On Physically Unacceptable Numerical Solutions Yielding Strong Chaotic Signals.

Physically unacceptable chaotic numerical solutions of nonlinear circuits and systems are discussed in this paper. First, as an introduction, a simple example of a wrong choice of a numerical solver to deal with a second-order linear ordinary differential equation is presented. Then, the main result follows with the analysis of an ill-designed numerical approach to solve and analyze a particular nonlinear memristive circuit. The obtained trajectory of the numerical solution is unphysical (not acceptable), as it violates the presence of an invariant plane in the continuous systems. Such a poor outcome is then turned around, as we look at the unphysical numerical solution as a source of strong chaotic sequences. The 0-1 test for chaos and bifurcation diagrams are applied to prove that the unacceptable (from the continuous system point of view) numerical solutions are, in fact, useful chaotic sequences with possible applications in cryptography and the secure transmission of data.

Identification of Ca2+ oscillations with the 0-1 test for periodic and chaotic time-series: One- and two-parameter bifurcations.

This paper examines the oscillatory responses (periodic and chaotic) of a biosystem store model for bursting and complex Ca2+ oscillations in which three compartments have been taken into consideration: the cytosol, endoplasmic reticulum (ER) and mitochondria. The oscillatory model is used to examine the reliability of the 0-1 test for chaos in the bifurcation analysis of continuous signals obtained when the frequencies of oscillatory responses vary significantly with a relatively small changes of the bifurcation parameters. The illustrative examples in both the one- and two-parameter cases are designed to show that for a periodic time-series the test's reliability may be questioned when a periodic series is classified as a chaotic one - the 'false-positive' case. To prevent the incorrect result an additional computational work is needed to examine the frequency spectrum of the periodic time-series. The illustrative examples utilize an autonomous dynamical model of cytosolic calcium oscillations with three dynamical variables and sixteen parameters. The dynamical model is such that the frequency of oscillations may change by the factor of about 200, when a certain dynamical system's parameter changes from its minimum to maximum values, making selection of the parameters in the 0-1 test extremely difficult. The extra computational work improves the test's reliability and eliminates the 'false-positive' outcomes of the test. The paper is focused on the computational aspects of the 0-1 test for periodic and chaotic oscillations rather than on the properties of the store model for bursting and complex Ca2+ oscillations.

Computational Analysis of Ca2+ Oscillatory Bio-Signals: Two-Parameter Bifurcation Diagrams.

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, Ls 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 Ca2+ signal regulation.

Comment on "Study of mixed-mode oscillations in a nonlinear cardiovascular system" [Nonlinear Dyn, doi: 10.1007/s11071-020-05612-8].

This note comments on the questionable approach and claims made in "Study of mixed-mode oscillations in a nonlinear cardiovascular system" [Nonlinear Dyn, doi: 10.1007/s11071-020-05612-8]. Although the author of the above paper attempts to discuss a nonlinear cardiovascular system and its dynamics, the conductance model used, is that of the well-known Cassie-Mayr model of electric arcs, known in the literature since the 1940s. No reference to that well-known model is made however. Moreover, the claim that the conductance model is taken "from the data of a large patient population" is baseless and has no grounds in reality. Thus, an extreme caution is advised, when analyzing the time responses and properties of the nonlinear model in the commented paper. It is questionable that they represent quantities of a human cardiovascular system.