# A New Approach of Asymmetric Homoclinic and Heteroclinic Orbits Construction in Several Typical Systems Based on the Undetermined Pade Approximation Method.

1. IntroductionHeteroclinic orbits (HOs) play a central role in nonlinear dynamics and many scholars have undertaken the study of HOs and heteroclinic bifurcation [1]. In mathematics, a HO (or heteroclinic connection) is a path in phase space which joins two different equilibrium points ([x.sub.0] and [x.sub.1])of a dynamical system. A HO is contained in the stable manifold of [mathematical expression not reproducible] and the unstable manifold of [mathematical expression not reproducible] [2] as shown in Figure 1. There are many physical applications which correspond to the homoclinic and heteroclinic orbits [3-5], such as the pull-in phenomenon in MEMS [6] and the self-contact problem of elastic rods [7].

Existing research of HOs is focused mainly on classic low-order [Z.sub.2] symmetric system. The study of [Z.sub.2] symmetric HOs in an unperturbed system has resulted in the development of a number of analytical methods. Some examples include the generalized harmonic multiple scales method [8], the generalized hyperbolic perturbation method [9], and the elliptic Lindstedt-Poincare (LP) method [10]. HOs in classic low-order autonomous systems, which consider perturbations of a [Z.sub.2] symmetric system, are also well understood. For example, analysis of weak nonlinear systems employs techniques such as the elliptic perturbation method [11], the hyperbolic perturbation method [12], and the method developed by Izydorek and Janczewska [13]. Similarly, in a strongly nonlinear system, the undetermined fundamental frequency method [14], the modification of the perturbation-incremental method [15], the hyperbolic LP method [16], and the generalized harmonic function perturbation method [17] are all applicable analysis techniques.

In the aforementioned techniques, the HOs all meet the characteristics that the saddle points are symmetric about a center point. The more general asymmetric case remains to be further explored where the non-[Z.sub.2] symmetric HOs are neither symmetric about x- (or y-) axis nor symmetric about the origin. Special non-[Z.sub.2] symmetric HOs with some complicated properties (orbit with cusps) were discovered when concerning the number of limit cycles and their relative locations in polynomial vector fields. The existence of these orbits has also been proved recently [18-20]. In fact, regardless of the number of cusps contained in the orbits, their properties can be discussed according to the symmetry along the x-axis and y-axis. In engineering applications, such as power systems, special non-[Z.sub.2] symmetric HOs have been found. Beyn [21] applied numerical methods to analyze the truncation error in bifurcation calculations and the non-[Z.sub.2] symmetric HOs, which rose the phenomenon of superconvergence, were discovered. Power system analysis by Pai and Laufenberg [22] revealed that local instabilities are due to the influence of non-[Z.sub.2] symmetric heteroclinic orbits, which means that the characteristics of the system will be a qualitative change. Petrisor [23] also found that non-[Z.sub.2] symmetric heteroclinic saddle points are present in the study of the dynamic characteristic of reversible magnetic vector fields. Since such orbits are more complicated than HOs with non-[Z.sub.2] symmetry in classic autonomous low-order systems, these orbits are still being solved by numerical methods [24-26]. To the authors' knowledge, existing analytical methods are rarely used for the analysis of these orbits due to the complexity of their application. Therefore, a more effective analytical method is necessary. A new analytical method based on the undetermined Pade approximation method is proposed in this paper. The application of this new method results in a higher accuracy of the achieved HOs. This method not only is applicable to totally asymmetric HOs but also shows its superiority in symmetric and conventional asymmetric HOs.

This paper is organized as follows: in Section 2, the symmetry characteristics of non-[Z.sub.2] symmetric HOs are categorized. After that, the proposed method, the "undetermined Pade approximant method," is introduced. In Section 3, analysis of HOs in three general systems (categorized by the complexity of the symmetry) demonstrates the validity and feasibility of the proposed method, which is verified against numerical simulations. Finally, the conclusions are presented in Section 4.

