., an increment in will have equal IJNS homepage:

6 18 International Journal of NonlinearScience,Vol.26(218),No.1,pp λ ma b) 1 m.1.2 c) d) e) log(n t ) log( ) Figure 4: ( The largest Lapunov eponent, λ ma, (b) the bifurcation diagram, m, as a function of coupling strength ϵ at b = 15. The (c) chaotic (ϵ =.8) (d) periodic (ϵ =.6) attractors. (e) The variation of average transient time log n t vs log(ϵ ϵ c ) (dots) for ω = 1 and α = 1 (see the tet for details). The solid line in e) is the linear fitting line to the data R 1 R b) c) 1.6 A15 Ω d) e) Figure 5: ( Phase diagram in parameter space ϵ for ω = 1 and α = 1 where the dashed line correspond to the contour at A =.4. The variation of ( A and (b) Ω as a function of. The orbits of (d) chaotic (ϵ =.2) and (e) periodic (ϵ = 2) at b = 15, i.e. =15 2 Eq. (5). IJNS for contribution: editor@nonlinearscience.org.uk

7 V. Varshne et al.: Targeting Periodic Solutions of Chaotic Sstems 19 increment in A for targeting in the region R 2. The computed sstem frequenc is plotted in Fig. 5(c) as a function of and shows that Ω ω. Fig. 5 (e) depicts the corresponding targeted attractor in region R 2. Similar results have been obtained in ϵ phase space for others parameter values of α, ϵ and k. λ ma log(n t ) b) m log( c ) Figure 6: Characteristics of the oscillator motion depicted in Fig. 5 (. The largest Lapunov eponent, λ ma (b) the bifurcation diagram m as a function of coupling strength ϵ at b = 1. (c) The log n t vs log(ϵ ϵ c ) for ω = 1 and α = 1 where n t is the averaged transient time after which the orbit falls in 1 4 of periodic orbit and ϵ c is the critical value after which chaotic orbit goes to periodic. c) Transition from chaos to periodic motion A detailed analsis of the transition from region R 1 to R 2 shown in Fig. 5 has also been carried out for the periodic forcing control of the Rössler sstem. We take b = 1 to show the transition corresponding to Fig. 5(. For b = 1, the largest Lapunov eponents and bifurcation diagram are shown in Fig. 6( and (b) respectivel. These show transition from chaos to periodic behavior around ϵ.72. In order to understand this transition we calculate the transient time (n t ) for a given initial condition which settle to the periodic orbit. The mean transient lifetime n t over 1 initial conditions as a function of (ϵ ϵ c ), where ϵ c =.72 is the critical parameter value is shown in Fig. 6(c). Similar to the periodic forcing of the Lorenz sstem, this also shows a power law behavior as n t (ϵ ϵ c ) β, where β is the critical eponent. Fitting the data with this scaling shows that the eponents β = 1.1. Hence this transition is also through boundar crises route as discussed earlier. 4 Summar We propose a new tpe of linear augmentation coupling through which a sstem can be stabilized around a target periodic orbit. Successful control of the sstem dnamics both for the target frequenc and the specific point around which the sstem is targeted to perform the periodic oscillation has been presented for two well known chaotic sstems, the chaotic Lorentz oscillator and Rössler oscillator. Further, it is shown that the transition from the chaotic to periodic dnamics occurs through a boundar crisis route. A critical eponent characterizing the transition is presented. In both the cases, the stabilization of the periodic orbit around the targeted point has been observed and the dnamics in different regions has been analsied b phase diagrams in - ϵ space. The results presented in this work are general as this scheme is independent of the choice of the variables in which coupling is applied. If we appl coupling in or z then also similar IJNS homepage:

