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Dynamics of the generalized KP-MEW-Burgers equation with external periodic perturbation
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Dynamics of the generalized KP-MEW-Burgers equation with external periodic perturbation Asit Saha Department of Mathematics, Sikkim Manipal Institute of Technology, Sikkim Manipal University, Majitar, Rangpo, East-Sikkim 737136, India
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Article history: Received 24 August 2016 Received in revised form 8 January 2017 Accepted 12 February 2017 Available online xxxx Keywords: KP-MEW-Burgers equation Bifurcation Periodic motion Chaotic motion
abstract The generalized Kadomtsev–Petviashvili modified equal width-Burgers (KP-MEW-Burgers) equation is introduced for the first time. The qualitative change of the traveling wave solutions of the KP-MEW-Burgers equation is studied using numerical simulations. Considering an external periodic perturbation, the periodic and chaotic motions of the perturbed KPMEW-Burgers equation are investigated by using the phase projection analysis, time series analysis, Poincare´ section and bifurcation diagram. The parameter a (nonlinear coefficient) plays a crucial role in the periodic motions and chaotic motions through period doubling route to chaos of the perturbed KP-MEW-Burgers equation. © 2017 Elsevier Ltd. All rights reserved.
1. Introduction During last few years, an appreciable attention has been paid to the study of nonlinear wave solutions (solitary wave solutions, periodic wave solutions, kink and anti-kink wave solutions) and integrability of some interesting nonlinear model equations, such as the KdV equation [1], generalized equal width (GEW) equation [2], sine–Gordon equation [3], nonlinear Schrödinger equation [4], which arise from many physical situations. Many of these nonlinear model equations are completely integrable. It is interesting to note that a completely integrable nonlinear model equation acquires some remarkable properties, such as Lax pair, N-soliton, infinite conservation laws, Painleve property and bi-Hamiltonian structures. Nguetcho et al. [5] investigated the bifurcations of phase portraits of a singular nonlinear equation of the second class and reported various sufficient conditions leading to the existence of propagating wave solutions. Sahu et al. [6] showed that the KdV equation describes the solitonic nature of the waves in small amplitude region and considering arbitrary amplitude waves in the fully nonlinear regime, the system exhibits possible existence of quasi-periodic behavior in quantum plasma. Recently, Selim et al. [7] studied bifurcations of nonlinear traveling waves in a multicomponent magnetoplasma with superthermal electrons. Very recently, Chai et al. [8] studied the effects of quantic nonlinearity on the propagation of the ultrashort optical pulses in a non-Kerr medium in the framework of a perturbed nonlinear Schrödinger equation with the power law nonlinearity as a planar dynamic-system view point. Chai et al. [9] also studied a (2 + 1)-dimensional breaking soliton equation that describes the (2 + 1)-dimensional interaction of a Riemann wave propagating along the y axis with a long wave along the x axis, where x and y are the scaled space coordinates. However, there exist often different types of periodic perturbations in many real physical situations. The nature of this external periodic perturbation may vary depending on the different physical situations. Recently, a remarkable attention has been paid to the study of nonlinear model equations considering external periodic perturbations [10,11]. Furthermore, a completely integrable nonlinear model equation cannot describe chaotic phenomenon, but the addition of a periodic
E-mail address: [email protected]. http://dx.doi.org/10.1016/j.camwa.2017.02.017 0898-1221/© 2017 Elsevier Ltd. All rights reserved.
