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# On Coexistence of Fractional-Order Hidden Attractors

[+] Author and Article Information
Manashita Borah

Department of Electrical Engineering,
Tezpur University,
Tezpur 784028, India
e-mail: manashitaborah@gmail.com

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received August 1, 2017; final manuscript received March 21, 2018; published online July 26, 2018. Assoc. Editor: Dumitru Baleanu.

J. Comput. Nonlinear Dynam 13(9), 090906 (Jul 26, 2018) (11 pages) Paper No: CND-17-1344; doi: 10.1115/1.4039841 History: Received August 01, 2017; Revised March 21, 2018

## Abstract

This paper proposes new fractional-order (FO) models of seven nonequilibrium and stable equilibrium systems and investigates the existence of chaos and hyperchaos in them. It thereby challenges the conventional generation of chaos that involves starting the orbits from the vicinity of unstable manifold. This is followed by the discovery of coexisting hidden attractors in fractional dynamics. All the seven newly proposed fractional-order chaotic/hyperchaotic systems (FOCSs/FOHSs) ranging from minimum fractional dimension ($nf$) of 2.76 to 4.95, exhibit multiple hidden attractors, such as periodic orbits, stable foci, and strange attractors, often coexisting together. To the best of the our knowledge, this phenomenon of prevalence of FO coexisting hidden attractors in FOCSs is reported for the first time. These findings have significant practical relevance, because the attractors are discovered in real-life physical systems such as the FO homopolar disc dynamo, FO memristive system, FO model of the modulation instability in a dissipative medium, etc., as analyzed in this work. Numerical simulation results confirm the theoretical analyses and comply with the fact that multistability of hidden attractors does exist in the proposed FO models.

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## Figures

Fig. 1

Bifurcation analysis for FOS I against the FO α ∈ [0.92,1] as the bifurcation parameter

Fig. 2

Bifurcation analysis for FOS II against the FO α ∈ [0.96,1] as the bifurcation parameter

Fig. 3

Bifurcation analysis for FOS III against the FO α ∈ [0.91,1] as the bifurcation parameter

Fig. 4

Bifurcation analysis for FOS IV against the FO α ∈ [0.95,1] as the bifurcation parameter

Fig. 5

Bifurcation analysis for FOS V against the FO α ∈ [0.91,1] as the bifurcation parameter

Fig. 6

Bifurcation analysis for FOS VI against the FO α ∈ [0.99,1] as the bifurcation parameter

Fig. 7

Bifurcation analysis for FOS VII against the FO α ∈ [0.905,1] as the bifurcation parameter

Fig. 8

FO coexisting hidden attractors and their time trajectories of FOS I at ICs at (0,−0.1,0) and (0,0.1,0): (a) 2 PO1 at α = 0.95, (b) 2 PO2 at α = 0.97, and (c) 2 CA at α = 0.98

Fig. 9

FO coexisting hidden attractors and their time trajectories (2 HCA) of FOS II for α=0.98 at ICs (0.1,0,0,25) and (0.1,0,0,−25)

Fig. 10

FO coexisting hidden attractors and their time trajectories (1 HCA and 1 spiral) of FOS III for α=0.95 at ICs (0,1,−0.5,0) and (0,1,−2,0)

Fig. 11

FO coexisting hidden attractors and their time trajectories of FOS IV at ICs (0,−10−9,0,0) and (0,10−9,0,0): (a) 2 spirals at α = 0.95 and (b) 2 HCAs at α = 0.98

Fig. 12

FO coexisting hidden attractors (1 HCA and 1 spiral) for α=0.94, and time trajectories of FOS V at ICs (0,1,−0.5,0) and (0,1,−1,0), respectively

Fig. 13

FO coexisting hidden attractors and time trajectories (1 CA and 2 PO) of FOS VI for α=0.996 at ICs 0.005,0.005,0.005,0.005,0.005,(−0.005,0.005,−0.005,0.005,−0.005) and (0.005,−0.005,0.005,−0.005,0.005), respectively

Fig. 14

FO coexisting hidden attractors and their time trajectories of FOS VII at ICs (0.05,−0.5,0.1,−1,2),(2,1,2,0,0) and (0,2,6,11,21), respectively: (a) 1 HCA and 2 POs α = 0.96 on x1x3 plane, (b) 1 HCA and 2 POs α = 0.96 on x2x5 plane, and (c) Time trajectories of the three hidden coexisting attractors

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