Dynamical behaviors for generalized pendulum type equations with p -Laplacian
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Dynamical behaviors for generalized pendulum type equations with p-Laplacian Yanmin NIU1 ,
Xiong LI2
1 School of Mathematical Sciences, Ocean University of China, Qingdao 266100, China 2 Laboratory of Mathematics and Complex Systems (Ministry of Education), School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China
c Higher Education Press 2020
Abstract We consider a pendulum type equation with p-Laplacian (φp (x0 ))0 + G0x (t, x) = p(t), where φp (u) = |u|p−2 u, p > 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 R1 infinitely many invariant tori whenever 0 p(t)dt = 0, which yields the boundedness 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 G0x (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 MSC 37C55, 37C75, 37B10 1
Introduction
Motivated by the periodically excited nonlinear equation of pendulum type, in this paper, we consider the equation with p-Laplacian of the form (φp (x0 ))0 + g(t, x) = 0, where φp : R → R is defined by φp (u) = |u|p−2 u,
p > 1,
and g(t + 1, x) = g(t, x + 1) = g(t, x). Received December 5, 2019; accepted August 26, 2020 Corresponding author: Xiong LI, E-mail: [email protected]
2
Yanmin NIU, Xiong LI
It is equivalent to the equation (φp (x0 ))0 + G0x (t, x) = p(t),
(1.1)
where G(t + 1, x) = G(t, x + 1) = G(t, x),
p(t + 1) = p(t).
For p = 2, (1.1) is of the form x00 + G0x (t, x) = p(t),
(1.2)
which is a well-known periodic differential equation. There have been many works on the study of the existence of periodic and quasi-periodic solutions, together with the boundedness of this equation, see [4,6–8,10,11] and references therein. Mawhin [7] and Willem [19] proved that (1.2) possesses at least two periodic solutions if G(t, x) is independent of t and p(t) ∈ C 0 (S1 ) with zero mean value. You [20] generalized the class of G(t, x), which belongs to C ∞ (T2 ), and obtained the existence of invariant tori and Lagrange stability of (1.2) whenever R1 0 p(t)dt = 0 by Moser’s twist theorem. If p 6= 2, the differential operator x → (φp (x0 ))0 is the so-called nonlinear one-dimensional p-Laplacian. However, different from the extensive study for pendulum-type equation (1.2), there are only few results on (1.1) with p-Laplacian. Equation (1.1) is equivalent to the system ( x0 = φq (y) = |y|q−2 y, (1.3) y 0 = −G0x (t, x) + p(t), where q > 1 satisfies p1 + 1q = 1, which is a Hamiltonian system with the Hamiltonian 1 H(t, x, y) = |y|q + G(t, x) − p(t)x. (1.4) q Since G(t, x) is periodic in x, syst
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