Vol. 8, No. 4 (2019): Regular Issue

This paper studies the stabilization based controller problem for Takagi-Sugeno fuzzy nonlinear systems. We give some new conditions to prove the global uniform stability of the closed-loop fuzzy control systems in presence of uncertainties. Furthermore, a numerical example is treated to validate our approach.

The synchronization of nodes is a widely reported phenomenon in a large variety of networks. However, the mechanisms that produce such behavior can not always be studied analytically. In this paper, certain non-smooth competitive networks with linear thresholds are studied with formal and numerical tools. We show that the symmetries of the network architecture allow to display and explain the synchronicity between the nodes. Finally it is demonstrated that in this type of non-smooth networks the synchronicity constitutes a bifurcation phenomenon related to the stability of limit cycles.

In this work, we investigate new (2+1) and (3+1)-dimensional Boussinesq equations. We show the complete integrability of these equations via using the Painlevé test. We derive multiple soliton solutions for each model by using the simplified Hirota’s method. Other exact solutions of physical structures are determined.

In this paper, we employ planar dynamical systems and invariant algebraic curves to characterize all algebraic traveling wave solutions to nonlinear evolution equations. In order to demonstrate the applicability and efficiency of the method, we apply the approach to four (2+1)-dimensional integrable extensions of the Kadomtsev–Petviashvili equation. The numerical simulations are also plotted for better understanding the physical phenomena.

In this article, we will investigate invariant intervals, periodic character, the character of semi cycles and global asymptotic stability of all positive solutions of a nonlinear rational difference equation of order k+1, zn+1 = azn+bzn−k+(α +β zn−k)/(A+Bzn−k), n = 0,1, ..., where the parameters a, b, α,β and A, B and the initial conditions z−k, z−k+1, z−k+2, ..., z0 are arbitrary positive real numbers. We also have studied the global stability of this equation through numerically solved examples and confirm our theoretical discussion through it.

In the current article, we consider a system of time fractional Kaup-Boussinesq shallow water equations. Using Lie group analysis, we obtain the symmetry group of transformations which reduces the system of fractional partial differential equations (FPDEs) to system of fractional ordinary differential equations (FODEs). Further, we investigate the exact explicit group invariant solution as well as power series solution of the given system of equations. Next the physical significance of the group invariant solution under the influence of fractional order is studied graphically. Lastly, conserved vectors for the FPDEs are obtained using the conservation theorem.

A three dimensional Kolmogorov problem with extended forcing term for Navier-Stokes equations is considered in the extended periodic domain. The Galerkin{Fourier method is applied and two problem congurations are considered: the symmetry preserving subset of solutions and the full set of solutions. The bifurcation patterns are revealed through the numerical analysis: the eigenvalues of the linearised perturbed system are analyzed (for stationary and periodic solutions) as well as the phase space trajectories that the system generates. Linear stability of the main solution is analyzed. For cubic domain we detected a fast transition to chaos through double Hopf bifurcation. Then the full bifurcation scenario is analyzed for rectangular parallelepiped with double side stretching. The initial stage of laminar-turbulent transition undergoes supercritical pitchfork bifurcation followed by subcritical Hopf bifurction. The system can either go through the series of cycles in Feigenbaum and Sharkovsky ordering or through the bifurcations on invariant tori up to the three and four dimensional tori. The transition to chaos goes either through the emergence of singular attractors of dimension greater than three or through the resonance of tori.

In this paper, dynamics of modified Leslie-Gower predator-prey model with Crowley-Martin functional response and nonlinear prey harvesting is discussed. We discuss the permanence analysis and local stability of all possible equilibrium points. Also we derive the conditions for occurrence of Hopf bifurcation around positive equilibrium point and its stability. The global stability of positive equilibrium point is derived by using proper Lyapunov functional. Further the Hutchinson’s delay is introduced to utilize the fact that prey takes some time lag to convert the food into its growth. It is noted that the Hopf bifurcation occurs when the time lag parameter τ crosses its critical value. The proposed theoretical results are verified with the help of numerical simulations.

In this paper we study a one-consumer-two-resources ecological system with simple mass action and Beddington-DeAngelis functional responses. The essential mathematical features of the present model have been analyzed thoroughly in terms of the local and the global stability and the bifurcations arising in some selected situations as well. The ranges of the significant parameters under which the system admits a Hopf bifurcation are investigated. The explicit formulae for determining the stability, direction and other properties of bifurcating periodic solutions are also derived with the use of both the normal form and the central manifold theory (cf. Carr [1], Hassard et al. [2]). Numerical illustrations are performed finally in order to validate the applicability of the model under consideration.

A mathematical model of steady laminar flow in a channel of varying cross section is studied under the effect of slip velocity at the permeable boundary. The fluid reabsorption at walls is taken care by the assumption of flow rate as a function of the axial coordinate at each cross section. An analytical solution of Navier-Stokes equations is determined by employing the perturbation technique. The graphical results are presented to illustrate the significance of slip velocity and various parameters on the velocity profiles, mean pressure drop, wall shear stress and stream function.

In this paper, we study spatiotemporal synchronization activity (STSA) of coupled Gaussian maps in a complex network. Our complex networks are changed stochastically with time. A coupled map lattice (CML) is adopted in Gaussian map as a prototype of a spatiotemporal chaotic systems with variation of parameters. A key motivation is that to determine (i) the effects of variation of randomness, (ii) the effects of variation of coupling strength ε , (iii) the effects of variation of parameter α when β is fixed and (iv) the effects of variation of β when parameter α is fixed on the synchronization behaviour. The variation of the basin size with respect to rewiring probability for different coupling strength and basin size with respect to coupling strength ε for different randomness, different parameters α and β are also plotted.

In this paper we discuss about the solution representation of abstract fractional differential equations with different conditions on the operator A. The solution representation obtained by using Mittag-Leffler function is correct when the operator A is bounded. However when the operator A is unbounded the representation derived in [1] seems to be incorrect. We also obtain a suitable form of solution to stochastic fractional differential equation with unbounded operators.