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Chua's Circuit: A Paradigm for Chaos
Abstract
For uninitiated researchers, engineers, and scientists interested in a quick entry into the subject of chaos, this book offers a timely collection of 55 carefully selected papers covering almost every aspect of this subject. Because Chua's circuit is endowed with virtually every bifurcation phenomena reported in the extensive literature on chaos, and because it is the only chaotic system which can be easily built by a novice, simulated in a personal computer, and tractable mathematically, it has become a paradigm for chaos, and a vehicle for illustrating this ubiquitous phenomenon. Its supreme
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In the citing paper
“Previously suggested chaotic circuits 19-22 , involving those with memristors 22 , involve several circuit elements.”
From Bistable Nonvolatile Elastic-Membrane Memcapacitor Exhibiting a Chaotic Behavior (2011) · cites this paper as [22]
In the citing paper
“Moreover, the membrane memcapacitor is a passive device in contrast with the active Chua's diode 21 and active memristors 22 used in existing chaotic circuits.”
From Bistable Nonvolatile Elastic-Membrane Memcapacitor Exhibiting a Chaotic Behavior (2011) · cites this paper as [22]
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Sorted by citations
Items 501–598 of 598
Chaotic Synchronization, Conditional Lyapunov Exponents and Lyapunov’s Direct Method
Jovic, B. (2011) · Signals and communication technology · doi:10.1007/978-3-642-21849-1_3 · Cited by 1
Robust Fault Detection during the Synchronization Process of Nonlinear Lur'e Dynamical Networks
Xu, S. Y., Liu, J., Sun, H. D., et al. (2013) · Advanced materials research · doi:10.4028/www.scientific.net/amr.718-720.1408 · Cited by 1
A Chemical Chaotic System Derived from Chua's Circuit
Xu, W. G., & Li, Q. S. (2002) · Journal of Chemical Research · doi:10.3184/030823402103172059 · Cited by 1
Saddle-Node Bifurcations in Classical and Memristive Circuits
de la Vega, I. G., & Riaza, R. (2016) · International Journal of Bifurcation and Chaos · doi:10.1142/s0218127416500644 · Cited by 1
Network Based Periodicity Detection of a Time Series
Vijesh, V., Drisya, G. V., & Kumar, K. S. (2018) · IEEE Recent Advances in Intelligent Computational Systems · doi:10.1109/raics.2018.8634897 · Cited by 1
“This makes the study of Chua’s circuit a universal problem [16], [17].”
BackgroundLv, Y., Zhou, J., Fu, J., et al. (2019) · IEEE International Conference on Industrial Technology (ICIT) · doi:10.1109/icit.2019.8755185 · Cited by 1
“1, which plays a double-scroll chaotic attractor [27].”
BackgroundStudy of Periodic Windows for the Chua's Circuit with a Cubic Nonlinearity
Galias, Z. (2020) · IEEE International Symposium on Circuits and Systems proceedings · doi:10.1109/iscas45731.2020.9180638 · Cited by 1
Three-Dimensional Autonomous Systems and Chaos
Lynch, S. (2018) · Springer eBooks · doi:10.1007/978-3-319-78145-7_8 · Cited by 1
Gale, E., Costello, B. D. L., & Adamatzky, A. (2019) · Springer eBooks · doi:10.1007/978-3-319-76375-0_27 · Cited by 1
Three-Dimensional Autonomous Systems and Chaos
Lynch, S. (2017) · Springer eBooks · doi:10.1007/978-3-319-61485-4_8 · Cited by 1
Circuits and Systems for the Synthesis of Chaotic Signals in Engineering Applications
Pareschi, F., Callegari, S., Setti, G., et al. (2009) · Studies in computational intelligence · doi:10.1007/978-3-540-95972-4_8 · Cited by 1
Feng, J., & Tse, C. K. (2004) · International Conference on Communications, Circuits and Systems · doi:10.1109/icccas.2004.1346314 · Cited by 1
A simple and robust method for controlling chaotic systems
Dedieu, H., & OGORZAŁEK, M. J. (2002) · IEEE International Symposium on Circuits and Systems proceedings · doi:10.1109/iscas.1994.409531 · Cited by 1
Motor Maps for Nonlinear Control
Arena, P. (2002) · AIP conference proceedings · doi:10.1063/1.1487520 · Cited by 1
Frequency hysteresis phenomena in the frequency switched Chua's circuit
Fortuna, L., Frasca, M., & Rizzo, A. (2002) · doi:10.1109/iscas.2001.921301 · Cited by 1
Periodic orbits and bifurcations in one-dimensional arrays of Chua's circuits
Gilli, M., Checco, P., & Corinto, F. (2003) · doi:10.1109/iscas.2003.1206429 · Cited by 1
CONTINUOUS FINITE TIME OBSERVER FOR CHAOTIC SYNCHRONIZATION
Perruquetti, W., & Floquet, T. (2006) · IFAC Proceedings Volumes · doi:10.3182/20060628-3-fr-3903.00033 · Cited by 1
On bounds of the parametric range of bifurcation of Chua's circuit
Huang, A. (1996) · IEEE Transactions on Circuits and Systems I Fundamental Theory and Applications · doi:10.1109/81.526686 · Cited by 1
“Furthermore, one wishes to know the conditions for when chaos disappears to avoid it. According to the two-parameter bifurcation diagram in the a-/3 plane, given in [ 3 ], one sees that chaos can be quenched by making Q sufficiently small or adjusting /3 appropriately when N is fixed.”