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A. Saha / Computers and Mathematics with Applications (
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perturbation to an integrable nonlinear model equation may lead to chaotic dynamics, such as perturbed sine–Gordon equation [12], perturbed KdV–Burgers equation [13], and perturbed Schrödinger equation [14]. Considering an external periodic perturbation, one can investigate chaotic features of a nonlinear physical system through different possible routes: (i) period doubling route to chaos, (ii) quasiperiodic route to chaos and (iii) intermittent bifurcation and crisis. The generalized KP-MEW equation [15] is given by
(qt + a(qn )x + bqxxt )x + rqyy = 0, (1) where a, b and r are constants. Wazwaz [15] conducted a comparative study between the tanh method and the sine–cosine method using the nonlinear wave equation (1). These two reliable schemes were used to back up his analysis to construct exact solutions with distinct physical structures. Recently, bifurcation of traveling wave solutions for the generalized KPMEW equations [16] and ZK-MEW equations [17] were investigated using the bifurcation theory of planar dynamical systems. Wei et al. [18] applied the qualitative theory of differential equations to the KP-MEW (2, 2) equation and obtained single peak solitary wave solutions for the generalized KP-MEW (2, 2) equation under boundary condition. Li and Song [19] derived compacton-like wave and kink-like wave solutions of the generalized KP-MEW (2, 2) equation. Zhong et al. [20] obtained compacton, peakon, cuspons, loop solutions and smooth solitons for the generalized KP-MEW equation. The Burgers equation is qt + aqqx = bqxx ,
(2)
which describes nonlinear ion-acoustic shock wave [21], dust-acoustic shock wave [22], dust-ion acoustic shock wave [23] and positron-acoustic shock wave [24] in plasmas, where x is a space variable, t is time variable, q is electrostatic potential, η the coefficient a involves some physical parameters depending on the plasma model and b = 2 , with η as the coefficient of viscosity. In this work, the generalized Kadomtsev–Petviashvili modified equal width-Burgers (KP-MEW-Burgers) equation is introduced for the first time, as
(qt + a(qn )x + µqxx − bqxxt )x + rqyy = 0, (3) where a, b and r are positive parameters and µ is a damping parameter. In Eq. (3), the first term is the evolution term, while the second term represents the nonlinear term, the third term is the dissipative term and the fourth term is the dispersion term. Eq. (3) is similar to the KP–Burgers equation [25,26] which describes nonlinear shock waves in plasmas. Eq. (3) may be applied as a nonlinear evolution equation for long-wave propagation in nonlinear media with dispersion and dissipation. We add the external periodic perturbation f to the KP-MEW-Burgers equation (3) and form the perturbed KP-MEW-Burgers equation as
(qt + a(qn )x + µqxx − bqxxt )x + rqyy = fxx .
(4)
Eq. (4) describes nonlinear shock waves in a dissipative system in the presence of external periodic perturbation. We investigate the qualitative change of traveling wave solutions of Eq. (3) and, periodic and chaotic motions of the perturbed KP-MEW-Burgers equation (4) for the first time. To study all possible periodic and chaotic motions of Eq. (4), we consider the transformation ξ = lx + my − ut, where l and m are direction cosines of the line of propagation of the traveling wave with velocity u in the xy-plane such that l2 + m2 = 1. Then Eq. (4) becomes
(−uqξ + nalqn−1 qξ + µl2 qξ ξ + bul2 qξ ξ ξ )ξ l + m2 rqξ ξ = l2 fξ ξ . Considering f = f∗ cos(ωξ ), and integrating twice Eq. (5) takes the form − uq + alqn + µl2 qξ + bul2 qξ ξ +
m2 r
q = lf∗ cos(ωξ ). l Then system (6) is equivalent to the following dynamical system: qξ = z , zξ = Aq − Bqn − Dz + f0 cos(ωξ ),
(5)
(6)
lu−(1−l2 )r
µ
(7)
a ∗ where A = , B = blu , D = bu and, f0 = blu is the strength of the external periodic perturbation and ω is the bl3 u frequency. Eq. (7) is similar to the well known duffing oscillator [27]. As an oscillator, it has tremendous applications in Biomathematical modeling [28] of the aneurysm of circle of Willis which simulates the blood flow inside aneurysm and practical engineering in case of vibration of plate spring in rare earth giant magnetostriction transducer. The remaining part of the paper is organized as follows: In Section 2, we consider unperturbed dynamical system and corresponding qualitative analysis. In Section 3, we study periodic and chaotic motions in the perturbed dynamical system. Section 4 is kept for conclusions. f
2. Unperturbed dynamical system and qualitative analysis In this section, we study the qualitative behavior of the unperturbed traveling wave system by using the bifurcation theory of planar dynamical system [29]. To do that, we assume f0 = 0 and the system (7) takes the form
qξ = z , zξ = Aq − Bqn − Dz .
(8)
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Fig. 1. (a) Phase portrait of the dynamical system (8), (b) plot of z vs. ξ , for n = 2, a = 0.08, b = 1, r = 0.08, l = 0.2, u = 0.4, and µ = 0.12.
Fig. 2. (a) Phase portrait of the dynamical system (8), (b) plot of z vs. ξ , for n = 3, a = 0.08, b = 1, r = 0.08, l = 0.2, u = 0.4, and µ = 0.12.
