Research Papers

On Phase and Anti-Phase Combination Synchronization of Time Delay Nonlinear Systems

[+] Author and Article Information
Gamal M. Mahmoud

Department of Mathematics,
Faculty of Science,
Assiut University,
Assiut 71516, Egypt
e-mails: gmahmoud@aun.edu.eg;

Ayman A. Arafa

Department of Mathematics,
Faculty of Science,
Sohag University,
Sohag 82524, Egypt
e-mail: ayman_math88@yahoo.com

Emad E. Mahmoud

Department of Mathematics,
Faculty of Science,
Sohag University,
Sohag 82524, Egypt
e-mail: emad_eluan@yahoo.com

1Corresponding author.

2Present address: Department of Mathematics, Faculty of Science, Taif University, Taif 888, Kingdom of Saudi Arabia.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received April 7, 2018; final manuscript received July 23, 2018; published online August 22, 2018. Assoc. Editor: Zaihua Wang.

J. Comput. Nonlinear Dynam 13(11), 111001 (Aug 22, 2018) (8 pages) Paper No: CND-18-1149; doi: 10.1115/1.4041033 History: Received April 07, 2018; Revised July 23, 2018

Extensive studies have been done on the phenomenon of phase and anti-phase synchronization (APS) between one drive and one response systems. As well as, combination synchronization for chaotic and hyperchaotic systems without delay also has been investigated. Thus, this paper aims to introduce the concept of phase and anti-phase combination synchronization (PCS and APCS) between two drive and one response time delay systems, which are not studied in the literature as far as we know. The analysis of PCS and APCS are carried out using active control technique. An example is given to test the validity of the expressions of control forces to achieve the PCS and APCS of time delay systems. This example is between three different systems. When there is no control, the PCS does not occur where the phase difference is unbounded. The bounded phase difference appears when the control is applied which means that PCS is achieved. The special case which is the combination synchronization is studied as well.

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Grahic Jump Location
Fig. 1

(a) Hyperchaotic attractor of delayed Rössler system (15). (b) Chaotic attractor of Burke–Shaw system (16). (c) Hyperchaotic attractor of delay Chen system (17).

Grahic Jump Location
Fig. 2

Phase combination synchronization of systems (15)(17) under the control (19) with pl=0(l=1,2,3) for the states (m1u1+n1v1,k1w1) in (a); (m2u2+n2v2,k2w2) in (b); (m3u3+n3v3,k3w3) in (c). Moreover, its zoom in small interval is plotted in (d)–(f).

Grahic Jump Location
Fig. 3

Phase combination synchronization errors of systems (15)(17): (a) (t,Δ1) diagram, (b) (t,Δ2) diagram, and (c) (t,Δ3) diagram

Grahic Jump Location
Fig. 4

Phase differences of PCS of systems (15)(17). They are unbounded before control: (a) (t,θ1d−θ1r) diagram, (b) (t,θ2d−θ2r) diagram, and (c) (t,θ3d−θ3r) diagram. On the other hand, they are bounded after control, (d) (t,θ1d−θ1r) diagram, (e) (t,θ2d−θ2r) diagram, and (f) (t,θ3d−θ3r) diagram.

Grahic Jump Location
Fig. 5

Amplitude differences of PCS of systems (15)(17): (a) (t,ρ1d−ρ1r) diagram, (b) (t,ρ2d−ρ2r) diagram, and (c) (t,ρ3d−ρ3r) diagram

Grahic Jump Location
Fig. 6

Combination synchronization between the two drive systems (15), (16) (solid blue curves) and the response system (17) (dashed red curves) with pl=−1(l=1,2,3): (a) m1u1+n1v1 and k1w1 versus t, (b) m2u2+n2v2 and k2w2 versus t, and (c) m3u3+n3v3 and k3w3 versus t

Grahic Jump Location
Fig. 7

Combination synchronization errors: (a) (t,Δ1) diagram, (b) (t,Δ2) diagram, (c) and (t,Δ3) diagram

Grahic Jump Location
Fig. 8

Anti-phase combination synchronization between the two drive systems (15), (16) and the response system (17) with pl=−1(l=1,2,3): (a) m1u1+n1v1 and k1w1 versus t, (b) m2u2+n2v2 and k2w2 versus t, and (c) m3u3+n3v3 and k3w3 versus t

Grahic Jump Location
Fig. 9

Anti-phase combination synchronization errors: (a) (t,Δ1∗) diagram, (b) (t,Δ2∗) diagram, and (c) (t,Δ3∗) diagram



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