Dynamical behaviors for generalized pendulum type equations with p-Laplacian

Abstract

We consider a pendulum type equation with p-Laplacian {({\varphi }_p(x^{\prime }))}^{\prime }+{G_x}^{\prime }(t,x)=p(t), where {{\displaystyle \varphi }}_p(u)={\left|u\right|}^{p-\mathrm{2}}u,p>\mathrm{1},G(t,x) and p(t) are 1-periodic about every variable. The solutions of this equation present two interesting behaviors. On the one hand, by applying Moser's twist theorem, we find infinitely many invariant tori whenever {\displaystyle {\int }_{\mathrm{0}}^{\mathrm{1}}p(t)}\mathrm{d}t=\mathrm{0}, which yields the bounded-ness of all solutions and the existence of quasi-periodic solutions starting at t = 0 on the invariant tori. On the other hand, if p(t) = 0 and {G_x}^{\prime }(t,x) has some specific forms, we find a full symbolic dynamical system made by solutions which oscillate between any two different trivial solutions of the equation. Such chaotic solutions stay close to the trivial solutions in some fixed intervals, according to any prescribed coin-tossing sequence.

Keywords

p-Laplacian / invariant tori / quasi-periodic solutions / boundedness / complex dynamics

References

[1] Alessio F, Jeanjean L, Montecchiari P. Stationary layered solutions in ℝ² for a class of non autonomous Allen-Cahn equations. Calc Var Partial Differential Equations, 2000, 11: 177–202

[2] Angenent S B,Mallet-Paret J, Peletier L A. Stable transition layers in a semilinear boundary value problem. J Differential Equations, 1987, 67: 212–242

[3] Byeon J, Rabinowitz P H. On a phase transition model. Calc Var Partial Differential Equations, 2013, 47: 1–23

[4] Ding T R, Zanolin F. Subharmonic solutions of second-order nonlinear equations: a time map approach. Nonlinear Anal, 1993, 20: 509–532

[5] Huang H, Yuan R. Boundedness of solutions and existence of invariant tori for generalized pendulum type equation. Chinese Sci Bull, 1997, 42: 1673–1675

[6] Levi M. KAM theory for particles in periodic potential. Ergodic Theory Dynam Systems, 1990, 10: 777–785

[7] Mawhin J,Willem M. Multiple solutions of the periodic boundary value problem for some forced pendulum-type equations. J Differential Equations, 1984, 52: 264–287

[8] Mawhin J, Willem M. Critical Point Theory and Hamiltonian Systems. Appl Math Sci, Vol 74. Berlin: Springer-Verlag, 1989

[9] Moser J. On invariant curves of area-preserving mappings of an annulus. Nachr Akad Wiss Göttingen Math Phys Kl, 1962, 2:1–20

[10] Moser J. Stable and Random Motions in Dynamical Systems with Special Emphasis on Celestial Mechanics. Ann of Math Stud, Vol 77. Princeton: Princeton Univ Press, 1973

[11] Moser J. Quasiperiodic solutions of nonlinear elliptic partial differential equations. Bull Braz Math Soc (N S), 1989, 20: 29–45

[12] Ortega R. Twist mappings, invariant curves and periodic differential equations. In: Grossinho M R, Ramos M, Rebelo C, Sanchez L, eds. Nonlinear Analysis and its Applications to Differential Equations. Progr Nonlinear Differential Equations Appl, Vol 43. New York: Springer, 2001, 85–112

[13] Papini D, Zanolin F. Periodic points and chaotic-like dynamics of planar maps associated to nonlinear Hill's equations with indefinite weight. Georgian Math J, 2002, 9: 339–366

[14] Papini D, Zanolin F. On the periodic boundary value problem and chaotic-like dynamics for nonlinear Hill's equations. Adv Nonlinear Stud, 2004, 4: 71–91

[15] Papini D, Zanolin F. Fix points, periodic points, and coin-tossing sequences for mappings defined on two-dimensional cells. Fixed Point Theory Appl, 2004, 2004: 113–134

[16] Papini D, Zanolin F. Some results on periodic points and chaotic dynamics arising from the study of the nonlinear Hill equations. Rend Semin Mat Univ Politec Torino, 2007, 65: 115{157

[17] Papini D, Zanolin F. Complex dynamics in a ODE model related to phase transition. J Dynam Differential Equations, 2017, 29: 1215–1232

[18] Rabinowitz P H, Stredulinsky E. On a class of infinite transition solutions for an Allen-Cahn model equation. Discrete Contin Dyn Syst, 2008, 21: 319–332

[19] Willem M. Oscillations forcées de systemes hamiltoniens. Publ Mat, 1981, (15 pp)

[20] You J G. Invariant tori and Lagrange stability of pendulum-type equations. J Differential Equations, 1990, 85: 54–65

RIGHTS & PERMISSIONS

Higher Education Press