The system (8) is a planar dynamical system with parameters a, b, r , l, µ and u. It is important to note that the phase orbits defined by the vector fields of Eq. (8) will determine all traveling wave solutions of Eq. (3). We study the bifurcations of phase portraits of Eq. (8) in the (q, z ) phase plane depending on the parameters a, b, r , l, µ and u. If n is an odd 1
integer, the dynamical system (8) has three equilibrium points at P0 (0, 0), P1 (q1 , 0) and P2 (q2 , 0), where q1 = ( AB ) n−1 and 1
q2 = −( AB ) n−1 . If n is an even integer, the dynamical system (8) has two equilibrium points at P0 (0, 0) and P1 (q1 , 0). The determinant of the Jacobian matrix of the linearized system of the traveling system (8) is
0 J = A − Bnqn−1 = −A + Bnqn−1 .
1 −D (9)
At P0 (0, 0), the value of the determinant is J (0, 0) = −A < 0. At P1,2 (q1,2 , 0), the value of the determinant is J (q1,2 , 0) =
√ −D± D2 +4A (n − 1)A > 0. The eigenvalues at P0 (0, 0) are λ = , which are real and are of opposite signs. Hence, the 2 equilibrium point P0 (0, 0) is a saddle point. The eigenvalues at P1,2 (q1,2 , 0) are λ = −α ± iβ , where α = D2 and β = 21 4A(n − 1) − D2 with 4A(n − 1) > D2 . Hence, the equilibrium points P1,2 (q1,2 , 0) are stable spirals. The phase portrait of the system (8) is shown in Fig. 1(a) for n = 2, a = 0.08, b = 1, r = 0.08, l = 0.2, u = 0.4, and µ = 0.12. In this case the dynamical system (8) has two equilibrium points at P0 (0, 0) and P1 (q1 , 0), where P0 (0, 0) is a saddle point and it represents a stable spiral at the equilibrium point P1 (q1 , 0). In Fig. 1(b), the variation of z with ξ is presented with same values of parameters as Fig. 1(a). In Fig. 2(a), the phase portrait of the dynamical system (8) is shown for n = 3 with same values of other parameters as Fig. 1. In this case the dynamical system (8) has three equilibrium points at P0 (0, 0), P1 (q1 , 0) and P2 (q2 , 0), where P0 (0, 0) is a saddle point and we have two stable spirals at the equilibrium points P1 (q1 , 0) and P2 (q2 , 0). In Fig. 2(b), we have plotted z vs. ξ with same values of parameters as Fig. 2(a). It is important to note that the wave particle moves to the right with stable oscillation which decays as ξ increases (see Figs. 1(b) and 2(b)). The system (8) with µ > 0 is dissipative and there is neither solitary nor periodic wave solutions as there does not exist any closed trajectory in the phase plane by Bendixson’s criterion [30] (see Figs. 1 and 2).
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Fig. 3. (a) Phase projection of the perturbed KP-MEW-Burgers equation (7), (b) plot z vs. ξ for n = 3, a = 0.015, b = 1, l = 0.2, r = 0.08, u = 0.4, µ = 0.12, ω = 1.25 and f0 = 0.5.
If there is no damping, i.e. µ = 0, then the system (8) takes the form
dq = z, dξ dz = Aq − Bqn , dξ
(10)
which is a planar Hamiltonian system with Hamiltonian function: H (q, z ) =
z2
−A
q2
+B
qn+1
.
2 2 n+1 The bifurcation of phase portraits of the system (10) is already reported in the work [16].
(11)
3. Periodic and chaotic motions in perturbed dynamical system In this section, we study the periodic and chaotic motions of the perturbed dynamical system (7) of the KP-MEWBurgers equation (3). In order to study all possible periodic and chaotic motions of the perturbed dynamical system (7), we apply different numerical tools that help to corroborate effectively periodic and chaotic phenomena [27] are: (i) the phase projection plots, (ii) time series plots, (iii) bifurcation diagram and (iv) the Poincaré section. The perturbed system (7) involves eight independent parameters a, b, l, r , u, µ, ω and f0 . Because of large number of parameters, it is really difficult to investigate the system for the complete range of parametric space. To simplify the task, we consider special fixed values of the parameters b, l, r , u, µ, ω, f0 varying parameter a associated with the nonlinear term. Thus, parameter a plays the role of bifurcation parameter. Of course one can vary any of the other parameters b, l, r , u, µ, ω, f0 , but, this will not give any significant different qualitative features. 