Bouncing solutions of an equation with attractive singularity

2004 ◽  
Vol 134 (1) ◽  
pp. 201-213 ◽  
Author(s):  
Dingbian Qian ◽  
Pedro J. Torres
Keyword(s):  
Periodic Solutions ◽  
Second Order ◽  
Attractive Singularity ◽  
Twist Theorem

For any n, m ∈ N, we prove the existence of 2mπ-periodic solutions, with n bouncings in each period, for a second-order forced equation with attractive singularity by using the approach of successor map and Poincaré-Birkhoff twist theorem.

scholarly journals Boundedness of Solutions for a Class of Second-Order Periodic Systems

2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
Shunjun Jiang ◽  
Yan Ding
Keyword(s):  
Periodic Solutions ◽  
Periodic System ◽  
Second Order ◽  
Periodic Systems ◽  
Boundedness Of Solutions ◽  
Small Twist Theorem ◽  
Twist Theorem

In this paper we study the following second-order periodic system:x′′+V′(x)+p(x,t)=0,whereV(x)has a singularity. Under some assumptions on theV(x)andp(x,t)by Ortega’ small twist theorem, we obtain the existence of quasi-periodic solutions and boundedness of all the solutions.

scholarly journals Periodic solutions to reversible second order autonomous DDEs in prescribed symmetric nonconvex domains

2021 ◽  
Vol 28 (4) ◽  
Author(s):  
Zalman Balanov ◽  
Norimichi Hirano ◽  
Wiesław Krawcewicz ◽  
Fangfang Liao ◽  
Adrian Murza
Keyword(s):  
Periodic Solutions ◽  
Second Order ◽  
Nonconvex Domains

scholarly journals Multiplicity of positive periodic solutions to second order differential equations

2006 ◽  
Vol 73 (2) ◽  
pp. 175-182 ◽  
Author(s):  
Jifeng Chu ◽  
Xiaoning Lin ◽  
Daqing Jiang ◽  
Donal O'Regan ◽  
R. P. Agarwal
Keyword(s):  
Periodic Solution ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Differential Equations ◽  
Periodic Solutions ◽  
Point Theorem ◽  
Positive Periodic Solution ◽  
Second Order ◽  
Positive Periodic Solutions ◽  
Second Order Differential Equations

In this paper, we study the existence of positive periodic solutions to the equation x″ = f (t, x). It is proved that such a equation has more than one positive periodic solution when the nonlinearity changes sign. The proof relies on a fixed point theorem in cones.

Synchronized Periodic Solutions of a Class of Periodically Driven Nonlinear Oscillators

1988 ◽  
Vol 55 (3) ◽  
pp. 721-728 ◽  
Author(s):  
Gamal M. Mahmoud ◽  
Tassos Bountis
Keyword(s):  
Periodic Solutions ◽  
Periodic Orbits ◽  
Nonlinear Oscillators ◽  
High Energy ◽  
Second Order ◽  
Multiple Time ◽  
Particle Beams ◽  
Periodically Driven ◽  
Time Scaling ◽  
A New Technique

We consider a class of parametrically driven nonlinear oscillators: x¨ + k1x + k2f(x,x˙)P(Ωt) = 0, P(Ωt + 2π) = P(Ωt)(*) which can be used to describe, e.g., a pendulum with vibrating length, or the displacements of colliding particle beams in high energy accelerators. Here we study numerically and analytically the subharmonic periodic solutions of (*), with frequency 1/m ≅ √k1, m = 1, 2, 3,…. In the cases of f(x,x˙) = x3 and f(x,x˙) = x4, with P(Ωt) = cost, all of these so called synchronized periodic orbits are obtained numerically, by a new technique, which we refer to here as the indicatrix method. The theory of generalized averaging is then applied to derive highly accurate expressions for these orbits, valid to the second order in k2. Finally, these analytical results are used, together with the perturbation methods of multiple time scaling, to obtain second order expressions for regions of instability of synchronized periodic orbits in the k1, k2 plane, which agree very well with the results of numerical experiments.

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