In this work, we propose a technique to study nonlinear dynamical systems with fractional-order. The main idea of this technique is to transform the fractional-order dynamical system to the integer one based on Jumarie’s modified Riemann–Liouville sense. Many systems in the interdisciplinary fields could be described by fractional-order nonlinear dynamical systems, such as viscoelastic systems, dielectric polarization, electrode-electrolyte polarization, heat conduction, resistance-capacitance-inductance (RLC) interconnect and electromagnetic waves. To deal with integer order dynamical system it would be much easier in contrast with fractional-order system. Two systems are considered as examples to illustrate the validity and advantages of this technique. We have calculated the Lyapunov exponents of these examples before and after the transformation and obtained the same conclusions. We used the integer version of our example to compute numerically the values of the fractional-order and the system parameters at which chaotic and hyperchaotic behaviors exist.
Reduction of nonlinear dynamical systems excited by random loads