Quantifying fractional characteristic exponent of fractional order systems and its applications

In many applications of interest, one of the typical problems that often occurs is that quantifying the bounded and unbounded solutions of fractional order systems remains very difficult and challenging. The location of eigenvalues of fundamental autonomous linear fractional order systems when they lie in the sector |arg(s)|>π2 often guarantees bounded solutions, but verifications demand different patterns on all fractional orders involved in such systems. When such systems become nonlinear, quantification of each solution seems entirely lost, and no definite theoretical knowledge is known about how to quantify predicting dynamics of solutions to such systems. We introduce a novel concept, fractional characteristic exponent (FCE), that gives an intuitive mathematical tool to obtain a finite number for any solutions arising in such systems. We prove that the FCE of any solution to general nonautonomous fractional-order systems remains finite. Our main result concerning a non-positive sign of FCE gives confirmation to boundedness, and a positive sign refers to unboundedness of solutions to such systems indicating applications to reality. New conditions to obtain fractional characteristic exponents are developed for such systems, guaranteeing a signature to solutions. We illustrate our novel findings with various advanced fractional order systems that give quantification to each solution.