An investigation of the parameter space for a family of dissipative mappings

Abstract

The parameter plane investigation for a family of two-dimensional, nonlinear, and area contracting map is made. Several dynamical features in the system such as tangent, period-doubling, pitchfork, and cusp bifurcations were found and discussed together with cascades of period-adding, period-doubling, and the Feigeinbaum scenario. The presence of spring and saddle-area structures allow us to conclude that cubic homoclinic tangencies are present in the system. A set of complex sets such as streets with the same periodicity and the period-adding of spring-areas are observed in the parameter space of the mapping.

Route to shrimps: Dissipation driven formation of shrimp-shaped domains

“…The rapidly increasing computational capacities have opened the way to examine bifurcation structures of various systems in many fields of sciences in multi-dimensional parameter space [2,3,4,5,6,7,8,9,10]. For example, based on the shooting method, several high-resolution studies have revealed the existence of shrimp-shaped domains (SSD) in bi-parametric planes [11,12,13,14,15,16,17,18,19,20,21]. They are a special class of codimension-two isoperiodic stable structures (ISS) [18], and they turned out to be an efficient "tool" to handle multi-stability [22] and to control chaos Email address: rvarga@hds.bme.hu (Roxána Varga) Preprint submitted to Chaos, Solitons and Fractals August 23, 2019 [21,23].…”

High-performance GPU computations in nonlinear dynamics: an efficient tool for new discoveries

“…That is, they require relatively low computational resources compared to an up to date personal computer. However, in order to explore the complex bifurcation structure in parameter space with high resolution [28][29][30][31][32], the necessary computational power can increase by orders of magnitude. For instance, even in a two dimensional parameter plane-employing an initial value problem solver (IVP) with a resolution of 1000 Â 1000-the computational requirements are increased by three orders of magnitude compared to conventional 1D bifurcation plots with the same resolution of 1000.…”