Hopf Bifurcation for a Fractional Differential

Bing Wang

Abstract This paper concerns with the fractional delay differential equations with convolution nonlinearity: Where with m and n being positive integers, are constants and fraction is ontinuous and such that the integral exists. Through linearize above problem at the

neighborhood of equilibrium point of its, we find the characteristic equation.

Keywords: Fractional calculus, Convolution Nonlinearity.

1 Introduction

Fractional calculus includes fractional order integral and fractional derivative, fractional differential, equations have gained enough importance due to their valuable applications in electroanalytical chemistry and many engineering and physical problems, see[1, 2, 3, 4, 5,] and the references therein.

It seems that the earliest systematic studies have been made in the 19th century by Liouville, Riemann, Leibniz,etc. In recent years, fractional calculus has been investigated in many areas of science and engineering, especially when we try to make models to describe the complex systems.

In this paper, we study the fractional differential equation:

where is sufficiently small, hence , substitute (3.4) into (3.3), it leads to (3.1). Using , and substitute it into

(3.1), we can show that (3.2) is a characteristic equation of (1.1), where

Reference

[1] Oldham, K. B., Spanier, I., The Fractional Calaulus. Academic Press, New York,1974

[2] Miller, K. B., Ross, B., An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York, 1993

[3] Podlubny, I., Fractional Differential Equations.Academic Press, New York, 1999

[4] Kilbas, A.A., Srivastava, H.M., Trujillo, J.J., Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, vol. 204. Elsevier, Amsterdam, 2006

[5] Hilfer, R., Applications of Fractional Calculus in Physics. Word Scientific, Singapore, 2000