MethodChaos Control of a Four Order Chua's Circuit
Liu, X., Wang, J., & Huang, L. (2006) · Chinese Control Conference (CCC) · doi:10.1109/chicc.2006.280958 · Cited by 1
“2 PRELIMINARIES 1 INTRODUCTION Chua's circuit is one ofthe most famous chaos circuits that exhibit a wide variety of nonlinear dynamics phenomena such as bifurcation and chaos [1-4].”
BackgroundAnalysis of regularization processes in an ensemble of coupled chaotic oscillators
Kuznetsov, A. S., & Shalfeev, V. D. (1998) · Radiophysics and Quantum Electronics · doi:10.1007/bf02676502 · Cited by 1
ON THE APPROXIMATE ONE-D MAP IN CHUA'S OSCILLATOR
FORBES, P., Boyle, S., O'DONOGHUE, K., et al. (2005) · International Journal of Bifurcation and Chaos · doi:10.1142/s0218127405013472 · Cited by 1
Robust dichotomy for nonlinear uncertain systems
Wang, J., Duan, Z., & Huang, L. (2006) · Chinese Control Conference (CCC) · doi:10.1109/chicc.2006.280928 · Cited by 1
“Consider Chua’s circuit system ([2], [3], [7])”
Chaos induced by noise in a four-frequency ring gas laser
Svirina, L. P. (2004) · Optics and Spectroscopy · doi:10.1134/1.1781298 · Cited by 1
A computer program for accurate time-domain analysis of 1D-arrays of Chua's oscillators
Biey, M., Gilli, M., Maio, I. A., et al. (2002) · IEEE International Symposium on Circuits and Systems proceedings · doi:10.1109/iscas.1998.704030 · Cited by 1
Robust dichotomy and controller design for a class of nonlinear systems
Wang, J., Huang, L., & Yang, Y. Z. (2007) · Proceedings of the IEEE Conference on Decision & Control, including the Symposium on Adaptive Processes · doi:10.1109/cdc.2007.4434312 · Cited by 1
“APPLICATIONS TO CHUA’S CIRCUIT AND EXTENDED CHUA’S CIRCUIT Chua’s circuit is a simple electronic circuit exhibiting a wide variety of bifurcation and chaotic phenomena [13], [14], [15].”
