Abstract
Two types of bifurcation diagrams of cytosolic calcium nonlinear oscillatory systems are presented in rectangular areas determined by two slowly varying parameters. Verification of the periodic dynamics in the two-parameter areas requires solving the underlying model a few hundred thousand or a few million times, depending on the assumed resolution of the desired diagrams (color bifurcation figures). One type of diagram shows period-n oscillations, that is, periodic oscillations having n maximum values in one period. The second type of diagram shows frequency distributions in the rectangular areas. Each of those types of diagrams gives different information regarding the analyzed autonomous systems and they complement each other. In some parts of the considered rectangular areas, the analyzed systems may exhibit non-periodic steady-state solutions, i.e., constant (equilibrium points), oscillatory chaotic or unstable solutions. The identification process distinguishes the later types from the former one (periodic). Our bifurcation diagrams complement other possible two-parameter diagrams one may create for the same autonomous systems, for example, the diagrams of Lyapunov exponents, diagrams for mixed-mode oscillations or the 0–1 test for chaos and sample entropy diagrams. Computing our two-parameter bifurcation diagrams in practice and determining the areas of periodicity is based on using an appropriate numerical solver of the underlying mathematical model (system of differential equations) with an adaptive (or constant) step-size of integration, using parallel computations. The case presented in this paper is illustrated by the diagrams for an autonomous dynamical model for cytosolic calcium oscillations, an interesting nonlinear model with three dynamical variables, sixteen parameters and various nonlinear terms of polynomial and rational types. The identified frequency of oscillations may increase or decrease a few hundred times within the assumed range of parameters, which is a rather unusual property. Such a dynamical model of cytosolic calcium oscillations, with mitochondria included, is an important model in which control of the basic functions of cells is achieved through the signal regulation.
Highlights
Periodic cycles—solutions of nonlinear planar systems—are covered by the celebrated Poincaré–Bendixson theorem proven with the use of the Green theorem [1]
The above issues are, to some extent, related to the results presented in the present paper, in which we follow the guidance expressed in the above quotations, and study two different bifurcation 2D diagrams for one particular non-linear model with complex Ca2+ oscillatory solutions
Typical visualization of those changes when one parameter varies slowly has the form of one-parameter bifurcation diagrams with the varying parameter representing the horizontal axis while the vertical axis is a certain quantity characterizing the changing responses
Summary
Periodic cycles—solutions of nonlinear planar (two variables) systems—are covered by the celebrated Poincaré–Bendixson theorem proven (under certain assumptions) with the use of the Green theorem [1]. The central issue is the fact that obtaining entirely identical long-term chaotic solutions via numerical solvers is practically impossible even when the same algorithm, time step and initial conditions are applied to a particular nonlinear ordinary differential equation (ODE) system. Reports on such an issue with interesting findings for the well-known chaotic systems are discussed in [8]. The above issues are, to some extent, related to the results presented in the present paper, in which we follow the guidance expressed in the above quotations, and study two different bifurcation 2D diagrams (of n-period and frequency types) for one particular non-linear model with complex Ca2+ oscillatory solutions. The 0–1 test for chaos (see Section 5) will still result in a number close to 1 indicating chaotic responses
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
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 call