2. The Undetermined Pade Approximation Method

2.1. Symmetry Analysis of Non-[Z.sub.2] Symmetry HOs. Consider the following general non-[Z.sub.2] symmetric nonlinear dynamical system:

[mathematical expression not reproducible]. (1)

Suppose that system (1) has a center 0(0, 0) and two saddle points [H.sub.1]([H.sup.l], 0) and [H.sub.2]([H.sup.r], 0) which are located on both sides of the center and [epsilon] represents the perturbation parameter. The corresponding HOs of system (1) are shown in Figure 2. When [epsilon] = 0, the unperturbed system is conservative and the orbit trajectories (the dotted line in Figure 2) have extreme values in the x-axis direction corresponding to the saddle points:

f([H.sup.l]) = 0, f([H.sup.r]) = 0. (2)

The extreme values of the trajectories in the vertical direction are

[??](O) = 0. (3)

In other words, the maximum value of the orbit in the x-axis direction lies in the center O.

When a small perturbation is applied ([epsilon] [not equal to] 0), HOs still exist in this autonomous system. There will be a critical orbit close to the saddle points, as shown by the solid line in Figure 2. In this case, the maximum value of the orbit in the [??]-axis direction does not lie on the center O; that is,

[??](O) [not equal to] 0. (4)

The non-[Z.sub.2] symmetric HOs of simple autonomous system have been widely studied, as shown by the solid line in Figure 2(a). The unperturbed system has [Z.sub.2] symmetric HOs, shown by the dotted line in Figure 2(a), where the saddle points fulfill the criteria

O[H.sup.r] = O[H.sup.l]. (5)

In other words, the distances between each saddle point and the center are equal.

However, in some complex systems, the unperturbed system will also result in a non-[Z.sub.2] symmetric HO phenomenon where the saddle points satisfy the following condition:

O[H.sup.r] [not equal to] O[H.sup.l]. (6)

In other words, the distances between each saddle point and the center are not equal. As shown in Figure 2(b), the HOs of the unperturbed system (dotted lines) and the HOs of the autonomous system (solid lines) satisfy condition (6); that is, all HOs are non-[Z.sub.2] symmetric. The solid line trajectories shown in Figure 2(b) correspond to those totally asymmetric HOs discussed previously in Section 1.

A trajectory that does not possess [Z.sub.2] symmetry characteristics is classified as non-[Z.sub.2] symmetry. This means that the saddle points are asymmetric, O[H.sup.r] [not equal to] O[H.sup.l], or the maximum points of the trajectory are not on the [+ or -]-axis, or both cases happened simultaneously. There is no guarantee that analytical methods which are suitable for solving [Z.sub.2] symmetric HOs will return the required level of accuracy if applied to a non-[Z.sub.2] symmetric HO. For example, when the saddle point [H.sup.r] does not coincide with -[H.sup.l], the calculations will show that the obtained results cannot pass through both saddle points. If the convergence conditions are that the heteroclinic solution finally tends to one saddle point [H.sup.r](-[H.sup.l]), then the determined trajectory will shift to this saddle point [H.sup.r](-[H.sup.l]). Thus, complete orbits cannot go through another saddle point--[H.sup.l]([H.sup.r]). As shown in Figure 3, (1) represents that the analytical results calculated by employing the method discussed in [27] and (2) are numerical results.

In order to resolve this problem, the Pade approximation method [28, 29] is used. The extended Pade approximation [27, 30-32] has proven its validity and reliability through its simple arithmetic process and accurate results in solving the non-[Z.sub.2] symmetric homoclinic orbit. The solving process of this superior method can be further enhanced and is presented in detail in the next section.

2.2. The Calculation Procedure. In this section, the solving process of this analytical method is proposed. The differential equation, including the entire nonlinear part, is solved directly. In addition, the magnitude of the perturbation parameter e does not need to be ascertained in advance. It has universality and can directly solve any symmetric case of critical orbits.

A generic case, such as the solid line orbits shown in Figure 2(b), is taken for example. This trajectory, which is completely nonsymmetric, simultaneously satisfies conditions (4) and (6). Therefore, the solution achieved in this example is universal and versatile.