8 2 International Journal of NonlinearScience,Vol.26(218),No.1,pp results are obtained. Validit of the same scheme for other sstems (e.g. Chua, Landau Stuart, etc) has also been verified. We believe that this control scheme is of considerable importance in situations where chaos is undesirable and we need to target the dnamics to a periodic orbit of desired frequenc around a specific target point in the phase space. Although proposed coupling scheme depends on a number of parameter (the coupling strength ϵ, the deca parameter k i, amplitude of eternal periodic forcing α, frequenc of eternal periodic forcing ω, and target point of oscillation b i ), we have eplored the entire parameter space of and ϵ. We have presented results for ω=1 and have obtained comparable results for ω=5 and ω=15. Analsis of the control scheme with the higher dimension of linear augmentation (r > 2) as well as unequal deca parameter k 1 k 2... k r is planned for future. Acknowledgement Authors thank SERB, Department of Science and Technolog, Government of India for the financial support. References [1] A. S. Pikovsk, M. G. Rosenblum and J. Kurths. Snchronization: A universal concept in Non linear sciences. Cambridge Universit Press, Cambridge. 21. [2] H. Fujisaka and T. Yamada. Stabilit Theor of Snchronized Motion in Coupled-Oscillator Sstems. Prog. Theor. Phs., 69(1983): [3] K. Yagasaki. Chaos in a Pendulum with Feedback Control. Nonlinear Dn., 6(1994): [4] Y. P. Luo and Y. C. Hung. Control snchronization and parameter identification of two different chaotic sstems. Nonlinear Dn., 73(213): [5] K. Mirus and J. C. Sprott. Controlling chaos in low- and high- dimensional sstems with periodic parametric perturbation. Phs. Rev. E., 59(1999): [6] Z. Qu, G. Yang and G. Qin. Phase Effect in Taming Nonautonomous Chaos b Weak Harmonic Perturbations. Phs. Rev. Lett., 74(1995): [7] X. Yuan, C. Li and T. Huang. Projective Lag Snchronization of Delaed Chaotic Sstems with Parameter Mismatch via Intermittent Control. International Journal of Nonlinear Science, 23(217): 3 1. [8] Z. Wang, J. Tang and Y. Luo. Controlling the Period-Doubling Bifurcation of Logistic Model. International Journal of Nonlinear Science, 2(215): [9] V. Piccirillo, J. M. Balthazar, B. R. Jr. Pontes and J. L. P. Feli. Chaos control of a nonlinear oscillator with shape memor allo using an optimal linear control: Part I: Ideal energ source. Nonlinear Dn., 55(29): [1] E. Braverman, and E. Liz. Global stabilization of periodic orbits using a proportional feedback control with pulses. Nonlinear Dn., 67(212): [11] E. Ott, C. Grebogi and J. A. Yorke. Controlling Chaos. Phs. Rev. Lett., 64(199): [12] S. Boccaletti, C. Grebogi, Y. C. Lai, H. Mancini and D. Maza. The Control of Chaos: theor and applications. Phs. Rep., 329(2): [13] H. Gritli, S. Belghith and N. Khraief. OGY-based control of chaos in semi-passive dnamic walking of a torso-driven biped robot. Nonlinear Dn., 79(215): [14] K. Pragas. Continuos control of chaos b self controlling feedback. Phs. Lett. A., 17(1992): [15] S. Bielawski, D. Derozier and P. Glorieu. Eperimental characterization of unstable periodic orbits b controlling chaos. Phs.Rev. A., 47(1993): [16] H. Nakajima and Y. Ueda. Half period dela feedback control for dnamical sstem with smmetries. Phs. Rev. E., 58(1998): [17] K. Pragas. Controll of Chaos via an Unstable Dela Feedback controller. Phs. Rev. Lett., 