Physics

Nonlinear system

Bifurcation

Mathematics

Agronomy

Dynamical systems theory

Biology

Partial differential equation

Quadratic equation

Phase portrait

Fractional calculus

Mathematical analysis

Artificial intelligence

Controller (irrigation)

Control (management)

Chaotic

Geometry

Dynamical system (definition)

Computer science

Quantum mechanics

Differential equation

Applied mathematics

Control theory (sociology)

Ordinary differential equation

Advances in Difference Equations

In this paper, we aim to analyze the complicated dynamical motion of a quarter-car suspension system with a sinusoidal road excitation force. First, we consider a new mathematical model in the form of fractional-order differential equations. In the proposed model, we apply the Caputo–Fabrizio fractional operator with exponential kernel. Then to solve the related equations, we suggest a quadratic numerical method and prove its stability and convergence. A deep investigation in the framework of time-domain response and phase-portrait shows that both the chaotic and nonchaotic behaviors of the considered system can be identified by the fractional-order mathematical model. Finally, we present a state-feedback controller and a chaos optimal control to overcome the system chaotic oscillations. Simulation results demonstrate the effectiveness of the proposed modeling and control strategies.

On a nonlinear dynamical system with both chaotic and nonchaotic behaviors: a new fractional analysis and control