Dimensional Sine Research Articles

Load frequency control of a microgrid employing a 2D Sine Logistic map based chaotic sine cosine algorithm

This paper proposes a maiden application of a two dimensional Sine Logistic map based chaotic sine cosine algorithm (2D-SLCSCA) optimized classical PID controller for load frequency control (LFC) of an islanded microgrid (MG). In comparison to random variables and 1D chaotic sequences, the 2D chaotic sequences are more ergodic and possess a wider chaotic range, thereby enhancing the global convergence speed and search capability of an algorithm. Initially, the proposed 2D-SLCSCA is tested on eight classical benchmark test functions and its performance is compared with 1D Logistic map based chaotic SCA (1D-LCSCA), 1D Sine map based chaotic SCA (1D-SCSCA), and the SCA incorporating random variables. Test results reveal that the proposed algorithm exhibits better convergence characteristics, statistics, and execution time. Finally, the proposed 2D-SLCSCA is implemented for the LFC analysis of the islanded MG. To establish the competence of the proposed algorithm in this regard, its performance is compared with 2D Hénon map based chaotic SCA, 2D Lozi map based chaotic SCA, improved salp swarm algorithm (ISSA), SCA, grey wolf optimizer (GWO), and particle swarm optimization (PSO) algorithm considering diverse load disturbance patterns in the MG. Simulation results affirm that the proposed control scheme augments the frequency response of the MG exhibiting a maximum percentage improvement of 78.89%, 78.86%, and 96.51% in peak overshoot (OSpeak), peak undershoot (USpeak) and objective function (OFITSE) value, respectively as compared to the other algorithms. Furthermore, sensitivity of the proposed 2D-SLCSCA is validated considering ± 30% variation in the MG parameters.

A Modified Image Encryption Technique Using Two Dimensional Sine Logistic Map

A Modified Image Encryption Technique Using Two Dimensional Sine Logistic Map

Multicavity formations and complexity modulation in a hyperchaotic discrete system

This paper introduces a novel and unified approach for controlling the directions and number of cavities of a two dimensional Sine ICMIC modulation map (2D-SIMM). Two controllers are added to the system for arranging the cavity fluctuations and translating the cavities respectively. Both the controllers can effectively redesign the dynamics of reproducing cavities in different directions with grid representations. The dynamics of the proposed controlled model are investigated with bifurcation, Lyapunov and FuzzyEn algorithms under various cavity formations in different directions. A relationship is established for the complexity of the phase space with the directional control and various arrangements of the sinusoidal cavities. The proposed model is overall hyperchaotic with the high complexity in the whole parameter plane. The proposed scheme is effective for a dynamical model to reproduce the self phase structure in various arrangements for the optimization and modulation of complexity.

Interpolating function and Stokes phenomena

When we have two expansions of physical quantity around two different points in parameter space, we can usually construct a family of functions, which interpolates the both expansions. In this paper we study analytic structures of such interpolating functions and discuss their physical implications. We propose that the analytic structures of the interpolating functions provide information on analytic property and Stokes phenomena of the physical quantity, which we approximate by the interpolating functions. We explicitly check our proposal for partition functions of zero-dimensional φ4 theory and Sine–Gordon model. In the zero dimensional Sine–Gordon model, we compare our result with a recent result from resurgence analysis. We also comment on construction of interpolating function in Borel plane.

Infinitely many nonlocal symmetries and conservation laws for the [formula omitted]-dimensional Sine–Gordon equation

Infinitely many nonlocal symmetries and infinitely many local and nonlocal conservation laws of the (1+1)-dimensional Sine–Gordon (SG) equation are derived in terms of its Bäcklund transformation (BT). Some special nonlocal symmetries and nonlocal conservation laws are obtained from the linearized equations of the SG equation and its BT. Furthermore, one can derive infinitely many nonlocal symmetries from a known nonlocal symmetry, but also infinitely many nonlocal conservation laws from a known nonlocal conservation law. In addition, infinitely many local and nonlocal conservation laws can be directly generated from the BT through the parameter expansion procedure.

