Non-integrability of Hamiltonian systems through high order variational equations: Summary of results and examples

2009 ◽  
Vol 14 (3) ◽  
pp. 323-348 ◽  
Author(s):  
R. Martínez ◽  
C. Simó
Keyword(s):  
Hamiltonian Systems ◽  
High Order ◽  
Variational Equations

scholarly journals Arbitrary High-order EQUIP Methods for Stochastic Canonical Hamiltonian Systems

2019 ◽  
Vol 23 (3) ◽  
pp. 703-725 ◽  
Author(s):  
Xiuyan Li ◽  
Chiping Zhang ◽  
Qiang Ma ◽  
Xiaohua Ding
Keyword(s):  
Hamiltonian Systems ◽  
High Order

scholarly journals EFFICIENT INTEGRATION OF THE VARIATIONAL EQUATIONS OF MULTIDIMENSIONAL HAMILTONIAN SYSTEMS: APPLICATION TO THE FERMI–PASTA–ULAM LATTICE

2012 ◽  
Vol 22 (09) ◽  
pp. 1250216 ◽  
Author(s):  
ENRICO GERLACH ◽  
SIEGFRIED EGGL ◽  
CHARALAMPOS SKOKOS
Keyword(s):  
Hamiltonian Systems ◽  
Coupled Oscillators ◽  
Taylor Series Expansion ◽  
Expansion Method ◽  
Variational Equations ◽  
Integration Schemes ◽  
Chaos Detection ◽  
Efficient Integration ◽  
Low Dimensional ◽  
Series Expansion Method

We study the problem of efficient integration of variational equations in multidimensional Hamiltonian systems. For this purpose, we consider a Runge–Kutta-type integrator, a Taylor series expansion method and the so-called "Tangent Map" (TM) technique based on symplectic integration schemes, and apply them to the Fermi–Pasta–Ulam β (FPU-β) lattice of N nonlinearly coupled oscillators, with N ranging from 4 to 20. The fast and accurate reproduction of well-known behaviors of the Generalized Alignment Index (GALI) chaos detection technique is used as an indicator for the efficiency of the tested integration schemes. Implementing the TM technique — which shows the best performance among the tested algorithms — and exploiting the advantages of the GALI method, we successfully trace the location of low-dimensional tori.

A NONINTEGRABILITY CRITERION FOR ADIABATIC SYSTEMS

2006 ◽  
Vol 16 (06) ◽  
pp. 1829-1833
Author(s):  
DESPINA VOYATZI ◽  
EFI MELETLIDOU
Keyword(s):  
Fixed Point ◽  
Time Dependence ◽  
Hamiltonian Systems ◽  
Additional Assumption ◽  
High Order ◽  
Degree Of Freedom ◽  
Unstable Fixed Point

In the present paper we investigate the nonintegrability of adiabatic one degree of freedom Hamiltonian systems, with the additional assumption that the frozen system possesses an unstable fixed point with two asymmetric homoclinic loops. We prove a criterion for the nonexistence of an integral for such systems, and therefore we prove the nonexistence of a quantity which is conserved in an arbitrarily high order on ε. A specific application is given in the asymmetric quartic oscillator with adiabatic time dependence.

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