First, without loss of generality, system (1) is rewritten as the following general form (considering a one-degree-of-freedom general nonlinear system):

[mathematical expression not reproducible], (7)

where F(X, dX/dt, [epsilon]) is a nonlinear term including damping and [[omega].sub.0] is the natural frequency of the system. It is obvious that system (7) is an autonomous system ([epsilon] [not equal to] 0) and we can assume that system (7) has HOs which are non-[Z.sub.2] symmetric.

Take a series solution of system (7) as

x(t) = [a.sub.0] + [a.sub.1]t + ... = [[infinity].summation over (n=0)] [a.sub.n][t.sup.n], (8)

with ([a.sub.0], [a.sub.1]) being the initial value of the trajectory point such that

x (0) = [a.sub.0], [??] (0) = [a.sub.1], (9)

where the undetermined parameters [a.sub.0] and [a.sub.1] are arbitrary nonzero constants.

The rest of the parameters of (8) can be expressed as a function of [a.sub.0] and [a.sub.1] as follows:

[mathematical expression not reproducible]. (10)

Usually, the initial point is taken as one of the maximum points on the trajectory where it also fulfills its tangent perpendicular to the [??]-axis; namely,

[a.sub.2] = 0. (11)

If system (7) is a conservative system ([epsilon] = 0), then the maximum point is on the [??]-axis. In this case, condition (11) above can be simplified to

[a.sub.0] = 0. (12)

According to the abovementioned initial conditions, we consider a new heteroclinic solution expression given by

[mathematical expression not reproducible]. (13)

An undetermined constant [omega] is introduced into expression (13). Compared with the traditional method, the undetermined constant [omega] not only reflects the original frequency [[omega].sub.0] but also contains the effect of the nonlinear term F(X, dX/dt, [epsilon]).

Expression (13) satisfied the following properties:

[mathematical expression not reproducible]. (14)

The form of the expression ensures its convergence. Meanwhile, the convergence speed of the whole calculation process is greatly accelerated, reducing the amount of computation time required. This expression can resolve the orbits in both autonomous and conservative systems with the boundary conditions [a.sub.0] and [[alpha].sub.n]/[[beta].sub.n].

When t tends to infinity, the heteroclinic solutions tend to [H.sup.l] and [H.sup.r]:

[mathematical expression not reproducible]. (15)

For the upper orbit, the boundary conditions can be expressed as left boundary [a.sub.0] [right arrow] [H.sup.l] and right boundary [[alpha].sub.n]/ [[beta].sub.n] [right arrow] [H.sup.r]. For the lower orbit, left boundary [[alpha].sub.n]/[[beta].sub.n] [right arrow] [H.sub.l]; right boundary [a.sub.0] [right arrow] [H.sup.r]. The abovementioned relationships are illustrated in Figure 4.

By equating the heteroclinic solution [QPA.sub.n] to series solution (8),

[[infinity].summation over (n=0)] [a.sub.n][t.sup.n] [??] [QPA.sub.n]. (16)

A functional relationship between the coefficients [[alpha].sub.i] and [[beta].sub.i] (heteroclinic solution [QPA.sub.n]) and coefficient [a.sub.i] (series solution) is achieved by comparing terms of (16) to the same order of t. This gives the following:

[mathematical expression not reproducible]. (17)

Equations (17) show that all unknown coefficients [[alpha].sub.i] and [[beta].sub.i] of [QPA.sub.n] can be expressed as a function of [a.sub.0] and [a.sub.1].

By applying boundary conditions (14) and (15), the initial values [a.sub.0] and [a.sub.1] and the undetermined constant [omega] can be calculated. When the determined values of [a.sub.0] and [a.sub.1] are substituted into (17), every coefficient of [QPA.sub.n] can be determined and even the higher order expressions are achievable. For instance, the specific expression of the fourth-order heteroclinic solution is given as follows:

[mathematical expression not reproducible]. (18)

The higher the approximation order, the better the accuracy. Ordinarily, a calculation to second-order approximation ([QPA.sub.2]) would meet the accuracy requirements (error range is [DELTA] < 1%) for low-order [Z.sub.2] symmetric systems (only with cubic nonlinearity). Similarly, a calculation to fourth-order approximation ([QPA.sub.4]) would meet very high accuracy, generally required for systems with non-[Z.sub.2] symmetric terms (i.e., with square or fourth nonlinearity) or systems with high-order nonlinear terms (i.e., with fifth nonlinearity and so on).

