| Citation: |
Sayed Allamah Iqbal, Md. Golam Hafez, Yu-Ming Chu, Choonkil Park. DYNAMICAL ANALYSIS OF NONAUTONOMOUS RLC CIRCUIT WITH THE ABSENCE AND PRESENCE OF ATANGANA-BALEANU FRACTIONAL DERIVATIVE[J]. Journal of Applied Analysis & Computation, 2022, 12(2): 770-789. doi: 10.11948/20210324
|
DYNAMICAL ANALYSIS OF NONAUTONOMOUS RLC CIRCUIT WITH THE ABSENCE AND PRESENCE OF ATANGANA-BALEANU FRACTIONAL DERIVATIVE
-
Abstract
The Fractional-order derivative (FOD) has a rich history in mathematical science. In comparison, the half derivative has not been extensively used in applied science and engineering. On the other hand, it is widely applicable in many branches of engineering modeling, especially control fractional-order proportional integral derivative(PID), semi-infinite lossy transmission, and many more. Thus, this work investigates the classical RLC circuit with the sense of the Atangana-Baleanu FOD. Concurrently, the Lyapunov spectral analysis is applied to determine whether or not stability and instability appear in the RLC circuit. In the first look, the RLC likes a straightforward dynamical system when the external driving force is zero. But, the dynamical system of the RLC evolves more complicated, and the wandering phase spaces do not illustrate the stability or instability when the time-dependent driving force is not zero. As a result, the Lyapunov exponents play a significant role to analyze the trait of the trajectories of the phase state. It is found from the bifurcation plots and Lyapunov exponent's spectral analysis that the non-autonomous dynamical systems lead to instability and the chaotic state with the manifestation of the local dynamical system, which switches to a stable spiral node with the presence Atangana-Baleanu FOD.
-
Keywords:
- Bifurcation features /
- chaos /
- PID /
- Atangana-Baleanu derivative /
- Caputo derivative /
- RLC circuit
-
-
References
[1] A. Atangana and T. R. Alqahtani, Numerical approximation of the space-time Caputo-Fabrizio fractional derivative and application to groundwater pollution equation, Adv. Differ. Equ., 2016, 2016, 156. doi: 10.1186/s13662-016-0871-x [2] A. Atangana and K. M. Owolabi, New numerical approach for fractional differential equations, Math. Model. Nat. Phenom., 2018, 13, 3. doi: 10.1051/mmnp/2018010 [3] A. Atangana and I. Koca, Chaos in a simple nonlinear system with Atangana-Baleanu derivatives with fractional order, Chaos, Solit. & Fract., 2016, 89, 447-454. [4] E. Afshari and A. Hajimiri, Nonlinear transmission lines for pulse shaping in silicon, IEEE J. Solid-State Circuits, 2005, 40, 744-752. doi: 10.1109/JSSC.2005.843639 [5] M. Al-Dhaifallah, N. Kanagaraj and K. S. Nisar, Fuzzy Fractional-Order PID Controller for Fractional Model of Pneumatic Pressure System, Math. Problems Eng., 2018, 2018, 1. [6] A. O. Almatroud, A. A. Khennaoui, A. Ouannas, G. Grassi, M. M. Al-sawalha and A. Gasri, Dynamical Analysis of a New Chaotic Fractional Discrete-Time System and Its Control, Entropy, 2020, 22, 12. DOI: 10.3390/e22121344. [7] S. Akter, M. G. Hafez, Y. Chu and M. D. Hossain, Analytic wave solutions of beta space fractional Burgers equation to study the interactions of multi-shocks in thin viscoelastic tube filled, Alexandr. Eng. J., 2021, 60, 877. doi: 10.1016/j.aej.2020.10.016 [8] H. M. Baskonus and H. Bulut, On the numerical solutions of some fractional ordinary differential equations by fractional Adams-Bashforth-Moulton method, Open Math. 2015, 13, 547-556. [9] S. Banerjee, Dynamics for Engineers, Wiley, 