BackgroundThe Parameter to Characterize Chaotic Dynamics
Bonasera, A. (2000) · International Joint Conference on Neural Network · Cited by 1
Kennedy, M. P. (1999) · Wiley Encyclopedia of Electrical and Electronics Engineering · doi:10.1002/047134608x.w2524 · Cited by 1
Grinits, E. V., & Bottura, C. P. (2005) · IEEE International Symposium on Intelligent Control · doi:10.1109/isic.2004.1387728 · Cited by 1
Analytical-numerical methods for finding hidden oscillations in dynamical systems
Vagaytsev, V. (2012) · Jyväskylä University Digital Archive (University of Jyväskylä) · Cited by 1
Optymalizacja statystycznych systemów uczących się
Jankowski, S. (2012) · Prace Naukowe Politechniki Warszawskiej. Elektronika
Signal transmission through a chain of Chua's circuits
Itoh, M., Murakami, H., & Chua, L. O. (2002) · IEEE International Symposium on Circuits and Systems proceedings · doi:10.1109/iscas.1995.519936
Dynamic analysis and chaos control of a modified Chua's circuit
Li, R., & Liu, C. (2013)
OGORZAŁEK, M. J. (1999) · Wiley Encyclopedia of Electrical and Electronics Engineering · doi:10.1002/047134608x.w2541
An approach for generating n×m-scroll attractors in hyperchaotic coupled Chua circuits
Cafagna, D., & Grassi, G. (2003) · Asia Pacific Conference on Circuits and Systems · doi:10.1109/apccas.2002.1115070
Baglio, S., Cristaudo, R., Fortuna, L., et al. (2002) · IEEE International Symposium on Circuits and Systems proceedings · doi:10.1109/iscas.1995.523857
A CNN implementation of a hysteresis chaos generator
Aissi, C., & Shams, A. (2003) · IEEE International Symposium on Circuits and Systems proceedings · doi:10.1109/iscas.1999.777651
Вадивасова, Т. Е. (2004) · AIP conference proceedings · doi:10.1063/1.1846463
Synchronization in Complex Networks: Properties and Tools
Biey, M., Corinto, F., Mishkovski, I., et al. (2012) · Understanding complex systems · doi:10.1007/978-3-642-33359-0_5
Mascolo, S., & Grassi, G. (2002) · doi:10.1109/iscas.2001.921295
Design of High-Order Chaotic Circuits
Tang, Z. L., & Yu, S. (2013) · Advanced materials research · doi:10.4028/www.scientific.net/amr.722.33
“Introduction The Chua's circuit processes an irreplaceable special status and plays a key role in the study of chaotic circuit, because of which it has become the model of chaotic theory and the experimental analysis of chaotic phenomenon [1].”
BackgroundA switching piecewise-linear control approach for generating of multi-scroll chaotic attractors
Wang, M., Sun, G., & Huang, L. (2010) · International Symposium on Systems and Control in Aeronautics and Astronautics · doi:10.1109/isscaa.2010.5634021
Impulsive control of Chua'S Oscillators with exponential decay rate
Jin, J., Yu, Y., Pu, X., et al. (2010) · International Conference on Communications, Circuits and Systems · doi:10.1109/icccas.2010.5581872
Research on a family of n-scroll chaos generators
Zhang, G., Yang, S.-Z., & He, L.-F. (2008) · Journal of Physics Conference Series · doi:10.1088/1742-6596/96/1/012202
“See the detailed basic characteristics of Chua’s circuit in [1-7].”
BackgroundModelling and quantifying bifurcation diagrams of Chua's oscillator
Dittmar, J., & Parlitz, U. (2007) · PAMM · doi:10.1002/pamm.200700735
Chua's Circuit Dynamics Controlled by Using a Water Electrolytic Cell Component
Buscarino, A., Famoso, C., Fortuna, L., et al. (2014) · International Journal of Bifurcation and Chaos · doi:10.1142/s0218127414300262
“The Chua circuit is one of the most studied chaotic circuits [Madan, 1993], with a topology and dynamics which makes it a plenty of nonlinear phenomena and a universal source of inspiration for the investigation of new implementation schemes, innovative realizations and the discovery of surprising…”
BackgroundINTRINSIC FLUCTUATION OF CHUA SYSTEM
Li, Q. S., & Lei, A. (2004) · International Journal of Bifurcation and Chaos · doi:10.1142/s021812740401117x
“Chua System and Chemical Chua System Chua system, also known as or Chua’s equation, is a canonical three-dimensional nonlinear autonomous system which is governed by the following deterministic differential equation [Madan, 1993; Pivka et al., 1996]: dx dt = α(y − bx(3) + cx) dy dt = β(x − y + z) dz dt = −γy (1)”
BackgroundFuzzy computational control for real Chua circuit
Garms, M. A., de Andrade, M. T. C., & Caldas, I. L. (2007) · Chaos Solitons & Fractals · doi:10.1016/j.chaos.2007.06.068
A Chua diode transmission line, for the propagation of a chaotic signal
Bai, E., & Lonngren, K. E. (2000) · Chaos Solitons & Fractals · doi:10.1016/s0960-0779(98)00259-8
“All rights reserved.”