3.1. Phase projection and time series analysis Phase projection analysis is one of the most important techniques for investigating the behavior of nonlinear systems, as there is no usual analytical solution for a nonlinear system. On the other hand, time series analysis consists of methods for analyzing time series data in order to bring out meaningful characteristics of the data. Time series forecasting is the use of a model to predict future values based on previously observed values. In Fig. 3(a), the phase space trajectory of the system (7) is presented for n = 3, a = 0.015, b = 1, l = 0.2, r = 0.08, u = 0.4, µ = 0.12, ω = 1.25 and f0 = 0.5. It shows the period-1 motion of the system (7) and the corresponding variation of z with ξ is depicted in Fig. 3(b). The system (7) shows period-2 motion for a = 0.029 with same values of other parameters as Fig. 3(a), depicted in Fig. 4(a). The corresponding variation of z vs. ξ is presented in Fig. 4(b) with same values of parameters as Fig. 4(a). In Fig. 5(a), the phase space trajectories of the system (7) are presented for a = 0.317 with same values of other parameters as Fig. 3(a). It shows period-4 motion of the system (7) and the corresponding variation of z with ξ is depicted in Fig. 5(b). In Fig. 6(a), the phase space trajectories of the system (7) are presented for a = 0.03502 with same values of other parameters as Fig. 3(a). It shows chaotic motion of the system (7) through period doubling route to chaos and the corresponding variation of z with ξ is depicted in Fig. 6(b). Furthermore, if we again increase the value of a (i.e., a = 0.043) keeping other parameters fixed as Fig. 3(a), then we reach to the state of period-3 motion, shown in Fig. 7(a) and the corresponding variation of z with ξ is presented in Fig. 7(b). In Fig. 8(a), the phase space trajectories of the system (7) are presented for a = 0.05 with same values of other parameters as Fig. 3(a). It shows period-5 motion of the system (7) and the corresponding variation of z with ξ is depicted in Fig. 8(b). If we further increase the value of a (i.e., a = 0.079) keeping other parameters fixed as Fig. 3(a), then we again reach to the chaotic state, shown in Fig. 9(a) and the corresponding variation of z with ξ is presented in Fig. 9(b).
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Fig. 4. (a) Phase projection of the perturbed KP-MEW-Burgers equation (7), (b) plot z vs. ξ for a = 0.029 with same values of other parameters as Fig. 3.
Fig. 5. (a) Phase projection of the perturbed KP-MEW-Burgers equation (7), (b) plot z vs. ξ for a = 0.0317 with same values of other parameters as Fig. 3.
Fig. 6. (a) Phase projection of the perturbed KP-MEW-Burgers equation (7), (b) plot z vs. ξ for a = 0.03502 with same values of other parameters as Fig. 3.
3.2. Poincaré return map analysis and bifurcation diagram A very useful tool for exploring chaotic attractors is Poincaré sections [31,32]. Poincaré sections provide us with images of the attractor in a (n − 1)-dimensional hyperplane of n-dimensional phase space. The Poincaré return map of the perturbed system (7) is shown in Fig. 10(a). In return map plots, Fig. 10(a) demonstrates that the points are densely scattered and exhibit irregular distributions without any known pattern for the system (7) for the parameter values f0 = 0.079 keeping other parameters fixed as in Fig. 3(a). Such return maps are characteristics of chaotic oscillations. Considering parameter a (coefficient of nonlinear term) as the bifurcation parameter, the bifurcation diagram in the (a − q) plane of the system (7) is shown in Fig. 10(b). One can easily verify different types of periodic motions (period-1, period-2, period-3, period-4 and period-5) and chaotic motions corresponding to the values of the parameter a (0.015, 0.0295, 0.043, 0.0317 and 0.05 respectively) in Fig. 10(b). 4. Conclusions The generalized KP-MEW-Burgers equation is introduced and investigated for the first time. The bifurcation behavior of the KP-MEW-Burgers equation is obtained. The stable oscillations of the KP-MEW-Burgers equation are presented. Considering an external periodic perturbation, the periodic and chaotic motions of the perturbed KP-MEW-Burgers equation
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Fig. 7. (a) Phase projection of the perturbed KP-MEW-Burgers equation (7), (b) plot z vs. ξ for a = 0.043 with same values of other parameters as Fig. 3.
Fig. 8. (a) Phase projection of the perturbed KP-MEW-Burgers equation (7), (b) plot z vs. ξ for a = 0.05 with same values of other parameters as Fig. 3.
Fig. 9. (a) Phase projection of the perturbed KP-MEW-Burgers equation (7), (b) plot z vs. ξ for a = 0.079 with same values of other parameters as Fig. 3.
Fig. 10. (a) Poincare section of the chaotic state in Fig. 9, (b) bifurcation diagram in the (a − q) plane with same values of other parameters as Fig. 9.