2005. [10] M. Caputo and F. Mauro, A new Definition of Fractional Derivative without Singular Kernel, PFDA, 2015, 1, 73-85. [11] M. Caputo and F. Mauro, Applications of New Time and Spatial Fractional Derivatives with Exponential Kernels, PFDA, 2016, 2, 1-11. doi: 10.18576/pfda/020101 [12] Z. Chen, K. Djidjeli and W. G. Price, Computing Lyapunov exponents based on the solution expression of the variational system, Applied Math. & Comput., 2006, 174, 982-996. [13] M. F. Danca and K. Nikolay, Matlab Code for Lyapunov Exponents of Fractional-Order Systems, Intl. Journal of Bifurcation and Chaos, 2018, 28, 1850067. doi: 10.1142/S0218127418500670 [14] K. Diethelm, N. J. Ford and A. D. Freed, A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations, Nonlinear Dyn., 2002, 29, 3-22. doi: 10.1023/A:1016592219341 [15] K. Diethelm, The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type, Springer-Verlag, Berlin Heidelberg, 2010. [16] W. Duan, Nonlinear waves propagating in the electrical transmission line EPL, 2004, 66, 192. doi: 10.1209/epl/i2003-10203-3 [17] R. Dhayal, M. Malik, S. Abbas, A. Kumar and R. Sakthivel, Approximation theorems for controllability problem governed by fractional differential equation, Evolution Equations & Cont. Theory, 2021, 10, 411-429. [18] T. M. Etehad, E. Y. K. Ng, C. Lucas, S. Sadri and M. Ataei, Nonlinear analysis using Lyapunov exponents in breast thermograms to identify abnormal lesions, Infr. Phys. & Tech., 2012, 55, 345-352. [19] K. Erenturk, Fractional-Order PIλDμ and Active Disturbance Rejection Control of Nonlinear Two-Mass Drive System, IEEE Trans. on Industrial Elect., 2012, 60, 3806-3813. [20] F. Gómez-AguilarJosé, J. Rosales-García, M. GuÃa-Calderón and J. Razo-Hernández, Fractional RC and LC Electrical Circuits, Ingeniería, Investigación y Tecnología, 2014, 15, 311-319. doi: 10.1016/S1405-7743(14)72219-X [21] J. Guckenheimer and P. Holmes, Introduction: Differential Equations and Dynamical Systems in Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, New York, Springer, 1983. [22] C. Holmes and P. Holmes, Second order averaging and bifurcations to subharmonics in duffing's equation, J. of Sound and Vib., 1981, 78, 161-174. DOI: 10.1016/S0022-460X(81)80030-2. [23] M. G. Hafez, S. A. Iqbal, Asaduzzaman and Z. Hammouch, Dynamical behaviors and oblique resonant nonlinear waves with dual-power law nonlinearity and conformable temporal evolution, DCDS-S, 2021, 14, 2245. doi: 10.3934/dcdss.2021058 [24] M. G. Hafez, S. A. Iqbal, S. Akhter and M. F. Uddin, Oblique plane waves with bifurcation behaviors and chaotic motion for resonant nonlinear Schrodinger equations having fractional temporal evolution, Results Phys., 2019, 15, 102778. doi: 10.1016/j.rinp.2019.102778 [25] M. G. Hafez, Nonlinear ion acoustic solitary waves with dynamical behaviours in the relativistic plasmas, Astrophys Space Sci., 2020, 365, 78. doi: 10.1007/s10509-020-03791-9 [26] S. A. Iqbal and M. G. Hafez, Dynamical Analysis of Nonlinear Electrical Transmission Line through Fractional Derivative, 2020 23rd International Conference on Computer and Information Technology, 2020, 1-5. [27] S. A. Iqbal, M. G. Hafez and S. A. A. Karim, Bifurcation analysis with chaotic motion of oblique plane wave for describing a discrete nonlinear electrical transmission line with conformable derivative, Results Phys., 2020, 18, 103309. doi: 10.1016/j.rinp.2020.103309 [28] S. A. Iqbal, Soliton Solutions: Discrete Dynamical Analysis of Nonlinear Vacuum Diode throughout the