**The Existence of Oscillatory Solutions in the Field–Noyes Model for the Belousov–Zhabotinskii Reaction**

The atomic number 58 particle catalyzed chemical reaction of malonic acid by bromate during a acid answer is basically the Belousov–Zhabotinskii reaction. Belousov [1] determined temporal oscillations within the concentrations of the atomic number 58 ions Ce(III) and Ce(IV) once the chemical agent was stirred. The quantitative model for the chemical change urged by Field and Noyes [3] is examined here and it’s shown that the solutions of the model equations square measure oscillating, of finite amplitude and should possess a minimum of one periodic answer.** [1]**

**On the existence of rapidly oscillatory solutions in the Nicholson blowflies equation**

The existence of speedily periodic solutions within the Nicholson blowflies equations were investigated. The dynamics of Nicholson blowflies was delineate mistreatment the delay equation. it had been shown that infinite set of positive solutions exists for delay equation that are speedily periodic a couple of purpose. it had been conjointly found that for p/δ=e case, all solutions of equation don’t oscillate. **[2]**

**Arnold Diffusion and Oscillatory Solutions in the Planar Three-Body Problem**

In this paper, the subsequent results regarding the Newtonian three-body drawback are obtained: (1) Arnold diffusion exists within the flattened three-body problem, as conjectured by V. 1. Arnold (1964, Dokl. Acad. Nauk SSSR156, 9). (2) The periodic solutions yet as capture and escaping solutions, among different categories of chaotic solutions, exist within the three-body drawback. (3) A special and fascinating development, that we have a tendency to decision the pseudo Arnold diffusion, arises close to time within the three-body drawback. (4) As a results of the existence of Arnold diffusion, the flattened three-body drawback is non-integrable and there are not any extra real analytic integrals besides the illustrious ones. **[3]**

**Geomagnetic field morphologies from a kinematic dynamo model**

THE Earth’s flux is generated by flow of liquid iron within the outer core, acting as a generator. within the easy kinematic theory a fluid flow is prescribed and tested for its ability to get flux, no account being taken of the forces needed to drive the flow. though incomplete, kinematic theory offers valuable insight into the tougher self-propelling drawback and produces field morphologies which may be compared with observations. we have a tendency to try a comprehensive study of three-dimensional kinematic generator action for a category of fluid flow typical of that driven by convection (G.S. and D.G., manuscript in preparation), and have already found similarities between steady dipole solutions and also the geomagnetic field1.** [4]**

**One Step Trigonometrically-fitted Third Derivative Method with Oscillatory Solutions**

A continuous one step Trigonometrically-fitted Third spinoff technique whose coefficients rely upon the frequency and step size comes victimization pure mathematics basis operate. the tactic obtained is use to resolve commonplace issues with periodic solutions. we tend to conjointly discuss the steadiness properties of the new technique . Numerical result obtained via the implementation of the strategies shows that the new technique performs higher than the one step Trigonometrically-fitted second spinoff method projected by Ngwane and Jator [1]. **[5]**

**Reference**

**[1]** Hastings, S.P. and Murray, J.D., 1975. The existence of oscillatory solutions in the Field–Noyes model for the Belousov–Zhabotinskii reaction. SIAM Journal on Applied Mathematics, 28(3), (Web Link)

**[2]** Gyori, I. and Trofimchuk, S.I., 2002. On the existence of rapidly oscillatory solutions in the Nicholson blowflies equation. Nonlinear analysis, 48(7), (Web Link)

**[3]** Xia, Z., 1994. Arnold diffusion and oscillatory solutions in the planar three-body problem. Journal of differential equations, 110(2), (Web Link)

**[4]** Geomagnetic field morphologies from a kinematic dynamo model

David Gubbins & Graeme Sarson

Nature volume 368, (Web Link)

**[5]** Adeniran, A. O. and Longe, I. O. (2017) “One Step Trigonometrically-fitted Third Derivative Method with Oscillatory Solutions”, Journal of Advances in Mathematics and Computer Science, 25(6), (Web Link)