Discontinuity, Nonlinearity, and Complexity 10(3) (2021) 547--570 | DOI:10.5890/DNC.2021.09.014

Fuhong Min$^1$ , Chen Wei$^{1}$, Hanyuan Ma$^{1}$, Yiping Dou$^{1}$, Chunbiao Li$^{2}$

$^1$ School of Electrical and Automation Engineering, Nanjing Normal University, Nanjing, 210023, China

$^ 2$ School of Electrical and Information Engineering, Nanjing University of Information Science and Technology, Nanjing, 210044, China

Abstract

In this paper, through the discontinuous dynamical system theory, the system interactions of two distinct Van der Pol-Duffing oscillators and a Memristor-Duffing oscillator is discussed under a switching nonlinear controller with symbolic functions. The interaction conditions of three chaotic systems are treated as separation boundaries which is time-varying. Thus the corresponding motion domains are constrained by the boundaries and studied, and the analytical conditions for function project synchronization of three nonautonomous system via the switch-ability and attractivity of edge flows are developed. The control parameter maps with different invariant sets are also studied under the analytical conditions. The partial and full function projective synchronization are carried out via numerical simulations, and the interactions of the control parameters on the synchronization has been investigated. The switching projective synchronization are experimentally realized via analog circuit, and the experimental results validate the theoretical analysis.

Acknowledgments

This work is supported by National Natural Science Foundation of China under Grant No. 61971228, 61871230.

References

1. [1]	Wen, S.P., Huang, T.W., Yu, X.H., Chen, M.Z.Q., and Zeng, Z. (2017), Sliding-mode control of memristive chuas systems via the event-based method. IEEE Transactions on Circuits and Systems II: Express Briefs, 64(1), 81-85.

2. [2]	Li, C.B, Joo-Chen Thio, W., Ho-Ching Lu, H., and Lu, T. (2018), A memristive chaotic oscillator with increasing amplitude and frequency, IEEE Access, 6, 12945-12950.

3. [3]	Pu, Y.F., Yuan, X., and Yu, B. (2018), Analog circuit implementation of fractional-order memristor: arbitrary-order lattice scaling fracmemristor. Circuits and Systems I: Regular Papers, IEEE Transactions on, 99, 1-14.

4. [4]	Wang, C., Liu, X., and Xia, H. (2017), Multi-piecewise quadratic nonlinearity memristor and its 2n-scroll and 2n+1-scroll chaotic attractors system, Chaos, 27(3), 033114.

5. [5]	Li, C.B, Thio, J.C., Sprott, J.C., Iu, H.H.C., and Xu, Y. (2018), Constructing infinitely many attractors in a programmable chaotic circuit, IEEE Access, 1-1.

6. [6]	Bao B.C., Yang Q.F., and Zhu L. (2019), Chaotic bursting dynamics and coexisting multi-stable firing patterns in 3D autonomous M-L model and microcontroller-based validations, Int J Bifurc Chaos, 29(10): 1950134.

7. [7]	Li, C.B., Sprott, J.C., and Mei, Y. (2017), An infinite 2-d lattice of strange attractors, Nonlinear Dynamics, 89(4), 2629-2639.

8. [8]	Pham, V.T., Kingni, S.T., and Volos, C. (2017), A simple three-dimensional fractional-order chaotic system without equilibrium: Dynamics, circuitry implementation, chaos control and synchronization, AEU, 78, 220-227.

9. [9]	Zhang, X., Li, Z.J., and Chang, D. (2017), Dynamics circuit implementation and synchronization of a new three-dimensional fractional-order chaotic system, AEU, 82, 435-445.

10. [10]	Aibhav, V., Sabarathinam, S., and Awadhesh, P. (2017), Infinite Number of Hidden Attractors in Memristor-Based Autonomous Duffing Oscillator, Int J Bifurc Chaos, 28(1), 1850013.

11. [11]	Dou, G., Duan, H.Y., and Yang, W.Y., (2019), Effects of Initial Conditions and Circuit Parameters on the SBT-Memristor-Based Chaotic Circuit, Int J Bifurc Chaos, 29(10), 1950171.

12. [12]	Sabarathinam, S., Volos, C.K., and Thamilmaran, K. (2017), Implementation and study of the nonlinear dynamics of a memristor-based Duffing oscillator, Nonlinear Dyn., 87(1), 37-49.

13. [13]	Wang, L.M., Dong, T.D., and Ge, M.F. (2019), Finite-time synchronization of memristor chaotic systems and its application in image encryption, Appl Math Model, 347, 293-305.

14. [14]	Liu, J. and Liu, S.T. (2017), Complex modified function projective synchronization of complex chaotic systems with known and unknown complex parameters, Appl Math Model, 48, 440-450.

15. [15]	Zhang, J.H. and Liao, X.F. (2019), Effects of initial conditions on the synchronization of the coupled memristor neural circuits, Nonlinear Dyn, 95, 1269--1282.

16. [16]	Xu, Y.H., Wu, X.Q., and Li, N. (2020), Fixed-time Synchronization of Complex Networks with a simpler nonchatter controller, IEEE Trans Circuits Syst II Exp Briefs, 67(4), 700--704.

17. [17]	Liao, X.F., and Zhang, J.H. (2017), Synchronization and chaos in coupled memristor-based FitzHugh-Nagumo circuits with memristor synapse, AEU, 75, 82-90.

18. [18]	Yang, S.J., Li, C.D., and Huang, T.W. (2017), Synchronization of coupled memristive chaotic circuits via state-dependent impulsive control, Nonlinear Dynamics, 88, 115--129.

19. [19]	Yao, W., Wang, C.H., and Cao, J.D. (2019), Hybrid multisynchronization of coupled multistable memristive neural networks with time delays, Neurocomputing, 363, 281-294.

20. [20]	Liu, Z.L., Wang, C.N., and Jin, W.Y. (2019), Capacitor coupling induces synchronization between neural circuits, Nonlinear Dynamics, 97, 2661--2673.

21. [21]	Luo, A.C.J. (2009), A theory for synchronization of dynamical systems, Commu. Nonlinear Science, 14(5), 1901-1951.

22. [22]	Luo, A.C.J. (2008), A theory for flow switchability in discontinuous dynamical systems, Nolinear Anal Hybri, 2(4), 1030-1061

23. [23]	Luo, A.C.J. and Min, F.H. (2011), Synchronization dynamics of two different dynamical systems, Chaos Solitons {$\&$ Fractals}, 44(6), 362-380

24. [24]	Min, F.H. and Luo, A.C.J. (2015), Complex dynamics of projective synchronization of Chua circuits with different scrolls, Int J Bifurc Chaos, 25(5), 1530016.

25. [25]	Min, F.H. and Ma, H.Y. (2020), The mechanism of switching combination synchronization for three distinct nonautonomous systems under sinusoidal constraints, Nonlinear Dynamics, 100(1), 475-492.

26. [26]	Vin\{\i}cius, W. and Rech, P.C. (2017), Multistability and organization of periodicity in a Van der Pol--Duffing oscillator, Chaos Solitons {$\&$ Fractals}, 103, 632-637.

27. [27]	Luo, A.C.J. and Lakeh, A.B. (2014), Period-m motions and bifurcation trees in a periodically forced van der Pol-Duffing oscillator, Int J Dyn. Control, 2(4), 474-493.