Discharging Capacitor, International Conference on Automation, Control and Mechatronics for Industry 4.0, Rajshahi-Bangladesh, 2021. DOI: 10.1109/ACMI53878.2021.9528137. [29] M. A. Iqbal, Y. Wang, M. M. Miah, M. S. Osman and Y. Chu, Study on Date-Jimbo-Kashiwara-Miwa Equation with Conformable Derivative Dependent on Time Parameter to Find the Exact Dynamic Wave Solutions, Fractal Fract., 2022, 6, 12. https://doi.org/10.3390/fractalfract6010004. doi: 10.3390/fractalfract6010004 [30] M. Inc et al., New solitary wave solutions for the conformable Klein-Gordon equation with quantic nonlinearity, Math., 2020, 5. DOI: 10.3934/math.2020447. [31] Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, Springer-Verlag, New York, 1997. [32] H. Kristian, Z. Dušan and M. C. Stevan, Fractional RLC circuit in transient and steady state regimes, Commun. Nonlinear Sci. Numer. Simulat., 2021, 96, 105670. doi: 10.1016/j.cnsns.2020.105670 [33] A. A. Khennaoui et al., Chaos, control, and synchronization in some fractional-order difference equations, Adv. Differ. Equ., 2029, 2019, 412. DOI: 10.1186/s13662-019-2343-6. [34] K. Karthikeyan, P. Karthikeyan, H. M. Baskonus, K. Venkatachalam and Y. Chu, Almost sectorial operators on Ψ-Hilfer derivative fractional impulsive integro-differential equations, Math. Methods Appl. Sci., 2021. https://doi.org/10.1002/mma.7954. doi: 10.1002/mma.7954 [35] T. D. Leta, W. Liu, A. E. Achab, H. Rezazadeh and A. Bekir, Dynamical Behavior of Traveling Wave Solutions for a (2+1)-Dimensional Bogoyavlenskii Coupled System, Qual. Theory Dyn. Syst., 2021, 20, 14. doi: 10.1007/s12346-021-00449-x. [36] A. Ouannas, A. A. Khennaoui, S. Momani, G. Grassi, and V. T. Pham, Chaos and control of a three-dimensional fractional order discrete-time system with no equilibrium and its synchronization, AIP. Adv., 2020, 10, 045310. DOI: 10.1063/5.0004884. [37] A. Ouannas, A. A. Khennaoui, S. Bendoukha and G. Grassi, On the Dynamics and Control of a Fractional Form of the Discrete Double Scroll, Int. J. of Bifurcation and Chaos, 2019, 29, 1950078. https://doi.org/10.1142/S0218127419500780. doi: 10.1142/S0218127419500780 [38] L. Perko, Differential Equations and Dynamical Systems, Springer-Verlag, 2001. [39] M. J. W. Rodwell, M. Kamegawa, R. Yu, M. Case, E. Carman and K. Giboney, GaAs Nonlinear transmission lines for picosecond pulse generation and millimeter-wave sampling, IEEE Trans. Microwave Theory Tech., 1991, 39, 1194-1204. doi: 10.1109/22.85387 [40] K. Ramasubramanian and M. S. Sriram, A comparative study of computation of Lyapunov spectra with different algorithms, Physica D: Nonlinear Phenomena, 2000, 16, 72-86. [41] H. Rezazadeh, M. Younis, Shafqat-Ur-Rehman, M. Eslami, M. Bilal and U. Younas, New exact traveling wave solutions to the (2+1)-dimensional Chiral nonlinear Schrödinger equation, Math. Model. Nat. Phenom., 2021, 16, 38. DOI: 10.1051/mmnp/2021001. [42] S. Rashid, S. Sultana, Y. Karaca, A. Khalid and Y. Chu, Some further extensions considering discrete proportional fractional operators, Fractals, 2022, 30, 12. https://10.1142/S0218348X22400266. [43] H. Steven and Strogatz, Nonlinear Dynamics And Chaos: With Applications To Physics, Biology, Chemistry, And Engineering (Studies in Nonlinearity), CRC Press, 2000. [44] M. Saqib, I. Khan and S. Shafie, Application of Atangana-Baleanu fractional derivative to MHD channel flow of CMC-based-CNT's nanofluid through a porous medium, Chaos, Solit. & Fract., 2018, 116, 79-85. [45] J. Sakai and T. Kawata, Analytical study for