Rotating periodic solutions for second‐order dynamical systems with singularities of repulsive type

Abstract

In this paper, we study the following second-order dynamical system:where c 0 is a constant, g 2 C 1 .R n n f0g, R/.n 2/ and e 2 C.R, R n /. When g admits a singularity at zero of repulsive type without the restriction of strong force condition, we apply the coincidence degree theory to prove that the system admits nonplanar collisionless rotating periodic solutions taking the form u.t C T/ D Qu.t/, 8t 2 R with T > 0 and Q an orthogonal matrix under the assumption of Landesman-Lazer type.We claim that there exist R > 0 independent of . , u/ and Q t 2 OE0, T such that ju. Q t/j < R.(2.4)

Rotating periodic solutions for second order systems with Hartman-type nonlinearity

“…Here we modify this idea and gluing two C 1 paths of solutions together to obtain a continuous path of solutions, by which we can track along the trajectory to obtain the rotating periodic solutions (specially, quasi-periodic solutions) of (1.1). For some recent work on rotating periodic solutions of ODEs, one can see [16][17][18][19] and the references.…”

Existence of rotating-periodic solutions for nonlinear second order vector differential equations

“…Then the operator T defined by (5) and the operator A defined by (19) are both compact operators from X to X. As in Sect.…”

“…Proof Let A be defined by (8) and Ω by (9). We suppose that T defined by (5) has no fixed point on ∂Ω; otherwise, Theorem 1 has been proven.…”

“…Recently, the existence of affine-periodic solutions and rotating-periodic solutions for nonlinear differential equations, which was firstly introduced in [24], has become a very interesting topic. Especially, Chang and Li [4,5] studied the existence of rotating-periodic solutions for second order dynamical systems by using the coincidence degree theory. In [21,23], the existence of affine-periodic solutions for nonlinear systems is obtained based on the existence of lower and upper solutions.…”

Rotating periodic solutions for asymptotically linear second‐order Hamiltonian systems with resonance at infinity

“…In 2016, Chang and Li studied the second‐order dissipative dynamical systems, and by using the coincidence degree, they obtained some existence results of rotating periodic solutions. Later, in one study, Chang and Li studied the second‐order dynamical systems with singularities of repulsive type; they proved that the system admits rotating periodic solutions under the assumption of Landesman‐Lazer type by applying the coincidence degree theory. In this paper, we shall study the existence of rotating periodic solutions for via Morse theory when V x ( t , x ) is asymptotically linear at infinity.…”