Heteroclinic orbits for a discrete pendulum equation

Abstract

About twenty years ago, Rabinowitz showed firstly that there exist heteroclinic orbits of autonomous Hamiltonian system joining two equilibria. A special case of autonomous Hamiltonian system is the classical pendulum equation. The phase plane analysis of pendulum equation shows the existence of heteroclinic orbits joining two equilibria, which coincide with the result of Rabinowitz. However, the phase plane of discrete pendulum equation is similar to that of the classical pendulum equation, which suggests the existence of heteroclinic orbits for discrete pendulum equation also. By using variational method and delicate analysis technique, we show that there indeed exist heteroclinic orbits of discrete pendulum equation joining every two adjacent points of {2kπ + π : k ∈ Z}.

Heteroclinic connections and Dirichlet problems for a nonlocal functional of oscillation type

“…We recall that existence of heteroclinic solutions to discrete recurrence relations such as (1.19) has been also considered in the literature, see e.g. [21,22]. A consequence of Proposition 1.7 is the following result.…”

Dynamics of a Higher-Order System of Difference Equations

“…0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 (2) 0.0006 0.0009 0.0000 0.0005 0.0007 0.0000 (3) 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 (2) 150.6878 359.6003 566.9402 167.4252 399.5398 629.9081 (3) 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 Table 5 95 96 97 98 99 100 (1) 0.0035 0.0315 0.0006 0.0031 0.0280 0.0005 (2) 0.0010 0.0014 0.0000 0.0008 0.0011 0.0000 Example 7. In (41), we take Θ = (1.1, 0.8, 1), Ξ = (2, 3, 1), Λ = (20,50,20,80,58,18,10,60,16). From Table 7 and Figure 3 x (1) n x (2) n x (3) x (1) n x (2) n x (3) 1.8…”

“…Not only does it provide us with some simple and useful mathematic models to help elucidate interesting phenomena in applications, but also it can kind of display some surprising complicated dynamics comparing with its analogue differential equations. Hence, the systems of difference equations and difference equations have attracted a lot of attention (see, e.g., the systems of difference equations [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16] and difference equations [17][18][19][20][21][22][23][24][25][26][27][28][29] and the references therein). Among them, symmetric and close to symmetric systems of difference equations have attracted a considerable interest.…”

Homoclinic orbits for discrete Hamiltonian systems with local super-quadratic conditions

“…Therefore, it is of both practical importance and theoretical significance to study the existence of homoclinic orbits of this kind of discrete systems. Since the first variational result about periodic solutions of difference equations was obtained by Guo and Yu [7] in 2003, critical point theory has been widely used in studying the existence of some special solutions of discrete systems, for example, see [3,20].…”