3. Some Examples

In this section, the analytical approach proposed is applied to several classical nonlinear systems.

3.1. The Nagumo System. The HO in this system meets the characteristics that the two distances between each saddle point and the center are not equal (condition (6)), and the maximum value of the orbit in the [??]-axis direction dose not lie in the center (condition (4)).

The Nagumo system [22] is defined as

[mathematical expression not reproducible], (19)

where [c.sub.1] = [square root of 2](1/2 - [c.sub.2]) and 0 < [c.sub.2] < 1/2. For all values of [c.sub.2], system (19) has one center 0([c.sub.2], 0) and a HO connecting the two saddle points ([H.sub.1](0, 0) and [H.sub.2](1, 0)). The phase diagram is shown in Figure 5 with [c.sub.2] = 2/5.

The maximum point of the trajectory of autonomous system (19) is no longer at the center O and the saddle points are asymmetric (O[H.sub.1] > O[H.sub.2]). Using the present method, coefficient expressions (10) can be written as the following series of coefficients:

[mathematical expression not reproducible], (20)

Using condition (11), we find that

[a.sub.1] = -[[a.sub.0] (-1 + [a.sub.0])([a.sub.0]--[c.sub.2])/[c.sub.1]]. (21)

The fourth-order heteroclinic solution can be achieved making use of (17). By applying boundary conditions (14) and (15), the initial value [a.sub.0] = 0.5 and undetermined constant [omega] = 0.49999 are calculated. The analytical heteroclinic solution (A.1) is also obtained. The superscript "+" indicates the solution with an initial value [a.sub.1] > 0, which is the upper half of the trajectory. Similarly, the superscript "-" indicates the solution with an initial value [a.sub.1] < 0 corresponding to the lower half of the trajectory. The corresponding phase diagram is shown in Figure 6 with [a.sub.0] = 0.5 and [a.sub.1] = 0.176776. The error of the maximum value of the orbit in the [??]-axis direction between analytical results and numerical results is 0.00057%.

3.2. General Non-[Z.sub.2] Symmetric Nonlinear Quintic System. The HOs (with one cusp) in this system meet the characteristics that the two distances between each saddle point and the center are not equal (condition (6)). When [epsilon] = 0 the maximum value of the orbit in the x-axis direction lies in the center (condition (3)). If [epsilon] [not equal to] 0, the maximum value of the orbit in the [??]-axis direction dose not lie in the center (condition (4)). Very few analytical methods exist for solving the problem with a fifth-order nonlinearity. Even fewer methods are available to analyze the problem of complex non-[Z.sub.2] symmetry systems. However, the undetermined Pade approximation can be employed to solve such problems with high efficiency and precision.

Consider the following autonomous system containing non-[Z.sub.2] symmetric term:

[mathematical expression not reproducible], (22)

which has a rich physical meaning and is widely used in many fields. Equation (22) contains nonlinear terms from secondary to quintic, which is a typical non-[Z.sub.2] symmetric system. System (22) has five equilibrium points, so the corresponding stable solution and its phase diagram have a variety of possible cases and complex dynamic behavior. Therefore, system (22) has a good generality, which encompasses the systems discussed in the literature [9, 16, 17, 30]. Here, we study the cases in which all nonlinear terms of (22) exist. In this scenario, its system has a pair of non-[Z.sub.2] symmetric homoclinic orbits and a pair of non-[Z.sub.2] symmetric heteroclinic orbits. Those heteroclinic orbits have asymmetric saddle points and a maximum point which intersects the [??]-axis.

