| [1] | Chow S N, Deng B, Terman D. The bifurcation of homoclinic and periodic orbits from two heteroclinic orbits[J]. SIAM J Math Anal,1990,21:179-204.
|
| [2] | Guckenheimer J, Holmes P. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields[M]. New York:Springer-Verlag,1983.
|
| [3] | Hale J K, Spezamiglio A. Perturbation of homoclinics and subharmonics in Duffing’s equation[J]. Nonlinear Anal:TMA,1985,9:181-192.
|
| [4] | He Z, Zhang W. Subharmonic bifurcations in a perturbed nonlinear oscillation[J]. Nonlinear Anal:TMA,2005,61:1057-1091.
|
| [5] | Zhu C. The coexistence of subharmonics bifurcated from homoclinic orbits in singular systems[J]. Nonlinearity,2008,21:285-303.
|
| [6] | Bulsara A R, Schieve W C, Jacobs E W. Homoclinic chaos in systems perturbed by weak Langevin noise[J]. Phys Rev,1990,A41:668-681.
|
| [7] | Deng G, Zhu D. Homoclinic and heteroclinic orbits for near-integrable coupled nonlinear Schrdinger equations[J]. Nonlinear Analysis:TMA,2010,73:817-827.
|
| [8] | Freddy D, Li C, Zhang Z. Unfolding of a quadratic integrable system with two centers and two unbounded heteroclinic loops[J]. J Diff Eqns,1997,139:146-193.
|
| [9] | Gan S, Wen L. Heteroclinic cycles and homoclinic closures for generic diffeomorphisms[J]. J Dyn Diff Eqns,2003,15:451-471.
|
| [10] | Han M, Hu S, Liu X. On the stability of double homoclinic and heteroclinic cycles[J]. Nonlinear Anal:TMA,2003,53:701-713.
|
| [11] | Li W, Lu K. Sternberg theorems for random dynamical systems[J]. Commun Pure Appl Math,2005,58:941-988.
|
| [12] | Lin X. Using Melnikov’s method to solve Silnikov’s problem[J]. Proc Roy Soc Edin,1990,A116:295-325.
|
| [13] | Lin X, Vivancos I B. Heteroclinic and periodic cycles in a perturbed convection model[J]. J Diff Eqns,2002,182:219-265.
|
| [14] | Liu B, Zanolin F. Boundedness of solutions of nonlinear differential equations[J]. J Diff Eqns,1998,144:66-98.
|
| [15] | Luo G, Liang J, Zhu C. The transversal homoclinic solutions and chaos for stochastic ordinary differential equations[J/OL]. J Math Anal Appl,2013,doi:10.1016/j.jmaa.2013.10.055.
|
| [16] | Palmer K J. Exponential dichotomies for almost periodic equation[J]. Proc Am Math Soc,1987,101:283-298.
|
| [17] | Palmer K J, Stoffer D. Chaos in almost periodic systems[J]. Z Angew Math Phys,1989,40:592-602.
|
| [18] | Palmer K J. Existence of transversal homoclinic points in a degenerate case[J]. Rocky Mount J Math,1990,20:1099-1118.
|
| [19] | Wiggins S. Global bifurcations and chaos-analytical methods[J]. New York:Springer-Verlag,1988.
|
| [20] | Zhang W. Bifurcation of homoclinics in a nonlinear oscillation[J]. Acta Math Sinica:Engl,1989,5:170-184.
|
| [21] | Deng B. The bifurcations of countable connections from a twisted heteroclinic loop[J]. SIAM J Math Anal,1991,22:653-679.
|
| [22] | Holmes P, Marsden J. Horseshoes in perturbations of Hamiltonian systems with two degrees of freedom[J]. Commun Math Phys,1981,82:523-544.
|
| [23] | Hale J K. Ordinary Differential Equations[M]. New York:Wiley-Interscience,1969.
|
| [24] | Chow S N, Hale J K. Methods of Bifurcation Theory[M]. New York:Springer-Verlag,1982.
|
| [25] | Palmer K J. Transversal heteroclinic orbits and Cherry’s example of a non-integrable hemiltonian system[J]. J Diff Eqns,1986,65:321-360.
|
| [26] | Zeng W. Exponential dichotomies and transversal homoclinic orbits in degenerate cases[J]. J Dyn Diff Eqns,1995,7:521-548.
