Stability Analysis of Nonlinear Caputo Cotangent Fractional Systems

In this manuscript, the stability characteristics of nonlinear nonautonomous dynamical systems with the newly defined Caputo cotangent fractional derivative (CCFD) are discussed. Conditions that guarantee stability and asymptotic stability of the system are developed through comparison methods for CCFD systems using Lyapunov functions. A quadratic inequality for the CCFD and a sign lemma are established as key analytical tools. The additional parameter r2 continuously recovers the classical Caputo derivative at r2=1 and introduces an exponential attenuation mechanism when 0<r2<1. Analytical examples and a reproducible numerical trajectory study illustrate the influence of r2 on the decay of solutions. The results extend classical Lyapunov stability theory to a broader class of nonlinear fractional systems and suggest modeling opportunities in systems where power-law memory and exponential attenuation coexist.