Firstly, we analyze unperturbed system (23) of system (22) where [epsilon] = 0:

[?] + ([c.sub.1]x + [c.sub.2][x.sup.2] + [c.sub.3][x.sup.3] + [c.sub.4][x.sup.4] + [c.sub.5][x.sup.5]) = 0. (23)

The case when conservative system (23) has five equilibrium points is studied, and any singular point can be selected as the zero point. For the parameters used in Figures 7 and 8, system (23) has three centers and two saddle points, a pair of non-[Z.sub.2] symmetric homoclinic orbits, and a pair of non-[Z.sub.2] symmetric HOs (with one cusp). There are two kinds of orbits when the parameters [c.sub.i] take different values: one is where [H.sup.r] > [absolute value of [H.sup.l]] shown in Figure 7 (cusp towards right) and the other one is [H.sup.r] < [absolute value of [H.sup.l]] shown in Figure 8 (cusp towards left). For brevity, the calculation process of Figures 7 and 8 is not presented here in this paper.

By applying this method, the initial values [a.sub.1] = [+ or -]0.904032 and [omega] = 1.20207 are obtained. The analytical heteroclinic solutions of system (23) are formulas (A.2) and (A.3). Similarly, analytical expressions of a pair of non-[Z.sub.2] symmetric homoclinic orbits can also be obtained by using this method. The analytical results for the upper HO in Figure 7 return the parameter values [a.sub.0] = 0 and [a.sub.1] = 0.904032 and the numerical results predict [a.sub.0] = 0 and [a.sub.1] = 0.900371. The error of the maximum value of the orbit in the [??]-axis direction between the analytical results and the numerical results is 0.4%. Results for the lower orbit are similar to the upper orbit with an error of 0.4%.

In the perturbed system ([epsilon] [not equal to] 0), trajectories in autonomous system (22) meet characteristics (4) and (6). It should be noted here that the perturbation parameter [epsilon] cannot be too large; otherwise, the phase diagram of system (22) will be significantly affected. When [[mu].sub.1] = 0, [[mu].sub.2] = -1, and [epsilon] = 0.01, the value of heteroclinic bifurcation parameter is [mu] = -0.155. By applying the analytical method developed in this paper, we obtain the analytic heteroclinic solutions (A.4) and (A.5) and the corresponding phase diagram of system (22) is shown in Figure 9. The analytical results for the upper orbit are [a.sub.0] = -0.000588768 and [a.sub.1] = 0.908729 with the corresponding numerical results giving [a.sub.0] = -0.000596024 and [a.sub.1] = 0.900226. The error of the maximum value of the orbit in the [??]-axis direction between the analytical results and the numerical results is 0.9%. Similarly, the analytical results for the lower orbit are [a.sub.0] = 0.000585934 and [a.sub.1] = -0.904105 with the numerical results returning [a.sub.0] = 0.000590165 and [a.sub.1] = -0.900514. The error of the maximum value of the orbit in the x-axis direction between the analytical results and the numerical results is 0.4%.

3.3. General [Z.sub.2] Symmetric System with High-Order Nonlinear Terms. The HOs (with two cusps) in this system meet the characteristics that the distances between each saddle point and the center are equal (condition (5)). If [epsilon] = 0, the maximum value of the orbit in the [??]-axis direction lies in the center (condition (3)). If [epsilon] [not equal to] 0, the maximum value of the orbit in the [??]-axis direction does not lie in the center (condition (4)).

Consider the following triple-well [Z.sub.2] symmetric autonomous system:

[mathematical expression not reproducible]. (24)

When [epsilon] = 0, this conservative system is [Z.sub.2] symmetric and has [Z.sub.2] symmetric HOs (with two cusps). As shown in Figure 10, when [c.sub.1] = 1, [c.sub.3] = -2, and [c.sub.5] = 1, the saddle points of this conservative system are [+ or -]1. The application of this method is also available and precise and the analytical heteroclinic solutions are (A.6) and (A.7). The analytical results for the upper orbit are [a.sub.0] = 0 and [a.sub.1] = 0.586815 and the numerical results are [a.sub.0] = 0 and [a.sub.1] = 0.57735. The error of the maximum value of the orbit in the x-axis direction between the analytical results and the numerical results is 1.6%.