|
| [27] | Melnikov V K. On the stability of the center for time periodic perturbations[J]. Trans Moscow Math Soc,1963,12:1-57.
|
| [28] | Neimark J I, Silnikov L P. A case of generation of periodic motions[J]. Soviet Math Docl,1965,6:1261-1264.
|
| [29] | Palmer K J. Exponential dichotomies and transversal homoclinic points[J]. J Diff Eqns,1984,55:225-256.
|
| [30] | Silnikov L P. A case of the existence of a countable number of periodic motions[J]. Soviet Math Dokl,1965,6:163-166.
|
| [31] | Silnikov L P. On a Poincaré-Birkhoff problem[J]. Math USSR-Sb,1967,3:353-371.
|
| [32] | Chow S N, Hale J K, Mallet-Paret J. An example of bifurcation to homoclinic orbits[J]. J Diff Eqns,1980,37:551-573.
|
| [33] | Hale J K. Bifurcation theory and applications[C]//Lecture Notes in Mathematics. Berlin:Springer-Verlag,1984,1057:106-151.
|
| [34] | Battelli F, Lazzari C. Exponential dichotomies, heteroclinic orbits, and Melnikov functions[J]. J Diff Eqns,1990,86:342-366.
|
| [35] | Battelli F, Palmer K J. Tangencies between stable and unstable manifolds[J]. Proc Roy Soc Edin,1992,A121:73-90.
|
| [36] | Fekan M. Bifurcation from degenerate homoclinics in periodically forced systems[J]. Discrete Contin Dyn Syst,1999,5:359-374.
|
| [37] | Gruendler J. Homoclionic solutions for autonomous systems in arbitrary dimension[J]. SIAM J Math Anal,1992,23:702-721.
|
| [38] | Gruendler J. Homoclinic solutions for autonomous ordinary differential equations with nonautonomous perturbations[J]. J Diff Eqns,1995,122:1-26.
|
| [39] | Gruendler J. The existence of transversal homoclinic solutions for higher order equations[J]. J Diff Eqns,1996,130:307-320.
|
| [40] | Hale J K, Lin X B. Heteroclinic orbits for retarded functional differential equations[J]. J Diff Eqns,1986,65:175-202.
|
| [41] | Knobloch J. Bifurcation of degenerate homoclinic orbits in reversible and conservative systems[J]. J Dyn Diff Eqns,1997,9:427-444.
|
| [42] | Battelli F, Palmer K J. Chaos in the Duffing equation[J]. J Diff Eqns,1993,101:276-301.
|
| [43] | Luo G, Zhu C. Transversal homoclinic orbits and chaos for functional differential equations[J]. Nonlinear Anal:TMA,2009,71:6254-6264.
|
| [44] | Zhu C, Luo G, Shu Y. The existences of transverse homoclinic solutions and chaos for parabolic equations[J]. J Math Anal Appl,2007,335:626-641.
|
| [45] | Awrejcewicz J, Holicke M M. Smooth and Nonsmooth High Dimensional Chaos and the Melnikov-Type Methods[M]. Singapore:World Scientific,2007.
|
| [46] | Zhu C, Zhang W. Linearly independent homoclinic bifurcations parameterized by a small function[J]. J Diff Eqns,2007,240:38-57.
|
| [47] | Zhu C, Luo G, Lan K. Multiple homoclinic solutions for singular differential equations[J]. Ann Inst H Poincare:AN,2010,27:917-936.
|
| [48] | Arnold L. Random Dynamical Systems[M]. New York:Springer-Verlag,1998.
|
| [49] | Jaeger L, Kantz H. Homoclinic tangencies and non-normal Jacobians-effects of noise in nonhyperbolic chaotic systems[J]. Physica,1997,D105:79-96.
|
| [50] | Kennedy J, York J. Topological horseshoes[J]. Trans Am Math Soc,2001,353:3513-2530.
|
| [51] | Lu K, Wang Q. Chaos in differential equations driven by a nonautonomous force[J]. Nonlinearity,2010,23:2935-2975.
|
| [52] | Lu K, Wang Q. Chaos behavior in differential equations driven by a Brownian motion[J]. J Diff Eqns,2011,251:2853-2895.
|