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are investigated with the help of phase projection analysis, time series analysis, Poincaré section and bifurcation diagram. The parameter a (nonlinear coefficient) plays as the bifurcation parameter in bifurcations, periodic motions and chaotic motions of the perturbed KP-MEW-Burgers equation. It is seen that the perturbed KP-MEW-Burgers equation possesses chaotic motions through period doubling route to chaos. Acknowledgments The constructive suggestions of the editor and the reviewers are gratefully acknowledged. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32]
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Contents lists available at ScienceDirect
Computers and Mathematics with Applications journal homepage: www.elsevier.com/locate/camwa
Dynamics of the generalized KP-MEW-Burgers equation with external periodic perturbation Asit Saha Department of Mathematics, Sikkim Manipal Institute of Technology, Sikkim Manipal University, Majitar, Rangpo, East-Sikkim 737136, India
article
info
Article history: Received 24 August 2016 Received in revised form 8 January 2017 Accepted 12 February 2017 Available online xxxx Keywords: KP-MEW-Burgers equation Bifurcation Periodic motion Chaotic motion
abstract The generalized Kadomtsev–Petviashvili modified equal width-Burgers (KP-MEW-Burgers) equation is introduced for the first time. The qualitative change of the traveling wave solutions of the KP-MEW-Burgers equation is studied using numerical simulations. Considering an external periodic perturbation, the periodic and chaotic motions of the perturbed KPMEW-Burgers equation are investigated by using the phase projection analysis, time series analysis, Poincare´ section and bifurcation diagram. The parameter a (nonlinear coefficient) plays a crucial role in the periodic motions and chaotic motions through period doubling route to chaos of the perturbed KP-MEW-Burgers equation. © 2017 Elsevier Ltd. All rights reserved.
1. Introduction During last few years, an appreciable attention has been paid to the study of nonlinear wave solutions (solitary wave solutions, periodic wave solutions, kink and anti-kink wave solutions) and integrability of some interesting nonlinear model equations, such as the KdV equation [1], generalized equal width (GEW) equation [2], sine–Gordon equation [3], nonlinear Schrödinger equation [4], which arise from many physical situations. Many of these nonlinear model equations are completely integrable. It is interesting to note that a completely integrable nonlinear model equation acquires some remarkable properties, such as Lax pair, N-soliton, infinite conservation laws, Painleve property and bi-Hamiltonian structures. Nguetcho et al. [5] investigated the bifurcations of phase portraits of a singular nonlinear equation of the second class and reported various sufficient conditions leading to the existence of propagating wave solutions. Sahu et al. [6] showed that the KdV equation describes the solitonic nature of the waves in small amplitude region and considering arbitrary amplitude waves in the fully nonlinear regime, the system exhibits possible existence of quasi-periodic behavior in quantum plasma. Recently, Selim et al. [7] studied bifurcations of nonlinear traveling waves in a multicomponent magnetoplasma with superthermal electrons. Very recently, Chai et al. [8] studied the effects of quantic nonlinearity on the propagation of the ultrashort optical pulses in a non-Kerr medium in the framework of a perturbed nonlinear Schrödinger equation with the power law nonlinearity as a planar dynamic-system view point. Chai et al. [9] also studied a (2 + 1)-dimensional breaking soliton equation that describes the (2 + 1)-dimensional interaction of a Riemann wave propagating along the y axis with a long wave along the x axis, where x and y are the scaled space coordinates. However, there exist often different types of periodic perturbations in many real physical situations. The nature of this external periodic perturbation may vary depending on the different physical situations. Recently, a remarkable attention has been paid to the study of nonlinear model equations considering external periodic perturbations [10,11]. Furthermore, a completely integrable nonlinear model equation cannot describe chaotic phenomenon, but the addition of a periodic
E-mail address: [email protected]. http://dx.doi.org/10.1016/j.camwa.2017.02.017 0898-1221/© 2017 Elsevier Ltd. All rights reserved.
