Vol. 9, No. 1 (2020): Regular Issue

In this article, the existence and uniqueness of local mild solution of a stochastic counterpart of Tosio Kato’s Quasi-linear partial differential equation with additive cylindrical wiener process in a separable Hilbert space is established using contraction mapping principle.

One-dimensional Bernoulli mapping with hole is suggested to describe the regularities of the appearance of a chaotic set under the saddle-node scenario of the birth of the Smale–Williams hyperbolic attractor. In such a mapping, a non-trivial chaotic set (with non-zero Hausdorff dimension) arises in the general case as a result of a cascade of period-adding bifurcations characterized by geometric scaling both in the phase space and in the parameter space. Numerical analysis of the behavior of models demonstrating the saddle-node scenario of birth of a hyperbolic chaotic Smale–Williams attractor shows that these regularities are preserved in the case of multidimensional systems. Limits of applicability of the approximate 1D model are discussed.

Bifurcations in the Thomas cyclic system leading from simple dynamics into chaotic regimes are considered. In particular, the existence of only one non-trivial fixed point of the system in parameter space implies that this point attractor may only be destabilized via a Hopf bifurcation as the single system parameter is varied. Saddle-node, transcritical and pitchfork bifurcations are precluded. The periodic orbit immediately following the Hopf bifurcation is constructed analytically by the method of multiple scales, and its stability is determined from the resulting normal form and verified by numerical simulations. The dynamically rich range of parameters past the Hopf bifurcation is next systematically explored. In particular, the period-doubling sequences there are found to be more complex than noted previously, and include period-three-like windows for instance. As the system parameter is decreased below these period-incrementing bifurcations, various additional features of the subsequent crises are also carefully tracked. Finally, we consider the effect of delay on the system, leading to the suppression of both the Hopf bifurcation as well as all of the subsequent complex dynamics. In modern terminology, this is an example of Amplitude Death, rather than Oscillation Death, as the complex system dynamics is quenched, with all of the variables settling to a fixed point of the original system.

In this paper, we establish some sufficient conditions for the existence of nonoscillatory solution for a class of mixed neutral fractional differential equations with Liouville fractional derivative of order α ≥ 0 on the halfaxis. Our results generalize some of the existing results in the literature. Some examples are given to illustrate our results.

The scaling formalism is applied to understand and describe the evolution towards the equilibrium at and near at a tangent bifurcation in the logistic map. At the bifurcation the convergence to the steady state is described by a homogeneous function leading to a set of critical exponents. Near the bifurcation the convergence is rather exponential whose relaxation time is given by a power law. We use two different approaches to obtain the critical exponents: (1) a phenomenological investigation based on three scaling hypotheses leading to a scaling law relating three critical exponents and; (2) an approximation that transforms the recurrence equations in a differential equation which is solved under appropriate conditions given analytically the scaling exponents. The numerical results give support for the theoretical approach.

The lubrication characteristics of bearings are significantly influenced by boundary conditions. The double Reynolds boundary conditions (DRBCs) are introduced to analyze the hydrodynamic lubrication characteristics of water-lubricated rubber bearings for turbulent flows, considering the elasticity of rubber liner. Differences in its steady state and dynamic characteristics arising from the DRBCs and the Reynolds boundary conditions (RBCs) are discussed based on the finite difference method. The results show that the water-film reformation boundary is significantly different between the two sets of conditions, however, the load capacity of the bearing is only slightly different. The attitude angle and friction are greater for DRBCs than for RBCs. In the horizontal direction, direct stiffness and direct damping coefficients are larger for DRBCs than for RBCs, but in the vertical direction the opposite result holds.

This paper presents a novel approach to analyze the dynamic effect of the fractional-order derivative of the two mutually coupled van der Pol oscillators. The stability analysis is presented by two complementary phase diagrams: the isospike diagrams and the two Lyapunov exponent spectra. These diagrams reveal precisely the Hubs, spirals bifurcation and chaos when the derivative order is fixed at q = 0.95. In addition, when the fractional-order is set as a control parameter, various methods for detecting chaos including bifurcation diagrams, spectrum of largest Lyapunov exponent are exploited to establish the connection between the system parameters and various complicated dynamics. A transition was also observed between a desynchronized state and a multistability situation. These diagrams displayed the coexistence of four disconnected attractors (two symmetric). We study the basins of attraction of the system in the multistability regime which thereby reveal the coexistence of attractors in the systems when the fractional-order derivative is taken as a function of initials conditions. Based on the parametric control, we have controlled this striking phenomenon in the system. Finally, the hardware circuit is implemented and the results are found to be in good agreement with the numerical investigations.

In this paper we study the renewable resources of a prey-predator system with prey refuge and nonlinear harvesting. Taxation is imposed to prevent over exploitations to maintain ecological balance. The steady state of the system are determined and various dynamical behavior are discussed in its steady states under certain parametric conditions. The boundedness, feasibility of interior equilibria, bionomic equilibrium have been studied. The main observation is that the taxation plays an important role in regulating the dynamics of the present system. Moreover the variation of the taxation change the system from periodic behaviors to chaos. Some numerical illustration are given in order to support of our analytical and theoretical findings.

The oblique impact phenomena is quite common in practical engineering. In this paper, by using the theory of discontinuous dynamical systems, we studied the complex dynamics behaviors of a two degrees of freedom oblique impact system. We can see that the dynamics of oblique impact is more complex than that of the direct impact. The occurrence or disappearance conditions of sticking motion and grazing motion on the separation boundaries are given in Section 3. The conditions here are necessary and sufficient, which generate better results than those obtained with only sufficient conditions. The results appropriately interpret the physical phenomenon of this oblique impact system, hence validate our conclusions. As a supplement, we also give the analytic conditions of the existence of periodic motions. Numerical simulations for sticking motion and grazing motion are presented at last.

Mathematical model of unsteady boundary layer flow and heat transfer is explored for analyzing the study of influence of thermal radiation on incompressible viscous electrically conducting fluid over continuous stretching surface embedded in a porous medium in the presence of heat source/sink. The scope of influencing parameters that describing phenomenon are determined and governing time dependent boundary layer equations are transformed to ordinary differential equations by using appropriate similarity transformation. Numerical computation of the problem was carried out by shooting iteration technique together with Runge-Kutta fourth order integration scheme. Effects of unsteadiness parameter, permeability parameter, magnetic parameter, thermal radiation parameter, Prandtl number and heat source/sink parameter on velocity and temperature profiles are computed and illustrated graphically, whereas local skin friction coefficient and local Nusselt number are represented numerically through tables. In nonmagnetic flow condition the result is found in concordance with earlier investigations.

In this paper, the influence of slip velocity on the fluid flow and solute transfer in a tube with permeable boundary is studied as a mathematical model for blood flow in glomerular capillaries. The viscous incompressible fluid flow across the permeable tube wall, as a result of differences in both hydrostatic and osmotic pressure, is considered. The solutions of differential equations governing the fluid flow and solute transfer are obtained using analytical and Crank-Nicolson type numerical methods. It is observed that the effect of slip on the hydrostatic and osmotic pressures, velocity profiles, concentration profile, solute mass flux and total solute clearance is significant and the results are presented graphically.

We study noise-induced intermittency in a synergetic model of two coupled oscillators with asymmetric nonlinear coupling. This model was previously used to simulate visual perception of ambiguous images. We show that additive noise induces preference for one of the coexisting unstable steady states. When the noise intensity exceeds a certain threshold value, the oscillations of one of the coupled subsystems are interrupted during some time intervals, resulting in intermittent oscillation death, while another subsystem exhibits noisy oscillations in the vicinity of an unstable fixed point.