the ability of nonlinear transmission lines to generate solitons, Chaos, Solit. & Fract., 2009, 39, 2125-2132. [46] J. Sakai and T. Kawata, Nonlinear Wave Modulation in the Transmission Line, J. Phys. Soc. Jpn., 1976, 41, 1819-1820. doi: 10.1143/JPSJ.41.1819 [47] P. Selvaraj, O. M. Kwon, S. H. Lee and R. Sakthivel, Cluster synchronization of fractional-order complex networks via uncertainty and disturbance estimator-based modified repetitive control, J. of the Franklin Instit., 2021, 358, 9951. doi: 10.1016/j.jfranklin.2021.10.008 [48] S. Sweetha, R. Sakthivel, V. Panneerselvam and Y. Ma, Nonlinear Fault-Tolerant Control Design for Singular Stochastic Systems With Fractional Stochastic Noise and Time-Delay, IEEE Access, 2021, 9, 153647-153655. DOI: 10.1109/ACCESS.2021.3128410. [49] S. Sweetha, R. Sakthivel and S. Harshavarthini, Finite-time synchronization of nonlinear fractional chaotic systems with stochastic actuator faults, Chaos, Solit. & Fract., 2012, 142, 110312. [50] I. Talbi, A. Ouannas, G. Grassi, A. A. Khennaoui, V. T. Pham and D. Baleanu, Fractional Grassi–Miller Map Based on the Caputo h-Difference Operator: Linear Methods for Chaos Control and Synchronization, Discrete Dyn. Nat. Soc., 2020, 2020, e8825694. DOI: 10.1155/2020/8825694. [51] M. F. Uddin, M. G. Hafez and S. A. Iqbal, Plane Wave Solutions With Dynamical Behaviors for Heisenberg Model of Ferromagnetic Spin Chain With Beta Derivative Evolution and Obliqueness, SSRN Electronic Journal, 2021. http://dx.doi.org/10.2139/ssrn.3893380. doi: 10.2139/ssrn.3893380 [52] Y. Ueda, Randomly transitional phenomena in the system governed by Duffing's equation, J. Stat. Phys., 1979, 20, 181-196. DOI: 10.1007/BF01011512. [53] M. F. Uddin, M. G. Hafez, I. Hwang and C. Park, Effect of Space Fractional Parameter on Nonlinear Ion Acoustic Shock Wave Excitation in an Unmagnetized Relativistic Plasma, Front. Phys., 2022, 9, 766035. DOI: 10.3389/fphy.2021.766035 [54] A. Wolf, J. B. Swift, H. L. Swinney and J. A. Vastano, Determining Lyapunov exponents from a time series, Physica D: Nonlinear Phenomena, 1985, 16, 285-317. doi: 10.1016/0167-2789(85)90011-9 [55] Z. U. A. Zafar, N. Sene, H. Rezazadeh and N. Esfandian, Tangent nonlinear equation in context of fractal fractional operators with nonsingular kernel, Math. Sci., 2021. DOI: 10.1007/s40096-021-00403-7. -
-
Figures(22) / Tables(1)
Export File
Citation
Sayed Allamah Iqbal, Md. Golam Hafez, Yu-Ming Chu, Choonkil Park. DYNAMICAL ANALYSIS OF NONAUTONOMOUS RLC CIRCUIT WITH THE ABSENCE AND PRESENCE OF ATANGANA-BALEANU FRACTIONAL DERIVATIVE[J]. Journal of Applied Analysis & Computation, 2022, 12(2): 770-789. doi: 10.11948/20210324
Format
Content
DownLoad:
-
Figure 1. Illustration of Second-order system (
$ RLC $ circuit) excited by sinusoidal source. -
Figure 2. Layout
$ RLC $ circuit equivalent mechanical mass spring damper system. -
Figure 3. Profile of periodic wave solution for the arbitrary values of parameters
$ L = 1H $ ;$ C = 1/6F $ ;$ R = 5\varOmega $ ;$ \omega = 1Hz $ ;$ V_m = 10V $ of equation (2.1). -
Figure 4. Illustration of 2D vector plot for the random values
$L = 1H$ ;$C = 0.1F$ ;$R = 0.5\varOmega$ ;$\omega = 0.5Hz$ ;$V_m = 1.5V$ of the equation (3.2) -
Figure 5. Portrays a 3D phase plot for the values of
$L = 1H$ ;$C = 0.1F$ ;$R = 0.5\varOmega$ ;$\omega = 0.5Hz$ ;$V_m = 1.5V$ of the equation (3.2). -
Figure 6. Representation of 2D phase plot for the values of
$L = 1H$ ;$C = 0.1F$ ;$R = 0.5\varOmega$ ;$\omega = 0.5Hz$ ;$V_m = 1.5V$ of the equation (3.2). -