ANALYSIS OF PATTERN FORMATION IN CHAIN OF COUPLED BISTABLE ELEMENTS
Kuznetsov, A. S., Forti, G. L., & Shalfeev, V. D. (2006) · International Journal of Bifurcation and Chaos · doi:10.1142/s021812740601512x
“In particular, it is the chain of bistable, trigger electronic circuits with special nonlinear couplings [Madan, 1993], or a chain of frequency controlled self-oscillators [Afraimovich et al., 1995].”
BackgroundThe Role of Input Flows in Microfluidic Experimentations
Sapuppo, F., Fortuna, L., Bucolo, M., et al. (2011) · IFAC Proceedings Volumes · doi:10.3182/20110828-6-it-1002.01604
ON THE ROLE OF RANDOM TRANSVERSAL UNCOUPLING ON RE-ENTRY FORMATION
Gómez-Gesteira, M., deCastro, M., & Pérez-Villar, V. (1999) · International Journal of Bifurcation and Chaos · doi:10.1142/s0218127499001681
“1), is a Chua's circuit [Madan, 1993; Chua, 1992], whose parameters are summarized in Table 1.”
BackgroundEncrypter Information Software Using Chaotic Generators
Cardoza-Avendaño, L., López-Gutiérrez, R. M., Inzunza-González, E., et al. (2009) · doi:10.5281/zenodo.1085825
Nucleation from the Boundaries in Excitable Media
deCastro, M., Gómez-Gesteira, M., Lorenzo, M. N., et al. (2003) · International Journal of Bifurcation and Chaos · doi:10.1142/s0218127403008235
“Chua's circuit is a nonlinear circuit [Madan, 1993; Chua, 1992] which works in an excitable state with the set of parameters shown in Table 1 [Gómez- Gesteira et al., 1999; deCastro et al., 1998; deCastro et al., 1999] .”
BackgroundChaotic and hyperchaotic dynamics in Chua’s circuits: The Adomian decomposition approach
Cafagna, D., & Grassi, G. (2007) · IEEE International Conference on Electro Information Technology · doi:10.1109/eit.2007.4374456
“INTRODUCTION In the field of electrical engineering, Chua’s circuit represents a paradigm for studying chaotic phenomena [1].”
BackgroundThe effects of state feedback and input transformation with application to Lur'e systems
Duan, Z., Wang, J., & Huang, L. (2005) · Proceedings of the ... American Control Conference/Proceedings of the American Control Conference · doi:10.1109/acc.2005.1470042
“Special Lur’e systems, Chua’s circuits, have also attracted a lot of interest[12], [15], [17], [23], [24].”
BackgroundDesign of a Hysteresis Chaotic Circuit
Aissi, C., & Kazakos, D. (2001)
Adaptive Synchronization of the Modified Chua's Circuit Systems
Tian, L., Li, D., & Sun, X. (2009) · doi:10.1109/csie.2009.258
“The Chua’s circuit is an extensively studied and well-understood chaotic system [ 12 ].”
BackgroundA Universal Circuit for Studying and Generating Chaos-Part I: Routes
Wu, C. W., & Huang, A. (1993)
Controlling double-scroll chaotic Lü attractor to difformity
Bao, B., Liu, Z., & Xu, J. (2009) · IEEE Conference on Industrial Electronics and Applications (ICIEA) · doi:10.1109/iciea.2009.5138328
“Therefore, there has been increasing interest in exploiting chaotic dynamics in engineering applications, where some attentions have been focused on generating chaos via physical systems such as electronic circuits [6]-[8] and switching piecewiselinear controllers [9]-[11].”
