A Novel Fractional-Order System: Chaos, Hyperchaos and Applications to Linear Control
A.E. Matouk
Abstract
Chaos and hyperchaos are generated from a new fractional-order system. Local stability of the system’s three equilibria is analyzed when the fractional parameter belongs to (0,2]. According to Hopf bifurcation theory in fractional-order systems, approximations to the periodic solutions around the system’s three equilibria are explored. Lyapunov exponents, Lyapunov spectrum and bifurcation diagrams are computed and chaotic (hyperchaotic) attractors are depicted. Furthermore, a linear control technique (LFGC) based on Lyapunov stability theory is implemented to derive the hyperchaotic states of the proposed system to its three equilibrium points. Numerical results are used to validate the theoretical results.
Topics & Concepts
Lyapunov exponentAttractorMathematicsFractional-order systemApplied mathematicsHopf bifurcationBifurcationBifurcation diagramChaoticStability (learning theory)Lyapunov functionControl theory (sociology)Order (exchange)Mathematical analysisFractional calculusNonlinear systemComputer scienceControl (management)PhysicsEconomicsArtificial intelligenceMachine learningQuantum mechanicsFinanceChaos control and synchronizationFractional Differential Equations Solutions