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perturbation to an integrable nonlinear model equation may lead to chaotic dynamics, such as perturbed sine–Gordon equation [12], perturbed KdV–Burgers equation [13], and perturbed Schrödinger equation [14]. Considering an external periodic perturbation, one can investigate chaotic features of a nonlinear physical system through different possible routes: (i) period doubling route to chaos, (ii) quasiperiodic route to chaos and (iii) intermittent bifurcation and crisis. The generalized KP-MEW equation [15] is given by
(qt + a(qn )x + bqxxt )x + rqyy = 0, (1) where a, b and r are constants. Wazwaz [15] conducted a comparative study between the tanh method and the sine–cosine method using the nonlinear wave equation (1). These two reliable schemes were used to back up his analysis to construct exact solutions with distinct physical structures. Recently, bifurcation of traveling wave solutions for the generalized KPMEW equations [16] and ZK-MEW equations [17] were investigated using the bifurcation theory of planar dynamical systems. Wei et al. [18] applied the qualitative theory of differential equations to the KP-MEW (2, 2) equation and obtained single peak solitary wave solutions for the generalized KP-MEW (2, 2) equation under boundary condition. Li and Song [19] derived compacton-like wave and kink-like wave solutions of the generalized KP-MEW (2, 2) equation. Zhong et al. [20] obtained compacton, peakon, cuspons, loop solutions and smooth solitons for the generalized KP-MEW equation. The Burgers equation is qt + aqqx = bqxx ,
(2)
which describes nonlinear ion-acoustic shock wave [21], dust-acoustic shock wave [22], dust-ion acoustic shock wave [23] and positron-acoustic shock wave [24] in plasmas, where x is a space variable, t is time variable, q is electrostatic potential, η the coefficient a involves some physical parameters depending on the plasma model and b = 2 , with η as the coefficient of viscosity. In this work, the generalized Kadomtsev–Petviashvili modified equal width-Burgers (KP-MEW-Burgers) equation is introduced for the first time, as
(qt + a(qn )x + µqxx − bqxxt )x + rqyy = 0, (3) where a, b and r are positive parameters and µ is a damping parameter. In Eq. (3), the first term is the evolution term, while the second term represents the nonlinear term, the third term is the dissipative term and the fourth term is the dispersion term. Eq. (3) is similar to the KP–Burgers equation [25,26] which describes nonlinear shock waves in plasmas. Eq. (3) may be applied as a nonlinear evolution equation for long-wave propagation in nonlinear media with dispersion and dissipation. We add the external periodic perturbation f to the KP-MEW-Burgers equation (3) and form the perturbed KP-MEW-Burgers equation as
(qt + a(qn )x + µqxx − bqxxt )x + rqyy = fxx .
(4)
Eq. (4) describes nonlinear shock waves in a dissipative system in the presence of external periodic perturbation. We investigate the qualitative change of traveling wave solutions of Eq. (3) and, periodic and chaotic motions of the perturbed KP-MEW-Burgers equation (4) for the first time. To study all possible periodic and chaotic motions of Eq. (4), we consider the transformation ξ = lx + my − ut, where l and m are direction cosines of the line of propagation of the traveling wave with velocity u in the xy-plane such that l2 + m2 = 1. Then Eq. (4) becomes
(−uqξ + nalqn−1 qξ + µl2 qξ ξ + bul2 qξ ξ ξ )ξ l + m2 rqξ ξ = l2 fξ ξ . Considering f = f∗ cos(ωξ ), and integrating twice Eq. (5) takes the form − uq + alqn + µl2 qξ + bul2 qξ ξ +
m2 r
q = lf∗ cos(ωξ ). l Then system (6) is equivalent to the following dynamical system: qξ = z , zξ = Aq − Bqn − Dz + f0 cos(ωξ ),
(5)
(6)
lu−(1−l2 )r
µ
(7)
a ∗ where A = , B = blu , D = bu and, f0 = blu is the strength of the external periodic perturbation and ω is the bl3 u frequency. Eq. (7) is similar to the well known duffing oscillator [27]. As an oscillator, it has tremendous applications in Biomathematical modeling [28] of the aneurysm of circle of Willis which simulates the blood flow inside aneurysm and practical engineering in case of vibration of plate spring in rare earth giant magnetostriction transducer. The remaining part of the paper is organized as follows: In Section 2, we consider unperturbed dynamical system and corresponding qualitative analysis. In Section 3, we study periodic and chaotic motions in the perturbed dynamical system. Section 4 is kept for conclusions. f
2. Unperturbed dynamical system and qualitative analysis In this section, we study the qualitative behavior of the unperturbed traveling wave system by using the bifurcation theory of planar dynamical system [29]. To do that, we assume f0 = 0 and the system (7) takes the form
qξ = z , zξ = Aq − Bqn − Dz .
(8)
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Fig. 1. (a) Phase portrait of the dynamical system (8), (b) plot of z vs. ξ , for n = 2, a = 0.08, b = 1, r = 0.08, l = 0.2, u = 0.4, and µ = 0.12.
Fig. 2. (a) Phase portrait of the dynamical system (8), (b) plot of z vs. ξ , for n = 3, a = 0.08, b = 1, r = 0.08, l = 0.2, u = 0.4, and µ = 0.12.