When [epsilon] [not equal to] 0, the autonomous system has non-[Z.sub.2] symmetric HOs (with two cusps). Bifurcation curves, as shown in Figure 11, can be obtained by applying this method. When [[mu].sub.1] = 0, [[mu].sub.2] = -1, and [epsilon] = 1, the value of the bifurcation parameter obtained by this method is [mu] = -0.172755 and by numerical methods [mu] = -0.1718. If [mu] = -0.1718 is used in the autonomous system (24), the analytic heteroclinic solutions (A.8) and (A.9) are obtained and are shown in the phase diagram (Figure 12). High precision results can be seen clearly. The analytical results for the upper orbit are [a.sub.0] = 0.101118 and [a.sub.1] = -0.609388 and the numerical results are [a.sub.0] = 0.0964347 and [a.sub.1] = -0.598408. The error of the maximum value of the orbit in the x-axis direction between the analytical results and the numerical results is 1.8%.

The results obtained from all of the examples presented demonstrate the very high precision of the analytical method developed. This is reflected in the phase diagram with the very close superposition of the analytic and numerical orbits.

4. Conclusions

The undetermined Pade approximation method is proposed to construct general non-[Z.sub.2] symmetric homoclinic and heteroclinic orbits. The geometry characteristics of the orbits and the original frequency effects due to nonlinear terms are all considered. An undetermined frequency parameter and new analytical expression are given. The convergence rate of progressive approximations is accelerated and the amount of computation time required is greatly reduced. The characteristics of the special heteroclinic orbits are discussed in detail and analytical expressions of those orbits are obtained by the method developed in this paper. Consequently, irrespective of whether the orbit is single or in pairs, [Z.sub.2] or non-[Z.sub.2] symmetric, and in a conservative or autonomous system, the proposed method demonstrates its superiority over other existing solutions and the scope of its application has been extended. In addition, when compared with numerical results, the analytic orbits and the values of the bifurcation parameters obtained by the method presented in this paper are almost identical.

Appendix

Consider the following

[mathematical expression not reproducible], (A.1)

[mathematical expression not reproducible], (A.2)

[mathematical expression not reproducible], (A.3)

[mathematical expression not reproducible], (A.4)

[mathematical expression not reproducible], (A.5)

[mathematical expression not reproducible], (A.6)

[mathematical expression not reproducible], (A.7)

[mathematical expression not reproducible], (A.8)

[mathematical expression not reproducible], (A.9)

http://dx.doi.org/10.1155/2016/8585290

Competing Interests

The authors declare that they have no competing interests.

Acknowledgments

This project is supported by National Natural Science Foundation of China (Grant nos. 11372210 and 11402172) and Tianjin Research Program of Application Foundation and Advanced Technology (16JCQNJC04700).

References

[1] A. H. Nayfeh and D. T. Mook, Nonlinear Oscillations, Wiley Press, New York, NY, USA, 2008.

[2] J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, vol. 42 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1983.

[3] S. C. Stanton, C. C. McGehee, and B. P. Mann, "Nonlinear dynamics for broadband energy harvesting: investigation of a bistable piezoelectric inertial generator," Physica D: Nonlinear Phenomena, vol. 239, no. 10, pp. 640-653, 2010.

[4] P. Yu, W. Zhang, and L. M. Wahl, "Dynamical analysis and simulation of a 2-dimensional disease model with convex incidence," Communications in Nonlinear Science & Numerical Simulation, vol. 37, pp. 163-192, 2016.

[5] M. Yao, W. Zhang, and D. M. Wang, "Modeling and chaotic dynamics of the laminated composite piezoelectric rectangular plate," Mathematical Problems in Engineering, vol. 2014, Article ID 345072, 19 pages, 2014.

[6] A. H. Nayfeh, M. I. Younis, and E. M. Abdel-Rahman, "Dynamic pull-in phenomenon in MEMS resonators," Nonlinear Dynamics, vol. 48, no. 1-2, pp. 153-163, 2007.

[7] J. Feng, Q. Zhang, and W. Wang, "Spatially complex localization in twisted elastic rods constrained to an elliptic cylinder," Journal of Physics: Conference Series, vol. 448, no. 1, pp. 140-144, 2013.

[8] Z. Xu and Y. K. Cheung, "A non-linear seales method for strongly non-linear oscillators," Nonlinear Dynamics, vol. 7, no. 3, pp. 285-299, 1995.

