Deterministic-Stochastic modeling: A new direction in modeling real world problems with crossover effect

Many real world problems depict processes following crossover behaviours. Modelling processes following crossover behaviours have been a great challenge to mankind. Indeed real world problems following crossover from Markovian to randomness processes have been observed in many scenarios, for example in epidemiology with spread of infectious diseases and even some chaos. Deterministic and stochastic methods have been developed independently to develop the future state of the system and randomness respectively. Very recently, Atangana and Seda introduced a new concept called piecewise di¤erentiation and integration, this approach helps to capture processes with crossover e¤ects. In this paper, an example of piecewise modelling is presented with illustration to chaos problems. Some important analysis including a piecewise existence and uniqueness and piecewise numerical scheme are presented. Numerical simulations are performed for di¤erent cases.Keywords and phrases: Stochastic-deterministic models, piecewise existence and uniqueness, piecewise numerical scheme, new classes of chaotic attractors.