Period three cycle and chaos in a dynamical system
Kulkarni P. R, Borkar V. C
The mathematical modeling of many phenomenon in the fields of physics, chemistry, astronomy, astrophysics, ecology, economics and many more has been done by scientists and the dynamics of such models has been studied all over the world. The modeling of the 'Predator and Prey' phenomenon in ecosystem in terms of the logistic family FÂµ (x) = Âµx(1 â€“ x) has played an important role in the development of the subject of dynamical system and chaos. The chaotic behavior of many families of mappings like the Tent family, Quadratic family, the logistic family, etc. has been already proved by many authors. The chaotic nature, in the sense of Devaney R. L., of the family of mappings fc (x) = x2 â€“ x + c through the period doubling cascade and using the concept of topological conjugacy has already been proved by Kulkarni P. R. and Borkar V. C. In this paper, we have proved the existence of the period three cycle in the family of mappings fc (x) = x2 â€“ x + c for c = -1.5 and thereby proved that this mapping exhibits chaos.