Effects of Volterra's formulation of heredity on nonlinear dynamical systems, bifurcation and chaos

In recent years, the effects of heredity on dynamical systems have been analyzed using the fractional derivative. But, another way of considering the heredity is the integral formulation proposed by Vito Volterra in 1912 and later almost forgotten. This semina, first generalizes this view on different fields of physics, biology and engineering. Then, it presents some results of this added integral component on the dynamics of a linear and nonlinear dynamical systems considering as example the linear and nonlinear vibration models. It is shown that the heredity amplifies the vibration amplitude, shifts the resonance frequency to higher values, generates self-sustained oscillations which are determined mathematically using the averaging method and confirmed numerically. The heredity can also be a source of chaos in a system free of chaos. We present how this integral component can be operated artificially for applications in mechanical and electrical engineering. Compared to the fractional derivative formulation of the heredity, this model is physical more understanble and can extended in different fields of physics and engineering.