TY - JOUR
T1 - Study of a dynamical system with a strange attractor and invariant tori
AU - Algaba, Antonio
AU - Merino, Manuel
AU - Qin, Bo-Wei
AU - RodrÃguez-Luis, Alejandro J.
PY - 2019/4/24
Y1 - 2019/4/24
N2 - A simple three-dimensional time-reversible system of ODEs with quadratic nonlinearities is considered in a recent paper by Sprott (2014). The author finds in this system, that has no equilibria, the coexistence of a strange attractor and invariant tori. The goal of this letter is to justify theoretically the existence of infinite invariant tori and chaotic attractors. For this purpose we embed the original system in a one-parameter family of reversible systems. This allows to demonstrate the presence of a Hopf-zero bifurcation that implies the birth of an elliptic periodic orbit. Thus, the application of the KAM theory guarantees the existence of an extremely complex dynamics with periodic, quasiperiodic and chaotic motions. Our theoretical study is complemented with some numerical results. Several bifurcation diagrams make clear the rich dynamics organized around a so-called noose bifurcation where, among other scenarios, cascades of period-doubling bifurcations also originate chaotic attractors. Moreover, a cross section and other numerical simulations are also presented to illustrate the KAM dynamics exhibited by this system.
AB - A simple three-dimensional time-reversible system of ODEs with quadratic nonlinearities is considered in a recent paper by Sprott (2014). The author finds in this system, that has no equilibria, the coexistence of a strange attractor and invariant tori. The goal of this letter is to justify theoretically the existence of infinite invariant tori and chaotic attractors. For this purpose we embed the original system in a one-parameter family of reversible systems. This allows to demonstrate the presence of a Hopf-zero bifurcation that implies the birth of an elliptic periodic orbit. Thus, the application of the KAM theory guarantees the existence of an extremely complex dynamics with periodic, quasiperiodic and chaotic motions. Our theoretical study is complemented with some numerical results. Several bifurcation diagrams make clear the rich dynamics organized around a so-called noose bifurcation where, among other scenarios, cascades of period-doubling bifurcations also originate chaotic attractors. Moreover, a cross section and other numerical simulations are also presented to illustrate the KAM dynamics exhibited by this system.
KW - Hopf-zero
KW - Invariant tori
KW - KAM theory
KW - Reversible system
KW - Strange attractor
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U2 - 10.1016/j.physleta.2019.01.066
DO - 10.1016/j.physleta.2019.01.066
M3 - 21_Publication in refereed journal
VL - 383
SP - 1441
EP - 1449
JO - Physics Letters, Section A: General, Atomic and Solid State Physics
JF - Physics Letters, Section A: General, Atomic and Solid State Physics
SN - 0375-9601
IS - 13
ER -