The system (8) is a planar dynamical system with parameters a, b, r , l, µ and u. It is important to note that the phase orbits defined by the vector fields of Eq. (8) will determine all traveling wave solutions of Eq. (3). We study the bifurcations of phase portraits of Eq. (8) in the (q, z ) phase plane depending on the parameters a, b, r , l, µ and u. If n is an odd 1
integer, the dynamical system (8) has three equilibrium points at P0 (0, 0), P1 (q1 , 0) and P2 (q2 , 0), where q1 = ( AB ) n−1 and 1
q2 = −( AB ) n−1 . If n is an even integer, the dynamical system (8) has two equilibrium points at P0 (0, 0) and P1 (q1 , 0). The determinant of the Jacobian matrix of the linearized system of the traveling system (8) is
0 J = A − Bnqn−1 = −A + Bnqn−1 .
1 −D (9)
At P0 (0, 0), the value of the determinant is J (0, 0) = −A < 0. At P1,2 (q1,2 , 0), the value of the determinant is J (q1,2 , 0) =
√ −D± D2 +4A (n − 1)A > 0. The eigenvalues at P0 (0, 0) are λ = , which are real and are of opposite signs. Hence, the 2 equilibrium point P0 (0, 0) is a saddle point. The eigenvalues at P1,2 (q1,2 , 0) are λ = −α ± iβ , where α = D2 and β = 21 4A(n − 1) − D2 with 4A(n − 1) > D2 . Hence, the equilibrium points P1,2 (q1,2 , 0) are stable spirals. The phase portrait of the system (8) is shown in Fig. 1(a) for n = 2, a = 0.08, b = 1, r = 0.08, l = 0.2, u = 0.4, and µ = 0.12. In this case the dynamical system (8) has two equilibrium points at P0 (0, 0) and P1 (q1 , 0), where P0 (0, 0) is a saddle point and it represents a stable spiral at the equilibrium point P1 (q1 , 0). In Fig. 1(b), the variation of z with ξ is presented with same values of parameters as Fig. 1(a). In Fig. 2(a), the phase portrait of the dynamical system (8) is shown for n = 3 with same values of other parameters as Fig. 1. In this case the dynamical system (8) has three equilibrium points at P0 (0, 0), P1 (q1 , 0) and P2 (q2 , 0), where P0 (0, 0) is a saddle point and we have two stable spirals at the equilibrium points P1 (q1 , 0) and P2 (q2 , 0). In Fig. 2(b), we have plotted z vs. ξ with same values of parameters as Fig. 2(a). It is important to note that the wave particle moves to the right with stable oscillation which decays as ξ increases (see Figs. 1(b) and 2(b)). The system (8) with µ > 0 is dissipative and there is neither solitary nor periodic wave solutions as there does not exist any closed trajectory in the phase plane by Bendixson’s criterion [30] (see Figs. 1 and 2).
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Fig. 3. (a) Phase projection of the perturbed KP-MEW-Burgers equation (7), (b) plot z vs. ξ for n = 3, a = 0.015, b = 1, l = 0.2, r = 0.08, u = 0.4, µ = 0.12, ω = 1.25 and f0 = 0.5.
If there is no damping, i.e. µ = 0, then the system (8) takes the form
dq = z, dξ dz = Aq − Bqn , dξ
(10)
which is a planar Hamiltonian system with Hamiltonian function: H (q, z ) =
z2
−A
q2
+B
qn+1
.
2 2 n+1 The bifurcation of phase portraits of the system (10) is already reported in the work [16].
(11)
3. Periodic and chaotic motions in perturbed dynamical system In this section, we study the periodic and chaotic motions of the perturbed dynamical system (7) of the KP-MEWBurgers equation (3). In order to study all possible periodic and chaotic motions of the perturbed dynamical system (7), we apply different numerical tools that help to corroborate effectively periodic and chaotic phenomena [27] are: (i) the phase projection plots, (ii) time series plots, (iii) bifurcation diagram and (iv) the Poincaré section. The perturbed system (7) involves eight independent parameters a, b, l, r , u, µ, ω and f0 . Because of large number of parameters, it is really difficult to investigate the system for the complete range of parametric space. To simplify the task, we consider special fixed values of the parameters b, l, r , u, µ, ω, f0 varying parameter a associated with the nonlinear term. Thus, parameter a plays the role of bifurcation parameter. Of course one can vary any of the other parameters b, l, r , u, µ, ω, f0 , but, this will not give any significant different qualitative features. 3.1. Phase projection and time series analysis Phase projection analysis is one of the most important techniques for investigating the behavior of nonlinear systems, as there is no usual analytical solution for a nonlinear system. On the other hand, time series analysis consists of methods for analyzing time series data in order to bring out meaningful characteristics of the data. Time series forecasting is the use of a model to predict future values based on previously observed values. In Fig. 3(a), the phase space trajectory of the system (7) is presented for n = 3, a = 0.015, b = 1, l = 0.2, r = 0.08, u = 0.4, µ = 0.12, ω = 1.25 and f0 = 0.5. It shows the period-1 motion of the system (7) and the corresponding variation of