Exponential Nonlinear Term Research Articles

Existence of non-Abelian vortices with product gauge groups

In this paper we establish several sharp existence and uniqueness theorems for some non-Abelian vortex models arising in supersymmetric gauge field theories. We prove these results by studying a family of systems of elliptic equations with exponential nonlinear terms in both doubly periodic-domain and planar cases. In the doubly periodic-domain case we obtain some necessary and sufficient conditions, each explicitly expressed in terms of a single inequality interestingly relating the vortex numbers, to coupling parameters and size of the domain, for the existence of solutions to these systems. In the planar case we establish the existence results for any vortex numbers and coupling parameters. Sharp decay estimates for the planar solutions are also obtained. Furthermore, the solutions are unique, which give rise to the quantized integrals in all cases.

Complete switched modified function projective synchronization of a five-term chaotic system with uncertain parameters and disturbances

A five-term three-dimensional (3D) autonomous chaotic system with an exponential nonlinear term is reported in this paper. Basic dynamical behaviours of the chaotic system are further investigated. Then a new synchronization phenomenon, complete switched modified function projective synchronization (CSMFPS), for this novel five-term chaotic system with uncertain parameters and disturbances is investigated. This paper extends previous work, where CSMFPS of chaotic systems means that all the state variables of the drive system synchronize with different state variables of the response system. As the synchronization scheme has many combined forms, it is a promising type of synchronization and can provide greater security in secure communication. Based on Lyapunov stability theory, a robust adaptive controller is contrived to acquire CSMFPS, parameter identification and suppress disturbances simultaneously. Finally, the Lorenz system and the new five-term chaotic system are taken as examples and the corresponding numerical simulations are presented to verify the effectiveness and feasibility of the proposed control scheme.

A reliable modification of the Adomian decomposition method for higher‐order nonlinear differential equations

PurposeThe purpose of this paper is to propose a new modification of the Adomian decomposition method for resolution of higher‐order inhomogeneous nonlinear initial value problems.Design/methodology/approachFirst the authors review the standard Adomian decomposition scheme and the Adomian polynomials for solving nonlinear differential equations. Next, the advantages of Duan's new algorithms and subroutines for fast generation of the Adomian polynomials to high orders are discussed. Then algorithms are considered for the solution of a sequence of first‐, second‐, third‐ and fourth‐order inhomogeneous nonlinear initial value problems with constant system coefficients by the new modified recursion scheme in order to derive a systematic algorithm for the general case of higher‐order inhomogeneous nonlinear initial value problems.FindingsThe authors investigate seven expository examples of inhomogeneous nonlinear initial value problems: the exact solution was known in advance, in order to demonstrate the rapid convergence of the new approach, including first‐ through sixth‐order derivatives and quadratic, cubic, quartic and exponential nonlinear terms in the solution and a sextic nonlinearity in the first‐order derivative. The key difference between the various modified recursion schemes is the choice of the initial solution component, using different choices to partition and delay the subsequent parts through the recursion steps. The authors' new approach extends this concept.Originality/valueThe new modified decomposition method provides a significant advantage for computing the solution's Taylor expansion series, both systematically and rapidly, as demonstrated in the various expository examples.

A novel fractional-order hyperchaotic system with a quadratic exponential nonlinear term and its synchronization

The dynamics of fractional-order systems have attracted increasing attention in recent years. In this paper a novel fractional-order hyperchaotic system with a quadratic exponential nonlinear term is proposed and the synchronization of a new fractional-order hyperchaotic system is discussed. The proposed system is also shown to exhibit hyperchaos for orders 0.95. Based on the stability theory of fractional-order systems, the generalized backstepping method (GBM) is implemented to give the approximate solution for the fractional-order error system of the two new fractional-order hyperchaotic systems. This method is called GBM because of its similarity to backstepping method and more applications in systems than it. Generalized backstepping method approach consists of parameters which accept positive values. The system responses differently for each value. It is necessary to select proper parameters to obtain a good response because the improper selection of parameters leads to inappropriate responses or even may lead to instability of the system. Genetic algorithm (GA), cuckoo optimization algorithm (COA), particle swarm optimization algorithm (PSO) and imperialist competitive algorithm (ICA) are used to compute the optimal parameters for the generalized backstepping controller. These algorithms can select appropriate and optimal values for the parameters. These minimize the cost function, so the optimal values for the parameters will be found. The selected cost function is defined to minimize the least square errors. The cost function enforces the system errors to decay to zero rapidly. Numerical simulation results are presented to show the effectiveness of the proposed method.

A Novel Three Dimension Autonomous Chaotic System with a Quadratic Exponential Nonlinear Term

A novel three dimension autonomous (3D) chaotic system with a quadratic exponential nonlinear term and a quadratic cross-product term is described in this paper. The basic dynamical properties of the new attractor are studied. The forming mechanism of its compound structure, obtained by merging together two simple attractors after performing one mirror operation, has been investigated by detailed numerical as well as theoretical analysis. Finally, the exponential operation circuit and its temperature-compensation circuit, which makes the new system more applicable from a practical engineering perspective, are investigated.

Inversion of nonlinear stochastic operators

The operator-theoretic method (Adomian and Malakian, J. Math. Anal. Appl. 76(1), (1980), 183–201) recently extended Adomian's solutions of nonlinear stochastic differential equations (G. Adomian, Stochastic Systems Analysis, in “Applied Stochastic Processes,” Nonlinear Stochastic Differential Equations, J. Math. Anal. Appl. 55(1) (1976), 441–452; On the modeling and analysis of nonlinear stochastic systems, in “Proceeding, International Conf. on Mathematical Modeling.” Vol. 1, pp. 29–40) to provide an efficient computational procedure for differential equations containing polynomial, exponential, and trigonometric nonlinear terms N( y). The procedure depends on the calculation of certain quantities A n and B n . This paper generalizes the calculation of the A n and B n to much wider classes of nonlinearities of the form N( y, y′,…). Essentially, the method provides a systematic computational procedure for differential equations containing any nonlinear terms of physical significance. This procedure depends on a recurrence rule from which explicit general formulae are obtained for the quantities A n and B n for any order n in a convenient form. This paper also demonstrates the significance of the iterative series decomposition proposed by Adomian for linear stochastic operators in 1964 and developed since 1976 for nonlinear stochastic operators. Since both the nonlinear and stochastic behavior is quite general, the results are extremely significant for applications. Processes need not, for example, be limited to either Gaussian processes, white noise, or small fluctuations.