The paper investigates the complex nonlinear behavior of a fractional order four dimension power system (FOFDPS). The discrete mathematical model of the FOFDPS is derived and presented. The equilibrium points along with the Eigen values of commensurate and incommensurate FOFDPS are presented. The existence of chaotic oscillations are supported by a positive Lyapunov exponent. Bifurcation plots are derived for both parameters and fractional orders to show the impact of the same on the dynamic behavior of FOFDPS. Having shown the existence of such complex behaviors in the FOFDPS, we present an adaptive fractional order sliding mode control (FOASMC) to suppress the chaotic oscillations. Numerical results are presented to support the theoretical results.

A fractional order model of a hydro turbine governing system (FOHTGS) is derived using the Grunwald‐Letnikov's (GL) method. This model is developed using its integer order. Dynamical properties of FOHTGS, such as equilibrium point and its stability, Hopf bifurcation, and Lyapunov exponents (LEs) are investigated. Different dynamics of FOHTGS with respect to changing fractional order of the system and its parameter are studied. The FOHTGS shows multistability and coexisting attractors. To support the existence of multistability, some coexisting attractors are presented. A fuzzy PID based chaos control algorithm is designed and proposed in a hydro turbine governing system (HTGS). Numerical simulations are conducted to validate the effectiveness of the derived controller.

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A two-degree-of-freedom shape memory oscillator derived using polynomial constitutive model is investigated. Periodic, quasiperiodic, chaotic, and hyperchaotic oscillations are shown by the shape memory alloy based oscillator for selected values of the operating temperatures and excitation parameters. Bifurcation plots are derived to investigate the system behavior with change in parameters. A fractional order model of the shape memory oscillator is presented and dynamical behavior of the system with fractional orders and parameters are investigated.

Chaotic behavior and bifurcation analysis of horizontal platform systems (HPS) have been investigated widely by many researchers. However, the multistable features of such systems have not been investigated, and hence we identified the multistable parameter and investigated the coexisting attractors of the HPS. To understand the effects of time delays on the nonautonomous and autonomous HPS, we introduced a constant time delay in the state feedback variable. Investigation of the bifurcation of the time delayed HPS with time delay and parameters reveals that the system behavior differs between the autonomous and nonautonomous HPS. To investigate the multistability existence in time delayed HPS, we plot the bifurcation of the autonomous HPS and show the multistability and coexisting attractors.