Abstract
Abstract We study behavior of the topological entropy as the function of parameters for two-parameter family of symmetric Lorenz maps T c,ɛ (x) = (−1 + c|x| 1−ɛ ) · sgn(x). This is the normal form for splitting the homoclinic loop in systems which have a saddle equilibrium with one-dimensional unstable manifold and zero saddle value. Due to L.P. Shilnikov results, such a bifurcation corresponds to the birth of Lorenz attractor (when the saddle value becomes positive). We indicate those regions in the bifurcation plane where the topological entropy depends monotonically on the parameter c, as well as those for which the monotonicity does not take place. Also, we indicate the corresponding bifurcations for the Lorenz attractors.
Highlights
The paper is devoted to the study of one-dimensional factor map for the geometric model of Lorenz attractors in the form of two-parameter family of Lorenz maps on the interval I = [−1, 1] given by
1. 0 < c ≤ 2, which guarantees the invariance of the interval I; 2. 0 < ε < 1, which corresponds to positive saddle value in the geometric Lorenz model and implies infinite one-sided derivatives at the discontinuity point
Applied Mathematics and Nonlinear Sciences 5(2020) 293–306. Such families of maps appear naturally in the studies of bifurcations related to the birth of Lorenz attractor from the separatrix loop with zero saddle value
Summary
Applied Mathematics and Nonlinear Sciences 5(2020) 293–306 Such families of maps appear naturally in the studies of bifurcations related to the birth of Lorenz attractor from the separatrix loop with zero saddle value. One of the results of those studies is the following fact: in the parameter plane (α, λ ), the boundary of the region corresponding to the existence of Lorenz attractor, contains the system having homoclinic figure-8 loop at a saddle with zero saddle value We show that in the case when Tc,ε is expanding ( (DT ≥ q > 1), the topological entropy is monotone increasing in c
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
