New stability theory for non-autonomous fractional time-varying order derivative systems and its applications

Extending beyond stability of equilibrium points in integer order dynamical systems remains a long-standing problem, and it was not known what would be for systems that use time-varying order derivatives. This paper develops a new stability theory that gives a way to predict the trajectory of such systems tending to equilibria under reasonable conditions. By using a fundamental initial-value problem and comparison principle, we put forward a so-called Lyapunov stability theory. We establish new generalized inequalities that make use of Lyapunov functions and allow a definitive way to be applicable to our Lyapunov theorems. We show that when the nonlinearity of a system is smooth and obeys a typical scaling majorization, the solution to such systems cannot blow up at any finite time. We give a few examples to illustrate the new theory’s importance in applicable mathematics, physical science, finance, and biology. The computational trajectories are roughly discovered to visualize scattering of state, either staying near to the stationary point or converging to it in a long time. • Consideration of time-varying order derivative to study systems. • Predicting dynamics without looking at true solutions. • Developing new knowledge gives new results. • Examples enable use of applicable results.