Some New Exact Solutions of (1+2)-Dimensional Sine-Gordon Equation

We use a generalized tanh function expansion method and a direct method to study the analytical solutions of the (1+2)-dimensional sine Gordon (2DsG) equation. We obtain some new interaction solutions among solitary waves and periodic waves, such as the kink-periodic wave interaction solution, two-periodic solitoff solution, and two-toothed-solitoff solution. We also investigate the propagation properties of these solutions.

Bifurcations of travelling wave solutions in the ( N + 1)-dimensional sine–cosine-Gordon equations

In this paper, the ( N + 1)-dimensional sine–cosine-Gordon equations are studied. The existence of solitary wave, kink and anti-kink wave, and periodic wave solutions are proved, by using the method of bifurcation theory of dynamical systems. All possible bounded exact explicit parametric representations of the above travelling solutions are obtained.

New (2 + 1)-dimensional Sine–Gordon equation with self-consistent sources derived by the nonlinear variable separation method

By using the Hirota’s bilinear transformation method and direct variable separation assumption, a new (2+1)-dimensional Sine–Gordon equation with self-consistent sources is derived for the first time. Correspondingly, a nonlinear variable separation solution included two lower-dimensional arbitrary functions is obtained.

A general design for one dimensional discrete sine transform

Implementation of discrete sine transform (DST) is proposed, by folding the input data. This results in recursive structure, besides saving in the overall computation cycles. A simplified architecture to derive DST kernel has also been suggested.

On [formula omitted]-dimensional [formula omitted]-semimetrics and simplex-type inequalities for high-dimensional sine functions

We show that high-dimensional analogues of the sine function (more precisely, the d -dimensional polar sine and the d -th root of the d -dimensional hypersine) satisfy a simplex-type inequality in a real pre-Hilbert space H . Adopting the language of Deza and Rosenberg, we say that these d -dimensional sine functions are d -semimetrics. We also establish geometric identities for both the d -dimensional polar sine and the d -dimensional hypersine. We then show that when d = 1 the underlying functional equation of the corresponding identity characterizes a generalized sine function. Finally, we show that the d -dimensional polar sine satisfies a relaxed simplex inequality of two controlling terms “with high probability”.

Some special types of solutions of a class of the (N+1)-dimensional nonlinear wave equations

Some special types of solutions with arbitrary functions for the (N+1)-dimensional sine–cosine-Gordon equation, the (N+1)-dimensional double sinh-Gordon equation, and the (N+1)-dimensional sinh–cosinh-Gordon equations are constructed by means of the separation transformation approach. As an illustrative sample, the properties of some solutions are shown by their figures.

Abundant new travelling wave solutions for the (2 + 1)-dimensional Sine–Gordon equation

Abundant new travelling wave solutions of the (2 + 1)-dimensional Sine–Gordon equation are obtained by the generalized Jacobi elliptic function method. The solutions obtained include the kink-shaped solutions, bell-shaped solutions, singular solutions and periodic solutions.

Bounded travelling wave solutions for the (n+1)-dimensional sine- and sinh-Gordon equations

Abstract For the (n + 1)-dimensional sine- and sinh-Gordon equations, by using the approach of dynamical systems to a class of travelling wave solutions, in 21 different regions of a five-parameter space, all possible 21 bounded solutions ( solitary breather solutions, kink and anti-kink breather solutions and double-periodic wave solutions) are obtained. Explicit exact parametric representations are given.