BackgroundA modular system for constructing dynamical systems
Freeman, I. (1998) · UC Research Repository (University of Canterbury) · doi:10.26021/2011
Estabilidade de sistemas dinâmicos: Estudo do memristor
Moreira, M. D. (2014) · Americanae (AECID Library)
Chaotic system with sine-like waveform
Hu, W. S., Bao, B., & Zhang, G. (2009) · International Conference on Communications, Circuits and Systems · doi:10.1109/icccas.2009.5250390
Weakly Connected Oscillatory Networks for Information Processing
Bonnin, M., Corinto, F., & GILLI, A. (2016) · CRC Press eBooks · doi:10.1201/b10781-25
Broadband RC chaotic oscillators
Efremova, E. V., & Kuzmin, L. V. (2003) · doi:10.1109/occsc.2002.1029101
Arrays of switched Chua's circuits
Caponetto, R., Fortuna, L., Frasca, M., et al. (2004) · IEEE International Symposium on Circuits and Systems proceedings · doi:10.1109/iscas.2004.1328679
“In a dimensionless form the Chua’s circuit can be represented by the following equations [3]:”
BackgroundFour Examples of Chaotic Circuits
Buscarino, A., Fortuna, L., Frasca, M., et al. (2014) · SpringerBriefs in applied sciences and technology · doi:10.1007/978-3-319-05900-6_1
Design Considerations for Integrated
Oscillators, C.-T. C., & iguez-Vazquez, A. R. ́. (1998)
TRANSITION TO CHAOS THROUGH QUASIPERIODICITY IN SYNCHRONIZED MATSUMOTO-CHUA CIRCUITS
dos Santos, E. P. (2006)
Controlling and Exploring Nonlinear Dynamics in Circuits by Sporadic Proportional Pulses
Ruiz, P. J., & Gutiérrez, J. M. (2001)
Analog Noise Generation via Non-Linear Device
Andò, B., & Graziani, S. (2000) · Springer eBooks · doi:10.1007/978-1-4615-4391-6_4
Hatjispyros, S. J., & Merkatas, C. (2015) · arXiv (Cornell University)
From Bifurcations to Chua's Periodic Table
Tonelli, R., Chua, L. O., & Meloni, F. (2002) · MRS Proceedings · doi:10.1557/proc-731-w5.8
On-off sensors based on strange attractors
Buscarino, A., Famoso, C., Fortuna, L., et al. (2014) · Proceedings of IEEE Sensors/Proceedings of IEEE Sensors ... · doi:10.1109/icsens.2014.6985264
“Starting from the Chua’s circuit [5], we identify one resistor of the circuit as the bifurcation”
MethodSychronisation of Chua's Oscillator via the State Observer Technique
Yin, X., & Cao, Y. (2003) · International Journal of Electrical Engineering Education · doi:10.7227/ijeee.40.1.4
Vibrational and Stochastic Resonances in Spatially Periodic Systems
Rajasekar, S., & Sanjuán, M. A. F. (2015) · Springer series in synergetics · doi:10.1007/978-3-319-24886-8_5
CHUA'S CIRCUIT: CHAOTIC PHENOMENA AND APPLICATIONS
Pivka, L., Wu, C. W., & Chua, L. O. (1995) · WORLD SCIENTIFIC eBooks · doi:10.1142/9789812830609_0010
PHASE AND COMPLETE SYNCHRONIZATIONS IN TIME-DELAY SYSTEMS
Senthilkumar, D. V., Shrii, M. M., & Kurths, J. (2013) · WORLD SCIENTIFIC eBooks · doi:10.1142/9789814434805_0032
Adaptive synchronization of chaotic systems with impulsive disturbances
Cheng, C.-F., & Wang, Z. (2013) · doi:10.1109/cacs.2013.6734145
The Behavior of Synchronized Frequency in Weakly Coupled Nonidentical Periodic Oscillators
Adhyapok, P., Patra, M., & Banerjee, S. (2017) · International Journal of Bifurcation and Chaos · doi:10.1142/s0218127417500080
“Slowly ideas in synchronization found its way into different fields — from biological systems [Mirollo & Strogatz, 1990; Rompala et al., 2007] and chemical oscillator systems [Neu, 1979] to electrical and electronics systems [Madan, 1993].”
BackgroundA Tabu Search algorithm for optimal sizing of Locally Recurrent Neural Networks
Cannas, B., Fanni, A., Marchesi, M., et al. (1999) · UNICA IRIS Institutional Research Information System (University of Cagliari)
On the Existence of the Double Scroll Attractor for the Chua's Circuit with a Smooth Nonlinearity
Galias, Z., & Tucker, W. (2018) · IEEE International Symposium on Circuits and Systems proceedings · doi:10.1109/iscas.2018.8351549
“The existence of various dynamical phenomena in this system has been studied extensively in the literature [2], [3], [4], [5], [6], [7], [8], [9], [10].”