[9] Y.-Y. Chen, L.-W. Yan, K.-Y. Sze, and S.-H. Chen, "Generalized hyperbolic perturbation method for homoclinic solutions of strongly nonlinear autonomous systems," Applied Mathematics and Mechanics, vol. 33, no. 9, pp. 1137-1152, 2012.

[10] M. Belhaq, B. Fiedler, and F. Lakrad, "Homoclinic connections in strongly self-excited nonlinear oscillators: the Melnikov function and the elliptic Lindstedt-Poincare method," Nonlinear Dynamics, vol. 23, no. 1, pp. 67-86, 2000.

[11] S. H. Chen, X. M. Yang, and Y. K. Cheung, "Periodic solutions of strongly quadratic non-linear oscillators by the elliptic perturbation method," Journal of Sound and Vibration, vol. 212, no. 5, pp. 771-780, 1998.

[12] S. H. Chen, Y. Y. Chen, and K. Y. Sze, "A hyperbolic perturbation method for determining homoclinic solution of certain strongly nonlinear autonomous oscillators," Journal of Sound and Vibration, vol. 322, no. 1-2, pp. 381-392, 2009.

[13] M. Izydorek and J. Janczewska, "Homoclinic solutions for a class of the second order Hamiltonian systems," Journal of Differential Equations, vol. 219, no. 2, pp. 375-389, 2005.

[14] Q.-C. Zhang, W. Wang, and W.-Y. Li, "Heteroclinic bifurcation of strongly nonlinear oscillator," Chinese Physics Letters, vol. 25, no. 5, pp. 1905-1907, 2008.

[15] Y. Y. Cao, K. W. Chung, and J. Xu, "A novel construction of homoclinic and heteroclinic orbits in nonlinear oscillators by a perturbation-incremental method," Nonlinear Dynamics, vol. 64, no. 3, pp. 221-236, 2011.

[16] S. Chen, Y. Chen, and K. Y. Sze, "Homoclinic and heteroclinic solutions of cubic strongly nonlinear autonomous oscillators by hyperbolic Lindstedt-Poincare method," Science China Technological Sciences, vol. 53, no. 3, pp. 692-702, 2010.

[17] Z. Li, J. Tang, and P. Cai, "A generalized harmonic function perturbation method for determining limit cycles and homoclinic orbits of Helmholtz--Duffing oscillator," Journal of Sound and Vibration, vol. 332, no. 21, pp. 5508-5522, 2013.

[18] M. Han, "Asymptotic expansions of Melnikov functions and limit cycle bifurcations," International Journal of Bifurcation and Chaos, vol. 22, no. 12, pp. 583-595, 2013.

[19] X. Sun, "Bifurcation of limit cycles from a Lienard system with a heteroclinic loop connecting two nilpotent saddles," Nonlinear Dynamics, vol. 73, no. 1-2, pp. 869-880, 2013.

[20] J. Wang, D. Xiao, and M. Han, "The number of zeros of abelian integrals for a perturbation of hyperelliptic Hamiltonian system with degenerated polycycle," International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 23, no. 3, Article ID 1350047, 18 pages, 2013.

[21] W.-J. Beyn, "Global bifurcations and their numerical computation," in Continuation and Bifurcations: Numerical Techniques and Applications, pp. 169-181, Springer, 1990.

[22] M. A. Pai and M. Laufenberg, "On the computation of heteroclinic orbits in dynamical systems," in Proceedings of the IEEE International Symposium on Circuits and Systems (ISCAS '95), pp. 151-154, May 1995.

[23] E. Petritor, "Heteroclinic connections in the dynamics of a reversible magnetic-type vector field," Physica D. Nonlinear Phenomena, vol. 112, no. 1-2, pp. 319-327, 1998.

[24] Y. A. Kuznetsov, "Computation of invariant manifold bifurcations," in Continuation and Bifurcations: Numerical Techniques and Applications, pp. 183-195, Springer, 1990.

[25] E. Doedel, H. B. Keller, and J. P. Kernevez, "Numerical analysis and control of bifurcation problems (I): bifurcation in finite dimensions," International Journal of Bifurcation and Chaos, vol. 1, no. 3, pp. B69-B72, 2012.

