TY - JOUR
T1 - Zero-Hopf Bifurcations in Three-Dimensional Chaotic Systems with One Stable Equilibrium
AU - Llibre, Jaume
AU - Messias, Marcelo
AU - De Carvalho Reinol, Alisson
N1 - Publisher Copyright:
© 2020 World Scientific Publishing Company.
PY - 2020/10/1
Y1 - 2020/10/1
N2 - In [Molaie et al., 2013] the authors provided the expressions of 23 quadratic differential systems in R3 with the unusual feature of having chaotic dynamics coexisting with one stable equilibrium point. In this paper, we consider 23 classes of quadratic differential systems in R3 depending on a real parameter a, which, for a = 1, coincide with the differential systems given by [Molaie et al., 2013]. We study the dynamics and bifurcations of these classes of differential systems by varying the parameter value a. We prove that, for a = 0, all the 23 considered systems have a nonisolated zero-Hopf equilibrium point located at the origin. By using the averaging theory of first order, we prove that a zero-Hopf bifurcation takes place at this point for a = 0, which leads to the creation of three periodic orbits bifurcating from it for a > 0 small enough: an unstable one and a pair of saddle type periodic orbits, that is, periodic orbits with a stable and an unstable manifold. Furthermore, we numerically show that the hidden chaotic attractors which exist for these systems when a = 1 are obtained by period-doubling route to chaos.
AB - In [Molaie et al., 2013] the authors provided the expressions of 23 quadratic differential systems in R3 with the unusual feature of having chaotic dynamics coexisting with one stable equilibrium point. In this paper, we consider 23 classes of quadratic differential systems in R3 depending on a real parameter a, which, for a = 1, coincide with the differential systems given by [Molaie et al., 2013]. We study the dynamics and bifurcations of these classes of differential systems by varying the parameter value a. We prove that, for a = 0, all the 23 considered systems have a nonisolated zero-Hopf equilibrium point located at the origin. By using the averaging theory of first order, we prove that a zero-Hopf bifurcation takes place at this point for a = 0, which leads to the creation of three periodic orbits bifurcating from it for a > 0 small enough: an unstable one and a pair of saddle type periodic orbits, that is, periodic orbits with a stable and an unstable manifold. Furthermore, we numerically show that the hidden chaotic attractors which exist for these systems when a = 1 are obtained by period-doubling route to chaos.
KW - Zero-Hopf bifurcation
KW - hidden chaotic attractors
KW - period-doubling route to chaos
KW - periodic orbits
UR - http://www.scopus.com/inward/record.url?scp=85095712518&partnerID=8YFLogxK
U2 - 10.1142/S0218127420501898
DO - 10.1142/S0218127420501898
M3 - Article
AN - SCOPUS:85095712518
SN - 0218-1274
VL - 30
JO - International Journal of Bifurcation and Chaos in Applied Sciences and Engineering
JF - International Journal of Bifurcation and Chaos in Applied Sciences and Engineering
IS - 13
M1 - 2050189
ER -