z with ξ is depicted in Fig. 3(b). The system (7) shows period-2 motion for a = 0.029 with same values of other parameters as Fig. 3(a), depicted in Fig. 4(a). The corresponding variation of z vs. ξ is presented in Fig. 4(b) with same values of parameters as Fig. 4(a). In Fig. 5(a), the phase space trajectories of the system (7) are presented for a = 0.317 with same values of other parameters as Fig. 3(a). It shows period-4 motion of the system (7) and the corresponding variation of z with ξ is depicted in Fig. 5(b). In Fig. 6(a), the phase space trajectories of the system (7) are presented for a = 0.03502 with same values of other parameters as Fig. 3(a). It shows chaotic motion of the system (7) through period doubling route to chaos and the corresponding variation of z with ξ is depicted in Fig. 6(b). Furthermore, if we again increase the value of a (i.e., a = 0.043) keeping other parameters fixed as Fig. 3(a), then we reach to the state of period-3 motion, shown in Fig. 7(a) and the corresponding variation of z with ξ is presented in Fig. 7(b). In Fig. 8(a), the phase space trajectories of the system (7) are presented for a = 0.05 with same values of other parameters as Fig. 3(a). It shows period-5 motion of the system (7) and the corresponding variation of z with ξ is depicted in Fig. 8(b). If we further increase the value of a (i.e., a = 0.079) keeping other parameters fixed as Fig. 3(a), then we again reach to the chaotic state, shown in Fig. 9(a) and the corresponding variation of z with ξ is presented in Fig. 9(b).
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Fig. 4. (a) Phase projection of the perturbed KP-MEW-Burgers equation (7), (b) plot z vs. ξ for a = 0.029 with same values of other parameters as Fig. 3.
Fig. 5. (a) Phase projection of the perturbed KP-MEW-Burgers equation (7), (b) plot z vs. ξ for a = 0.0317 with same values of other parameters as Fig. 3.
Fig. 6. (a) Phase projection of the perturbed KP-MEW-Burgers equation (7), (b) plot z vs. ξ for a = 0.03502 with same values of other parameters as Fig. 3.
3.2. Poincaré return map analysis and bifurcation diagram A very useful tool for exploring chaotic attractors is Poincaré sections [31,32]. Poincaré sections provide us with images of the attractor in a (n − 1)-dimensional hyperplane of n-dimensional phase space. The Poincaré return map of the perturbed system (7) is shown in Fig. 10(a). In return map plots, Fig. 10(a) demonstrates that the points are densely scattered and exhibit irregular distributions without any known pattern for the system (7) for the parameter values f0 = 0.079 keeping other parameters fixed as in Fig. 3(a). Such return maps are characteristics of chaotic oscillations. Considering parameter a (coefficient of nonlinear term) as the bifurcation parameter, the bifurcation diagram in the (a − q) plane of the system (7) is shown in Fig. 10(b). One can easily verify different types of periodic motions (period-1, period-2, period-3, period-4 and period-5) and chaotic motions corresponding to the values of the parameter a (0.015, 0.0295, 0.043, 0.0317 and 0.05 respectively) in Fig. 10(b). 4. Conclusions The generalized KP-MEW-Burgers equation is introduced and investigated for the first time. The bifurcation behavior of the KP-MEW-Burgers equation is obtained. The stable oscillations of the KP-MEW-Burgers equation are presented. Considering an external periodic perturbation, the periodic and chaotic motions of the perturbed KP-MEW-Burgers equation
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Fig. 7. (a) Phase projection of the perturbed KP-MEW-Burgers equation (7), (b) plot z vs. ξ for a = 0.043 with same values of other parameters as Fig. 3.
Fig. 8. (a) Phase projection of the perturbed KP-MEW-Burgers equation (7), (b) plot z vs. ξ for a = 0.05 with same values of other parameters as Fig. 3.
Fig. 9. (a) Phase projection of the perturbed KP-MEW-Burgers equation (7), (b) plot z vs. ξ for a = 0.079 with same values of other parameters as Fig. 3.
Fig. 10. (a) Poincare section of the chaotic state in Fig. 9, (b) bifurcation diagram in the (a − q) plane with same values of other parameters as Fig. 9.
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are investigated with the help of phase projection analysis, time series analysis, Poincaré section and bifurcation diagram. The parameter a (nonlinear coefficient) plays as the bifurcation parameter in bifurcations, periodic motions and chaotic motions of the perturbed KP-MEW-Burgers equation. It is seen that the perturbed KP-MEW-Burgers equation possesses chaotic motions through period doubling route to chaos. Acknowledgments The constructive suggestions of the editor and the reviewers are gratefully acknowledged. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32]
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