BackgroundA note on a piecewise-linear Duffing-type systemPreprint
Tigan, G., & Astolfi, A. (2005) · arXiv (Cornell University) · doi:10.48550/arxiv.math/0512341
“In fact, Chua and collaborators discovered chaotic behavior in piecewise linear systems [9].”
BackgroundEmergent Spiking in Non-Ideal Memristor NetworksPreprint
Gale, E., Costello, B. D. L., & Adamatzky, A. (2012) · arXiv (Cornell University) · doi:10.48550/arxiv.1210.8024
Transformation and Equivalence
Mathis, W. M. (2005) · CRC Press eBooks · doi:10.1201/9781420037081.ch4
Frasca, M., Gambuzza, L. V., Buscarino, A., et al. (2018) · SpringerBriefs in applied sciences and technology · doi:10.1007/978-3-319-75957-9_2
The Focus-Center-Limit Cycle Bifurcation in 3D Continuous Piecewise Linear Systems with Two Zones
Ponce, E., Ros, J., & Vela, E. (2022) · RSME Springer series · doi:10.1007/978-3-031-21135-5_8
Chaos in the Piecewise Linear Saddle-Focus with Helicoidal Return
Freire, E., Ponce, E., & Ros, J. (2024) · IFAC-PapersOnLine · doi:10.1016/j.ifacol.2024.07.064
Three-Dimensional Autonomous Systems and Chaos
Lynch, S. (2025) · Springer eBooks · doi:10.1007/978-3-031-89067-3_16
Emergent synchronization in inductively coupled heterogeneous oscillators
Faruque, S. A. K. M., Chaudhury, S., Chatterjee, K., et al. (2025) · Nonlinear Dynamics · doi:10.1007/s11071-025-11724-w
Reservoir observer with extended input spacePreprint
Liu, Y., & Chu, T. (2026) · SSRN Electronic Journal · doi:10.2139/ssrn.6505802
Recovering the Governing Equations of Chua’s Circuit by Sparse Identification from Experimental Data
Venkatachalam, A., Gambuzza, L. V., Famoso, C., et al. (2026) · International Journal of Bifurcation and Chaos · doi:10.1142/s0218127426300260
Nonlinear Dynamical Modeling of Questioning
Fortuna, L., Garraffa, G., Sferlazza, A., et al. (2026) · International Journal of Bifurcation and Chaos · doi:10.1142/s0218127426502044
Çimen, M. E. (2026) · Chaos Theory and Applications · doi:10.51537/chaos.1942344
Spatial Diversity in Reaction-Diffusion Fuzzy Cellular Networks
Bucolo, M., Cutuli, M. C., Fortuna, L., et al. (2001) · Physica-Verlag eBooks · doi:10.1007/978-3-7908-1818-5_4
“In particular, the chaotic generator has been obtained by using the sampling of the output of a Chua Circuit [10] with the following parameters a = 10, b = 14.”
Method
Metadata
- Authors
- Madan, R. N.
- Affiliations
- Office of Naval Research (US)
- Corresponding author
- Madan, Rabinder N
- Venue
- World Scientific Series on Nonlinear Science Series B
- Publisher
- WORLD SCIENTIFIC
- Published
- 1993-11
- ISSN
- 1793-1029 (ISSN-L)
- DOI
- 10.1142/1997
- OpenAlex
- W1599681710
- Cited by
- 602
- References
- 0
- Citations by year
- 2026: 7 · 2025: 3 · 2024: 7 · 2023: 4 · 2022: 7 · 2021: 4 · 2020: 11 · 2019: 14 · 2018: 13 · 2017: 12 · 2016: 13 · 2015: 15 · 2014: 19 · 2013: 16 · 2012: 19
- FWCI
- 6.41
- Type
- book
- Language
- en
- Open access
- closed
- Locations
- World scientific series on nonlinear science, series B (publishedVersion)
- CERN Document Server (European Organization for Nuclear Research) (submittedVersion)
- Topics
- Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Chaos control and synchronization
- Physical Sciences › Mathematics › Mathematical Physics › Mathematical Dynamics and Fractals
- Physical Sciences › Mathematics › Geometry and Topology › Advanced Differential Equations and Dynamical Systems
- Keywords
- Chua's circuit, chaos, nonlinear dynamics, bifurcation, chaotic systems, circuit simulation
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