Figure 7. Time series of current in RLC circuit for the parameters
$L = 1H$ ,$C = 0.1F$ ,$R = 0.5 \varOmega$ ,$\omega = 0.5Hz$ ,$V_m = 1.5V$ for the equation (3.2). -
Figure 8. Time series of the rate of change of current
$L = 1H$ ,$C = 0.1F$ ,$R = 0.5\varOmega$ ,$\omega = 0.5Hz$ ,$V_m = 1.5V$ . -
Figure 9. Illustration of 3D phase plot for the values of
$L = 20H$ ,$C = 0.1F$ ,$R = 0.5\varOmega$ ,$\omega = 0.5Hz$ ,$V_m = 1.5V$ . -
Figure 10. Representation of 2D phase plot for the values of
$L = 20H$ ,$C = 0.1F$ ,$R = 0.5\varOmega$ ,$\omega = 0.5Hz$ ,$V_m = 1.5V$ . -
Figure 11. Time series of current in RLC circuit for the parameters
$L = 20H$ ,$C = 0.1F$ ,$R = 0.5\varOmega$ ,$\omega = 0.5Hz$ ,$V_m = 1.5V$ . -
Figure 12. Time series of the rate of change of current
$L = 20H$ ,$C = 0.1F$ ,$R = 0.5\varOmega$ ,$\omega = 0.5Hz$ ,$V_m = 1.5V$ . -
Figure 13. Representation of Poincaré bifurcation of the implemented frequency force for the
$L = 1H, C = 0.1F,R = 0.5\varOmega,\omega = 0.5Hz,$ and$ V_m = 1.5V$ values on the iteration$:=25000$ , step size$:=0.0001$ for the initial state$(i=0,\varUpsilon=0)$ . -
Figure 14. Representation of Poincaré bifurcation of the implemented frequency(
$\omega$ ) force for the$L = 1H, C = 0.1F,R = 0.5\varOmega,\omega = 0.5Hz,$ and$ V_m = 1.5V$ values on the iteration$:=25000$ , step size$:=0.0001$ for the initial state$(i=0.1,\varUpsilon=0.1)$ . -
Figure 15. Representation of Poincaré bifurcation of the applied voltage(
$V_m$ ) amplitude force for the$L = 20H, C = 0.1F,R = 0.5\varOmega,\omega = 0.5Hz,$ and$ V_m = 1.5V$ values on the iteration$:=25000$ , step size$:=0.0001$ for the opening state$(i=0,\varUpsilon=0)$ . -
Figure 16. Representation of Poincaré bifurcation of the applied voltage(
$V_m$ ) amplitude force for the$L = 20H, C = 0.1F,R = 0.5\varOmega,\omega = 0.5Hz,$ and$ V_m = 1.5V$ values on the iteration$:=25000$ , step size$:=0.0001$ for the opening state$(i=0.1,\varUpsilon=0.1)$ . -
Figure 17. Illustration of Poincaré map of equation (3.2) for the
$L = 1H, C = 0.1F,R = 0.5\varOmega,\omega = 0.5Hz,$ and$ V_m = 1.5V$ values on trajectories intersection of$1501$ points for the opening state$(i=0,\varUpsilon=0)$ . -
Figure 18. Illustration of Poincaré map of equation (3.2) for the
$L = 20H, C = 0.1F,R = 0.5\varOmega,\omega = 0.5Hz,$ and$ V_m = 1.5V$ values on trajectories intersection of$1501$ points for the opening state$(i=0.1,\varUpsilon=0.1)$ . -
Figure 19. Representation of the Lyponve spectrum pfrofile of equation (3.2) for the values of
$L = 20H, C = 0.1F,R = 0.5\varOmega,\omega = 0.5Hz, V_m = 1.5V$ . -
Figure 20. Atangana-Baleanu FOD order
$\alpha_1=0.97$ and$\alpha_2=0.98$ , individually, describe the phase state profile for the values of$L = 20H, C = 0.1F, R = 0.5\varOmega,\omega = 0.5Hz,$ and$V_m = 1.5V$ concerning the values of$A_B(\alpha_1)=1$ ,$A_B(\alpha_2)=1$ for the equation (3.1). -
Figure 21. Atangana-Baleanu FOD order
$\alpha_1=0.97$ and$\alpha_2=0.98$ , individually, describe the time series of current in RLC circuit for the values of$L = 20H, C = 0.1F, R = 0.5\varOmega,\omega = 0.5Hz,$ and$V_m = 1.5V$ concerning the values of$A_B(\alpha_1)=1$ ,$A_B(\alpha_2)=1$ for the equation (3.1). -
Figure 22. Atangana-Baleanu FOD order
$\alpha_1=0.97$ and$\alpha_2=0.98$ , individually, describe time series of the rate of change of current in RLC circuit for the values of$L = 20H, C = 0.1F, R = 0.5\varOmega,\omega = 0.5Hz,$ and$V_m = 1.5V$ concerning the values of$A_B(\alpha_1)=1$ ,$A_B(\alpha_2)=1$ for the equation (3.1).