Product and Rational Decompositions of Theta Functions Representations for Nonlinear Periodic Waves

A class of periodic solutions of nonlinear evolution equations is expressed as products and rational expressions of theta/elliptic functions. Examples of equations treated include a coupled system of nonlinear Schrodinger (NLS) equations, the (2+1) dimensional sine Gordon system and the Sasa-Satsuma equation. Coupled modified Korteweg-de Vries and NLS systems show that these product periodic waves can be expanded as an infinite sum of solitary waves arising from the coupling. Brief consideration of discrete evolution equations show similar trends but some quantitative difference with the continuous counterpart.

On the geometry of an integrable (2+1)–dimensional sine–Gordon system

It is recorded that Darboux9s method of linking the classical Lame system governing triply orthogonal systems of surfaces with an integrable (2+1)–dimensional sine–Gordon equation may be extended and applied to the integrable two–component generalization of the latter introduced by Konopelchenko and Rogers. Thus, in a reinterpretation, this (2+1)–dimensional sine–Gordon system is shown to define particular (integrable) motions of surfaces.

Superposition principles associated with the Moutard transformation: an integrable discretization of a (2+1)–dimensional sine–Gordon system

Superposition principles, both linear and nonlinear, associated with the Moutard transformation are found. On suitable reinterpretation, these constitute an integrable discrete nonlinear system and its associated linear system. Further, it is shown that, in a particular form, this system is an integrable discretization of a (2+1)–dimensional sine–Gordon system. Solutions of the discrete nonlinear system are constructed by means of a discrete analogue of the Moutard transformation. Included in these solutions are discrete analogues of the kink solutions of the continuous system.

On travelling waves and double-periodic structures in two-dimensional sine - Gordon systems

Exact travelling-wave solutions of the (2 + 1)-dimensional sine - Gordon equation possessing a velocity smaller than the velocity of the linear waves in the correspondent model system are obtained. The dependence of their dispersion relations and allowed areas for the wave parameters on the wave amplitude are discussed. The obtained waves contain as particular cases static structures consisting of elementary cells with zero topological charge. The self-consistent parameters of one static structure are calculated. The obtained structures require minima spatial system sizes for their existence. As an illustration the obtained results are applied for a description of structures in spin systems with an anisotropy created by a magnetic field or by a crystal anisotropy field.

The (2 + 1)-dimensional sine - Gordon equation; integrability and localized solutions

In this paper, the (2 + 1)-dimensional sine - Gordon equation (2DSG) introduced by Konopelchenko and Rogers is investigated and is shown to satisfy the Painlevé property. A variable coefficient Hirota bilinear form is constructed by judiciously using the Painlevé analysis with a non-conventional choice of the vacuum solutions. First the line kinks are constructed. Then, exact localized coherent structures in the 2DSGI equation are generated by the collision of two non-parallel ghost solitons, which drive the two non-trivial boundaries present in the system. Also the reason for the difficulty in identifying localized solutions in the 2DSGII equation is indicated. We also highlight the significance of the asymptotic values of the boundaries of the system.

Multigrid monte carlo in the sine gordon model

We pose two questions about the dynamical critical behavior of multigrid Monte Carlo: Will a multigrid Monte Carlo simulation of the two dimensional Sine Gordon model exhibit critical slowing down, as expected by a theoretical analysis of Metropolis acceptance rates? Can we reduce critical slowing down caused by decreasing acceptance rates on large blocks by performing more updates on coarser lattices? To this end we simulate the model with a W-cycle ( γ = 2) and a higher cycle with γ = 4 using piecewise constant interpolation. The answer to the first question is positive, the answer to the second one is negative.

Multigrid Monte Carlo algorithm with higher cycles in the sine-Gordon model.

We study the dynamical critical behavior of multigrid Monte Carlo for the two dimensional Sine Gordon model on lattices up to 128 x 128. Using piecewise constant interpolation, we perform a W-cycle (gamma=2). We examine whether one can reduce critical slowing down caused by decreasing acceptance rates on large blocks by doing more work on coarser lattices. To this end, we choose a higher cycle with gamma = 4. The results clearly demonstrate that critical slowing down is not reduced in either case.