• Sheela Jacob

Bifurcation theory is a subject with classical mathematical origin. The modern development of this subject dates back to the pioneering work of Poincare, and over the past four decades this theory has witnessed rapid developments with new ideas and methods from dynamical systems theory, singularity theory and many others. In general, in a dynamical system, a parameter is allowed to vary, and then the differential system may change. Equilibrium can become unstable and a periodic solution may appear or a new stable equilibrium may appear making the previous equilibrium unstable. The value of parameter at which these changes occur is known as “bifurcation value” and the parameter that is varied is known as the “bifurcation parameter”. In recent years many types of bifurcations of flow and maps have been studied and classified including saddle node, Hopf, umbilic, zip, Homo clinic tangencies, period doubling and cusp bifurcations. It is our belief that in the years to come the bifurcation theory plays a more active role in various application domains of Science and Technology. In the first section several types of bifurcations have shown through examples. In the second section the structural stability is introduced and studied. The most important two bifurcations have dealt with in detail in the next two sections, where also sufficient conditions are given for these bifurcations. Applications of bifurcation will be given in fifth section.

Bifurcation, Equilibrium, structural stability, saddle node bifurcation, Hopf bifurcation

"A Study of Bifurcation Theory and its Applications in Differential Equations", International Journal of Emerging Technologies and Innovative Research (www.jetir.org), ISSN:2349-5162, Vol.6, Issue 6, page no.820-832, June-2019, Available :http://www.jetir.org/papers/JETIR1906S55.pdf

"A Study of Bifurcation Theory and its Applications in Differential Equations", International Journal of Emerging Technologies and Innovative Research (www.jetir.org | UGC and issn Approved), ISSN:2349-5162, Vol.6, Issue 6, page no. pp820-832, June-2019, Available at : http://www.jetir.org/papers/JETIR1906S55.pdf