Adaptive Sliding Mode Control of a Novel Class of Fractional Chaotic Systems

Table 1

List of published fractional chaotic systems covered by (7).

Name	Model	𝑓(⋅)	𝑔(⋅)	ℎ(⋅)
Lorenz system	𝐷^𝑞1𝑥=−𝑎𝑥−𝑦	𝑎𝑦	𝑟−𝑧	𝑦
	𝐷^𝑞2𝑦=𝑟𝑥−𝑦−𝑥𝑧
	𝐷^𝑞3𝑧=−𝑏𝑧+𝑥𝑦
Chen system^a	𝐷^𝑞1𝑥=(𝑐−𝑎)𝑦−𝑦𝑧+𝑐𝑥	(𝑐−𝑎)𝑦−𝑦𝑧+𝑐𝑥	𝑎	𝑦
	𝐷^𝑞2𝑦=𝑎𝑥−𝑦
	𝐷^𝑞3𝑧=𝑥𝑦−𝑏𝑧
Liu system	𝐷^𝑞1𝑥=−𝑎𝑥−𝑦	𝑎𝑦	𝑏−𝑘𝑧	𝑑𝑥
	𝐷^𝑞2𝑦=𝑏𝑥−𝑘𝑥𝑧
	𝐷^𝑞3𝑧=−𝑐𝑧+𝑑𝑥^2
Lü system^a	𝐷^𝑞1𝑥=𝑐𝑥−𝑦𝑧	𝑐𝑥−𝑦𝑧	𝑎	𝑦
	𝐷^𝑞2𝑦=𝑎𝑥−𝑦
	𝐷^𝑞3𝑧=𝑥𝑦−𝑏𝑧
Lotka Volterra system	𝐷^𝑞1𝑥=𝑎𝑥−𝑏𝑥𝑦+𝑒𝑥^2−𝑠𝑥^2𝑧	𝑎𝑥−𝑏𝑥𝑦𝑧+𝑒𝑥^2−𝑠𝑥^2	𝑑𝑥	𝑠𝑥𝑧
	𝐷^𝑞2𝑦=−𝑐𝑦+𝑑𝑥𝑦
	𝐷^𝑞3𝑧=−𝑝𝑧+𝑠𝑥^2𝑧
Rucklidge system^a	𝐷^𝑞1𝑥=𝑦	𝑦	𝜆−𝑧	𝑥
	𝐷^𝑞2𝑦=−𝑘𝑥+𝜆𝑥−𝑥𝑧
	𝐷^𝑞3𝑧=−𝑧+𝑥^2

aWe have replaced 𝑥 with 𝑦 just to adopt the general class (6). To propose control scheme for systems in this case, we only need to add the control input to the second state equation of the dynamical systems of the original form, see the second example in Section 5.