Mirroring and nonlinear perturbation of a circuit's system with multiple attractors
Abstract
We infix the duality-symmetric and the mirror symmetry conversion processes into a dynamical system representing an electric circuit diagram with three input (or output) as shown in Figure 2. Hence, a new non-linear variable order initial value problem is obtained and solved using the Haar wavelet numerical method (HWNM). Error, stability and entropy analyzes show the reliability of the method. Numerical simulations are then implemented and show for the new system, existence of various attractors’ types (point attractors (PAs), limit cycles, strange attractors (SAs), double attractor (DA), coexisting attractors (CoAs)) with their mirror reflections. Both are in a symmetrical structure in which they face each other, separated by a changing symmetry line and exhibiting similar properties. The circuit implementation using a Field Programmable Gate Array (FPGA) is performed and concur with the expected results.
Keyword : electric circuit diagram, limit cycles, mirror symmetry conversion process, perturbation, circuit implementation
How to Cite
Doungmo Goufo, E. F. (2024). Mirroring and nonlinear perturbation of a circuit’s system with multiple attractors. Mathematical Modelling and Analysis, 29(4), 731–752. https://doi.org/10.3846/mma.2024.21033
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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S. Temme and P. Brunet. A new method for measuring distortion using a multitone stimulus and noncoherence. Journal of the Audio Engineering Society, 56(3):176–188, 2008.
E. Tlelo-Cuautle, J.D. Díaz-Munõz, A.M. González-Zapata, R. Li W.D., León-Salas, F.V. Fernández, O. Guillén-Fernández and I. Cruz-Vega. Chaotic image encryption using hopfield and hindmarsh–rose neurons implemented on FPGA. Sensors, 20(5):1326, 2020. https://doi.org/10.3390/s20051326
L. Tonelli. Sull’integrazione per parti. Rend. Acc. Naz. Lincei, 5(18):246–253, 1909.
A. Banerjee, D. Singh, S. Sahana and I. Nath. Impacts of metaheuristic and swarm intelligence approach in optimization. In Cognitive Big Data Intelligence with a Metaheuristic Approach, pp. 71–99. Elsevier, 2022. https://doi.org/10.1016/B978-0-323-85117-6.00008-X
M. Caputo. Linear models of dissipation whose Q is almost frequency independent–II. Geophysical Journal International, 13(5):529–539, Reprinted in: Fract. Calc. Appl. Anal. 11, No 1 (2008), 3–14., 1967. https://doi.org/10.1111/j.1365-246X.1967.tb02303.x
A. Chen, J. Lu, J. Lü and S. Yu. Generating hyperchaotic Lü attractor via state feedback control. Physica A: Statistical Mechanics and its Applications, 364:103–110, 2006. https://doi.org/10.1016/j.physa.2005.09.039
A. Chithra and I. Raja Mohamed. Multiple attractors and strange nonchaotic dynamical behavior in a periodically forced system. Nonlinear Dynamics, 105(4):3615–3635, 2021. https://doi.org/10.1007/s11071-021-06608-8
J. Crank and P. Nicolson. A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type. Advances in Computational Mathematics, 6(1):207–226, 1996. https://doi.org/10.1007/BF02127704
G. Fubini. Opere scelte. II. Cremonese, Roma, 1958.
V. Gallese, P.F. Ferrari and M.A. Umiltà. The mirror matching system: A shared manifold for intersubjectivity. Behavioral and Brain Sciences, 25(1):35–36, 2002. https://doi.org/10.1017/S0140525X02370018
E.F. Doungmo Goufo. Stability and convergence analysis of a variable order replicator–mutator process in a moving medium. Journal of theoretical biology, 403:178–187, 2016. https://doi.org/10.1016/j.jtbi.2016.05.007
E.F. Doungmo Goufo. Strange attractor existence for non-local operators applied to four-dimensional chaotic systems with two equilibrium points. Chaos: An Interdisciplinary Journal of Nonlinear Science, 29(2):023117, 2019. https://doi.org/10.1063/1.5085440
E.F. Doungmo Goufo. The proto-Lorenz system in its chaotic fractional and fractal structure. International Journal of Bifurcation and Chaos, 30(12), 2020. https://doi.org/10.1142/S0218127420501801
K.J. Havens and E. Sharp. Thermal imaging techniques to survey and monitor animals in the wild: a methodology. Academic Press, 2015.
K.H. Kim and S.J. Kim. A wavelet-based method for action potential detection from extracellular neural signal recording with low signal-to-noise ratio. IEEE Transactions on Biomedical Engineering, 50(8):999–1011, 2003. https://doi.org/10.1109/TBME.2003.814523
Y. Li, Z. Li, M. Ma and M. Wang. Generation of grid multi-wing chaotic attractors and its application in video secure communication system. Multimedia Tools and Applications, 79(39):29161–29177, 2020. https://doi.org/10.1007/s11042-020-09448-7
E.N. Lorenz. Deterministic nonperiodic flow. Journal of the Atmospheric Sciences, 20(2):130–141, 1963. https://doi.org/10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2
J. Lü and G. Chen. Generating multiscroll chaotic attractors: theories, methods and applications. International Journal of Bifurcation and Chaos, 16(04):775– 858, 2006. https://doi.org/10.1142/S0218127406015179
Z.T. Njitacke, L.K. Kengne et al. Antimonotonicity, chaos and multiple coexisting attractors in a simple hybrid diode-based jerk circuit. Chaos, Solitons & Fractals, 105:77–91, 2017. https://doi.org/10.1016/j.chaos.2017.10.004
J.S. Richman and J.R. Moorman. Physiological time-series analysis using approximate entropy and sample entropy. American Journal of Physiology-Heart and Circulatory Physiology, 278(6):H2039–H2049, 2000. https://doi.org/10.1152/ajpheart.2000.278.6.H2039
A. Rupani and G. Sujediya. A review of FPGA implementation of internet of things. International Journal of Innovative Research in Computer and Communication Engineering, 4(9):16203–16203, 2016.
M.-Z. Shieh and S.-C. Tsai. Computing the ball size of frequency permutations under Chebyshev distance. Linear algebra and its applications, 437(1):324–332, 2012. https://doi.org/10.1016/j.laa.2012.02.016
A. Simon and E.G. Boyer. Mirrors for behavior III. An anthology of observation instruments. ERIC, 1974.
S. Temme and P. Brunet. A new method for measuring distortion using a multitone stimulus and noncoherence. Journal of the Audio Engineering Society, 56(3):176–188, 2008.
E. Tlelo-Cuautle, J.D. Díaz-Munõz, A.M. González-Zapata, R. Li W.D., León-Salas, F.V. Fernández, O. Guillén-Fernández and I. Cruz-Vega. Chaotic image encryption using hopfield and hindmarsh–rose neurons implemented on FPGA. Sensors, 20(5):1326, 2020. https://doi.org/10.3390/s20051326
L. Tonelli. Sull’integrazione per parti. Rend. Acc. Naz. Lincei, 5(18):246–253, 1909.