[26] M. J. Friedman and E. J. Doedel, "Numerical computation and continuation of invariant manifolds connecting fixed points," SIAM Journal on Numerical Analysis, vol. 28, no. 3, pp. 789-808, 1991.

[27] J. J. Feng, Q. C. Zhang, and W. Wang, "Chaos of several typical asymmetric systems," Chaos, Solitons & Fractals, vol. 45, no. 7, pp. 950-958, 2012.

[28] E. Emaci, A. F. Vakakis, I. V. Andrianov, and Y. Mikhlin, "Study of two-dimensional axisymmetric breathers using Pade approximants," Nonlinear Dynamics, vol. 13, no. 4, pp. 327-338, 1997.

[29] G. V. Manucharyan and Yu. V. Mikhlin, "The construction of homo- and heteroclinic orbits in non-linear systems," Journal of Applied Mathematics and Mechanics, vol. 69, no. 1, pp. 39-48, 2005.

[30] Q. C. Zhang, J. J. Feng, and W. Wang, "Construction of homoclinic and heteroclinic orbits in two-dimensional nonlinear systems based on the quasi-Pade approximation," Chinese Journal of Theoretical and Applied Mechanics, vol. 43, no. 5, pp. 914-921, 2011.

[31] J.-J. Feng, Q.-C. Zhang, and W. Wang, "The construction of homoclinic and heteroclinic orbitals in asymmetric strongly nonlinear systems based on the Pade approximant," Chinese Physics B, vol. 20, no. 9, Article ID 090202, 2011.

[32] L. Liu, G. Moore, and R. D. Russell, "Computation and continuation of homoclinic and heteroclinic orbits with arclength parameterization," SIAM Journal on Scientific Computing, vol. 18, no. 1, pp. 69-93, 1997.

Jingjing Feng, (1) Qichang Zhang, (2) Wei Wang, (2) and Shuying Hao (1)

(1) Tianjin Key Laboratory for Control Theory & Applications in Complicated Systems, School of Mechanical Engineering, Tianjin University of Technology, Tianjin 300384, China

(2) Tianjin Key Laboratory of Nonlinear Dynamics and Control, Tianjin University, Tianjin 300072, China

Correspondence should be addressed to Jingjing Feng; jjfeng@tju.edu.cn

Received 24 December 2015; Revised 20 June 2016; Accepted 4 July 2016

Academic Editor: Stefano Lenci

Caption: Figure 1: Heteroclinic connection is a path joining two different equilibrium points.

Caption: Figure 2: HOs of system (1) with (a) symmetric saddle points and (b) asymmetrical saddle points.

Caption: Figure 3: Comparison of the asymmetric heteroclinic orbits found using (1) classic analytical techniques (O) and (2) numerical computation (solid).

Caption: Figure 4: The boundary condition for (a) the upper orbit and (b) the lower orbit.

Caption: Figure 5: The phase diagram of Nagumo system (19).

Caption: Figure 6: Comparison of HO calculated using the proposed analytical method ([omicron]) and a numerical Runge-Kutta algorithm (solid).

Caption: Figure 7: Comparison of the orbits calculated using the analytical method ([omicron]) and a numerical Runge-Kutta technique (solid), for [H.sup.r] > [absolute value of [H.sup.l]].

Caption: Figure 8: Comparison of the orbits calculated using the analytical method ([degrees]) and a numerical Runge-Kutta technique (solid), for [H.sup.r] < [absolute value of [H.sup.l]].

Caption: Figure 9: Comparison of HOs for the present results and the Runge-Kutta procedure.

Caption: Figure 10: HOs for the present results and the Runge-Kutta procedure comparison.

Caption: Figure 11: Heteroclinic bifurcation curve comparison.

Caption: Figure 12: HOs for the present results and the Runge-Kutta procedure comparison.

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Title Annotation: | Research Article |
---|---|

Author: | Feng, Jingjing; Zhang, Qichang; Wang, Wei; Hao, Shuying |

Publication: | Mathematical Problems in Engineering |

Date: | Jan 1, 2016 |

Words: | 5293 |

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