Sliding Bifurcations in the Memristive Murali–Lakshmanan–Chua Circuit and the Memristive Driven Chua Oscillator

In this paper, we report the occurrence of sliding bifurcations admitted by the memristive Murali–Lakshmanan–Chua circuit [Ishaq & Lakshmanan, 2013] and the memristive driven Chua oscillator [Ishaq et al., 2011]. Both of these circuits have a flux-controlled active memristor designed by the authors in 2011, as their nonlinear element. The three-segment piecewise-linear characteristic of this memristor bestows on the circuits two discontinuity boundaries, dividing their phase spaces into three subregions. For proper choice of parameters, these circuits take on a degree of smoothness equal to one at each of their two discontinuities, thereby causing them to behave as Filippov systems. Sliding bifurcations, which are characteristic of Filippov systems, arise when the periodic orbits in each of the subregions, interact with the discontinuity boundaries, giving rise to many interesting dynamical phenomena. The numerical simulations are carried out after incorporating proper zero time discontinuity mapping (ZDM) corrections. These are found to agree well with the experimental observations which we report here appropriately.

An Overview of Karl Barth's Theology: Focused on the Doctrines of God, Jesus Christ, and the Holy Spirit

ABSTRACT The purpose of this study is to have an overview of the theology of Karl Barth who is considered as one of the most influential theologians in contemporary Christian world. This study is of worthy in order to have an accurate grasp of the trend of modern Chriatian theology. After a brief survey of his life and works, this study provides an overview of Barth’s theology focusing on three major areas of his theology: the doctrines of God, Jesus Christ, and the Holy Spirit. Barth’s emphasis upon the transcendence of God, the centrality of Jesus Christ in Christian theology, and the importance of the Holy Spirit in the Trinity should not be ignored for better understanding of the modern Christian theology. In a word, Barth’s theology has continuity of, and, at the same time, discontinuity from liberal theology. Keywords: Karl Barth, morder Christian theology, transcendence of God, centraliy of Jesus Christ, importance of the Holy Spirit, neo-orthodoxy, liberal theology

Can Quantum Mechanics and Relativity be considered per se complete? - A discussion on Quantum Mechanics and Relativity in Full Space-time Domain

This paper points out the incompleteness of the traditional quantum mechanics and relativity, which is embodied in space-time domains of definition, not in physical quantities for description. The real time and space are not continuous. The phenomena called “ghost-like long-range action” by Einstein in fact occur in the time discontinuity points, that is, Time Quantum Worm Holes put forward by Hawking. This paper also gives an essential difference between the macroscopic random motion and the microscopic random motion, which is critical for understanding wave-particle duality.

Simulating of the Flood Runoff in Case of Heavy Rains in the Zone of Many-year Frozen Earths

Special features of the rain floods runoff formation and an approach to their simulating for small catchments of the zone of many-year frozen earths’ spreading are discussed. Information about heavy rains and resulted floods obtained from the observation data from the experimental catchments of the Kolyma and Bomnak runoff stations has been used as initial data. On the basis of the available concepts of the many-year frozen earths’ active layer dynamics and the outcomes of the different years’ experimental investigations, a mathematical model has been proposed. It enables to take into consideration the accumulating role of the soil melted layer and the moss cover. The НЕС-HMS simulating system able to reproduce the flood runoff course with the 1 hour time discontinuity in specific conditions of slope regulation on the basis of the Clark unit hydrograph method was used to take in account special features of the rain runoff formation.

Discontinuity Induced Hopf and Neimark–Sacker Bifurcations in a Memristive Murali–Lakshmanan–Chua Circuit

We report using Clarke’s concept of generalized differential and a modification of Floquet theory to nonsmooth oscillations, the occurrence of discontinuity induced Hopf bifurcations and Neimark–Sacker bifurcations leading to quasiperiodic attractors in a memristive Murali–Lakshmanan–Chua (memristive MLC) circuit. The above bifurcations arise because of the fact that a memristive MLC circuit is basically a nonsmooth system by virtue of having a memristive element as its nonlinearity. The switching and modulating properties of the memristor which we have considered endow the circuit with two discontinuity boundaries and multiple equilibrium points as well. As the Jacobian matrices about these equilibrium points are noninvertible, they are nonhyperbolic, some of these admit local bifurcations as well. Consequently when these equilibrium points are perturbed, they lose their stability giving rise to quasiperiodic orbits. The numerical simulations carried out by incorporating proper discontinuity mappings (DMs), such as the Poincaré discontinuity map (PDM) and zero time discontinuity map (ZDM), are found to agree well with experimental observations.

NONSMOOTH BIFURCATIONS, TRANSIENT HYPERCHAOS AND HYPERCHAOTIC BEATS IN A MEMRISTIVE MURALI–LAKSHMANAN–CHUA CIRCUIT

In this paper, a memristive Murali–Lakshmanan–Chua (MLC) circuit is built by replacing the nonlinear element of an ordinary MLC circuit, namely the Chua's diode, with a three-segment piecewise-linear active flux controlled memristor. The bistability nature of the memristor introduces two discontinuity boundaries or switching manifolds in the circuit topology. As a result, the circuit becomes a piecewise-smooth system of second order. Grazing bifurcations, which are essentially a form of discontinuity-induced nonsmooth bifurcations, occur at these boundaries and govern the dynamics of the circuit. While the interaction of the memristor-aided self oscillations of the circuit and the external sinusoidal forcing result in the phenomenon of beats occurring in the circuit, grazing bifurcations endow them with chaotic and hyperchaotic nature. In addition, the circuit admits a codimension-5 bifurcation and transient hyperchaos. Grazing bifurcations as well as other behaviors have been analyzed numerically using time series plots, phase portraits, bifurcation diagram, power spectra and Lyapunov spectrum, as well as the recent 0–1 K test for chaos, obtained after constructing a proper Zero Time Discontinuity Map (ZDM) and Poincaré Discontinuity Map (PDM) analytically. Multisim simulations using a model of piecewise linear memristor have also been used to confirm some of the behaviors.