86(21): [18] J. M. Jonnalagadda, D. Purnima and G. V. S. R. Deekshitulu. Discrete Control Sstems of Fractional Order. International Journal of Nonlinear Science, 21(216): [19] E. Liz. How to control Chaotic behavior and population size with proportional feedback. Phs. Lett. A., 374(21): [2] J. L. Breeden. Open-loop control of nonlinear sstems. Phs. Lett. A, 19(1994): [21] S. Sinha, R. Ramaswam and J. Subba Rao. Adaptive Controll in Non Linear dnamics. Phsica D, 43(199): IJNS for contribution: editor@nonlinearscience.org.uk

9 V. Varshne et al.: Targeting Periodic Solutions of Chaotic Sstems 21 [22] B. A. Huberman and E. Lumer. Dnamics of adaptive sstems. IEEE Trans. Circuits Sst, 37(199): [23] T. Tamura, N. Inaba and J. Miamichi. Mechanism of tamming chaos b weak harmonic perturbation. Phs. Rev. Lett., 83(1999): [24] H. T. Su, Y. H. chen and R. R. Hsu. Master Slave Scheme and controlling chaos in the Braiman- Goldhirsch method. Phs. Rev. E., 59(1999): [25] R. R. HSu, H. T. Hsu, J. L. Chern and C. C.Chen. Conditions to control chaotic dnamics b weak periodic perturbation. Phs. Rev. Lett., 78(1997): [26] S. Zhang, H. Lu, X. Chen and Z. Zhang. Bursting Snchronization of Two Fast-slow Oscillators with Mismatch in Parameters. International Journal of Nonlinear Science, 21(216): [27] J. Llibre and M. R. Candid. ZeroHopf Bifurcations in A Hperchaotic Lorenz Sstem II. International Journal of Nonlinear Science, 25(217): [28] H. J. Li and J. L. Chern. Goal-oriented scheme for taming chaos with a weak periodic perturbation: Eperiment in a diode resonator. Phs. Rev. E., 54(1996): [29] M. Shoji and M. Streiff. Non linear characteristics in dripping faucet. 5th Int. Workshop on Chaos Turbulence Comple sstems, Tsukuba, November 6, [3] P. R. Sharma, A. Sharma, M. D. Shrimali and A. Prasad. Targeting fied point solutions in non linear oscillators through linear augmentation. Phs. Rev. E., 83(211): [31] P. R. Sharma, M. D. Shrimali, A. Prasad and U. Feudel. Controlling bistabilit b linear augmentation. Phs. Lett. A., 377(213): [32] P. R. Sharma, A. Singh, A. Prasad and M. D. Shrimali. Controlling dnamical behavior of drive-response sstem through linear augmentation. Eur. Phs. J. Special Topics., 223(214): [33] P. R. Sharma, M. D. Shrimali, A. Prasad, N. V. Kuznetsov and G. A. Leonov. Control of multistabilit in hidden attractors. Eur. Phs. J. Special Topics., 224(215): [34] P. R. Sharma, M. D. Shrimali, A. Prasad, N. V. Kuznetsov and G. A. Leonov. Controlling Dnamics of Hidden Attractors. Int. J. Bifur. chaos., 25(215): [35] R. Karnatak. Linear Augmentation for Stabilizing Stationar Solutions: Potential Pitfalls and Their Application. PLoS ONE 1 (11), e [36] F. T. Arecchi, R. Meucci, G. Pucciono and J. Tredicce. Eperimental Evidence of Subharmonic Bifurcations, Multistabilit, and Turbulence in a Q-Switched Gas Laser. Phs. Rev. Lett., 49(1982): [37] M. D. Shrimali, A. Prasad,R. Ramaswam and U. Feudel. The nature of attractor basins in multistable sstems. Int. J. Bifur. chaos., 6(28): [38] G. Saena, A. Prasad and R. Ramaswam. Amplitude death: The emergence of stationarit in coupled nonlinear sstems. Phs. Rep., 521(212): [39] V. Resmi, G. Ambika and R. E. Amritkar. General mechanism for amplitude death in coupled sstems. Phs. Rev. E., 84(211): [4] M. Tabor. Chaos and Integrabilit in Non Linear Dnamics A Wile Interscience Publication (1988). [41] K. Kiono and N. Fuchikami. Bifurcations Induced b Periodic Forcing and Taming Chaos in Dripping Faucets. J. Ph. Soc. Japan., 71(22): IJNS homepage: