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1

Stability Analysis of Three Coupled Kerr Oscillators: Implications for Quantum Computing

Chmielewski Kuba, Grygiel Krzysztof, Bartkiewicz Karol

Computational Methods in Science and Technology

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2025

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Vol. 31, No. 1-3

33--47

EN

We investigate the classical dynamics of optical nonlinear Kerr couplers, focusing on their potential relevance to quantum computing applications. The system consists of three Kerr-type nonlinear oscillators arranged in two configurations: a triangular arrangement, where each oscillator is coupled to the others, and a sandwich arrangement, where only the middle oscillator interacts with the two outer ones. The system is driven by an external periodic field and subject to dissipative processes. Its evolution is governed by six non-autonomous differential equations derived from a Kerr Hamiltonian with nonlinear coupling terms. We demonstrate that even for identical Kerr media, the interplay between nonlinear couplings and mismatched fundamental and pump frequencies gives rise to rich and complex dynamics, including the emergence of multiple stable attractors. These attractors are highly sensitive to both the coupling configuration and initial conditions. A key contribution of this work is a detailed stability analysis based on numerical calculation of Lyapunov exponents, revealing transitions from regular to chaotic dynamics as damping is reduced. We identify critical damping thresholds for the onset of chaos and characterize phenomena such as chaotic beats. These findings offer insights for potential experimental realizations and are directly relevant to emerging quantum technologies, where Kerr parametric oscillators play a central role in quantum gates, error correction protocols, and quantum neural network architectures.

2

Lyapunov Instability in Cell Models With and Without Time-Reversible Heat Transfer

Hoover William Graham, Hoover Carol Griswold

Computational Methods in Science and Technology

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2025

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Vol. 31, No. 1-3

5--11

EN

Two-temperature pedagogical cell models, extensions of the equilibrium Einstein model of solid state physics, can allow nonequilibrium hot-to-cold heat transfer. These heat-flow models can be driven by thermostatted temperature differences, for instance between horizontal and vertical degrees of freedom. We present pedagogical benchmark Lyapunov exponents for four equilibrium cell models: isoenergetic Hamiltonian, constrained Isokinetic, and two Isothermal NoséHoover models. We also compute representative Lyapunov exponents for two members of a family of two-temperature dissipative Nosé-Hoover cell models. Further exploration of such dissipative models is the subject of the 2024–2025 $1000 Snook Prize.

3

A novel 5D hyperchaotic system with multi-layer attractors: sliding mode control and electronic circuit implementation

Mechekef Manal, Meddour Lotfi

Archives of Control Sciences

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2025

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Vol. 35, No. 4

635--653

EN

This paper introduces a novel 5D hyperchaotic system derived from a 3D chaotic system. The system has a unique equilibrium point at the origin and exhibits two positive Lyapunov exponents, demonstrating a complex multi-layer attractor structure. Its rich dynamical behavior suggests strong potential for applications in secure communications and control systems engineering. In line with these capabilities, we propose a sliding mode control strategy to stabilize the system at the origin. The designed controllers drive system states toward a predefined sliding surface and maintain them there, with convergence and stability rigorously proven using Lyapunov theory. Furthermore, we realize the system as an electronic circuit in Multisim and verify its hyperchaotic behavior through phase portrait analysis in MATLAB.

4

Influence of arc ignition voltage on vibrations generated in autonomous RLC circuits. Part. 2. Oscillations in circuits with arc of undefined and specified ignition voltage values

Sawicki Antoni

Materials Science and Welding Technologies / Technologie Materiałowe i Spawalnicze

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2024

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Vol. 68, No. 2

art. no. 6

EN

The determination of parameters and initial conditions responsible for the formation of chaotic vibrations necessitated the considering of mathematical models of simple circuits with electric arc of unspecified ignition voltage. The use of numerical calculations made it possible to identify ranges of circuit element parameters, where at least one of the Lyapunov exponents of chaotic attractors had a positive value. The subsequent stage of the research work discussed in the article involved the reduction of arc ignition voltage until all exponents reached negative values. The above-named situation corresponds to the disappearance of chaotic vibrations in the circuit. The tests revealed low relative current values of the arc model causing the aforementioned bifurcation.

PL

W celu ustalenia parametrów oraz warunków początkowych powstawania drgań chaotycznych, rozpatrzono modele matematyczne prostych obwodów z łukiem elektrycznym o nieokreślonej wartości napięcia zapłonu. Korzystając z obliczeń numerycznych, uzyskano przedziały wartości parametrów elementów obwodu, w których przynajmniej jeden z wykładników Lapunowa atraktorów chaotycznych ma wartość dodatnią. Następnie zmniejszano wartość napięcia zapłonu łuku aż wszystkie wykładniki uzyskały ujemne wartości. Taka sytuacja odpowiada zanikowi drgań chaotycznych w obwodzie. W badaniach stwierdzono małe wartości względne prądu modelu łuku powodującego taką bifurkację.

5

A new modified WINDMI jerk system with exponential and sinusoidal nonlinearities, its bifurcation analysis, multistability, circuit simulation and synchronization design

Mohamed Mohamad Afendee, Vaidyanathan Sundarapandian, Hannachi Fareh, Sambas Aceng, Darwin P.

Archives of Control Sciences

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2023

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Vol. 33, No. 4

711--735

EN

In this work, a new 3-D modified WINDMI chaotic jerk system with exponential and sinusoidal nonlinearities is presented and its dynamical behaviours and properties are investigated. Firstly, some properties of the system are studied such as equilibrium points and their stability, Lyapunov exponents and Kaplan-Yorke dimension. Also, we study the new jerk system dynamics using numerical simulations and analyses, including phase portraits, Lyapunouv exponent spectrum, bifurcation diagram and Poincaré map, 0-1 test. Next, we exhibit that the new 3-D chaotic modified WINDMI jerk system has multistability with coexisting chaotic attractors. Moreover, we design an electronic circuit using MultiSim 14.1 for real implementation of the modified WINDMI chaotic jerk system. Finally, we design an active synchronization scheme for the complete synchronization of the modified WINDMI chaotic jerk systems via backstepping control.

6

Dynamical study of Lyapunov exponents for Hide's coupled dynamo model

Alresheedi Teflah, Allahem Ali

Demonstratio Mathematica

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2021

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Vol. 54, nr 1

189--195

EN

In this paper, we introduced the Lyapunov exponents (LEs)as a significant tool that is used to study the numerical solution behavior of the dynamical systems. Moreover, Hide’s coupled dynamo model presents a valuable dynamical study. We simulate the convergence of the LEs of the model in three cases by means of periodic flow, regular flow, and chaos flow. In addition, we compared these cases in logic connections and proved them in a mathematical way.

7

Response of Lyapunov exponents to diffusion state of biological networks

Altuntas Volkan, Gok Murat, Kocal Osman Hilmi

International Journal of Applied Mathematics and Computer Science

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2020

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Vol. 30, no. 4

689--702

EN

The topologies of protein-protein interaction networks are uncertain and noisy. The network topology determines the reliability of computational knowledge acquired from noisy networks and can impose the deterministic and non-deterministic character of the resulting data. In this study, we analyze the effect of the network topology on Lyapunov exponents and its relationship with network stability. We define the methodology to convert the network data into signal data and obtain the Lyapunov exponents for a variety of networks. We then compare the Lyapunov exponent response and the stability results. Our technique can be applied to all types of network topologies as demonstrated with our experiments, conducted on both synthetic and real networks from public databases. For the first time, this article presents findings where Lyapunov exponents are evaluated under topological mutations and used for network analysis. Experimental results show that Lyapunov exponents have a strong correlation with network stability and both are correlatively affected by the network model. Hence we develop a novel coefficient, termed LEC, to measure the robustness of biological networks. LEC can be applied to real or synthetic biological networks rapidly. Results are a striking indication that the Lyapunov exponent is a potential candidate measure for network analysis.

8

Synchronization of fractional order Rabinovich-Fabrikant systems using sliding mode control techniques

Kumar Sanjay, Singh Chaman, Prasad Sada Nand, Shekhar Chandra, Aggarwal Rajiv

Archives of Control Sciences

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2019

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Vol. 29, No. 2

307--322

EN

In this research article, we present the concepts of fractional-order dynamical systems and synchronization methodologies of fractional order chaotic dynamical systems using slide mode control techniques. We have analysed the different phase portraits and time-series graphs of fractional order Rabinovich-Fabrikant systems. We have obtained that the lowest dimension of Rabinovich-Fabrikant system is 2.85 through utilization of the fractional calculus and computational simulation. Bifurcation diagrams and Lyapunov exponents of fractional order Rabinovich-Fabrikant system to justify the chaos in the systems. Synchronization of two identical fractional-order chaotic Rabinovich-Fabrikant systems are achieved using sliding mode control methodology.

9

The Φ4 Model, Chaos, Thermodynamics, and the 2018 SNOOK Prizes in Computational Statistical Mechanics

Hoover W. G., Hoover C. G.

Computational Methods in Science and Technology

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2018

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Vol. 24, No. 2

159--163

EN

The one-dimensional Φ4 Model generalizes a harmonic chain with nearest-neighbor Hooke’s-Law interactions by adding quartic potentials tethering each particle to its lattice site. In their studies of this model Kenichiro Aoki and Dimitri Kusnezov emphasized its most interesting feature: because the quartic tethers act to scatter long-wavelength phonons, Φ4 chains exhibit Fourier heat conduction. In his recent Snook-Prize work Aoki also showed that the model can exhibit chaos on the three-dimensional energy surface describing a two-body two-spring chain. That surface can include at least two distinct chaotic seas. Aoki pointed out that the model typically exhibits different kinetic temperatures for the two bodies. Evidently few-body Φ4 problems merit more investigation. Accordingly, the 2018 Prizes honoring Ian Snook (1945-2013) will be awarded to the author(s) of the most interesting work analyzing and discussing few-body Φ4 models from the standpoints of dynamical systems theory and macroscopic thermodynamics, taking into account the model’s ability to maintain a steady-state kinetic temperature gradient as well as at least two coexisting chaotic seas in the presence of deterministic chaos.

10

Symmetry, Chaos and Temperature in the One-dimensional Lattice Φ4 Theory

Aoki K.

Computational Methods in Science and Technology

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2018

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Vol. 24, No. 2

83--95

EN

The symmetries of the minimal Φ4 theory on the lattice are systematically analyzed. We find that symmetry can restrict trajectories to subspaces, while their motions are still chaotic. The chaotic dynamics of autonomous Hamiltonian systems are discussed in relation to the thermodynamic laws. Possibilities of configurations with non-equal ideal gas temperatures in the steady state in Hamiltonian systems, are investigated, and examples of small systems in which the ideal gas temperatures are different within the system are found. The pairing of local (finite-time) Lyapunov exponents are analyzed, and their dependence on various factors, such as the energy of the system, the characteristics of the initial conditions are studied and discussed. We find that for the Φ4 theory, higher energies lead to faster pairing times. We also find that symmetries can impede the pairing of local Lyapunov exponents and the convergence of Lyapunov exponents.

11

On Local Lyapunov Exponents of Chaotic Hamiltonian Systems

Hofmann T., Merker J.

Computational Methods in Science and Technology

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2018

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Vol. 24, No. 2

97--111

EN

Chaos in conservative systems, particularly in Hamiltonian systems, is different from chaos in dissipative systems. For example, not only the eigenvalues of the symmetric Jacobian, but also the global Lyapunov exponents of Hamiltonian systems occur in pairs (λ;-λ). In this article, we even show that appropriately defined local Lyapunov exponents occur in pairs, and in turn this allows to give a new and easily accessible proof of the pairing property for global Lyapunov exponents. As examples of low dimensional chaotic Hamiltonian systems, we discuss the classical Hénon-Heiles system and a sixth order generalisation. For the latter, there is numerical evidence of two disjoint chaotic seas.

12

A new chaotic system with axe-shaped equilibrium, its circuit implementation and adaptive synchronization

Vaidyanathan S., Sambas A., Mamat M.

Archives of Control Sciences

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2018

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Vol. 28, No. 3

443--462

EN

In the recent years, chaotic systems with uncountable equilibrium points such as chaotic systems with line equilibrium and curve equilibrium have been studied well in the literature. This reports a new 3-D chaotic system with an axe-shaped curve of equilibrium points. Dynamics of the chaotic system with the axe-shaped equilibrium has been studied by using phase plots, bifurcation diagram, Lyapunov exponents and Lyapunov dimension. Furthermore, an electronic circuit implementation of the new chaotic system with axe-shaped equilibrium has been designed to check its feasibility. As a control application, we report results for the synchronization of the new system possessing an axe-shaped curve of equilibrium points.

13

Instantaneous Pairing of Lyapunov Exponents in Chaotic Hamiltonian Dynamics and the 2017 Ian Snook Prizes

Hoover W. G., Hoover C. G.

Computational Methods in Science and Technology

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2017

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Vol. 23, No. 1

73--79

EN

The time-averaged Lyapunov exponents, f_ig, support a mechanistic description of the chaos generated in and by nonlinear dynamical systems. The exponents are ordered from largest to smallest with the largest one describing the exponential growth rate of the (small) distance between two neighboring phase-space trajectories. Two exponents, γ1 + γ2, describe the rate for areas defined by three nearby trajectories. γ1 + γ2 + γ3 is the rate for volumes defined by four nearby trajectories, and so on. Lyapunov exponents for Hamiltonian systems are symmetric. The time-reversibility of the motion equations links the growth and decay rates together in pairs. This pairing provides a more detailed explanation than Liouville’s for the conservation of phase volume in Hamiltonian mechanics. Although correct for long-time averages, the dependence of trajectories on their past is responsible for the observed lack of detailed pairing for the instantaneous “local” exponents, {γi(t)}. The 2017 Ian Snook Prizes will be awarded to the author(s) of an accessible and pedagogical discussion of local Lyapunov instability in small systems. We desire that this discussion build on the two nonlinear models described here, a double pendulum with Hooke’s-Law links and a periodic chain of Hooke’s-Law particles tethered to their lattice sites. The latter system is the ϕ4 model popularized by Aoki and Kusnezov. A four-particle version is small enough for comprehensive numerical work and large enough to illustrate ideas of general validity.

14

Two-dimensional Time-reversible Ergodic Maps with Provisions for Dissipation

Patra P. K.

Computational Methods in Science and Technology

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2016

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Vol. 22, No. 2

61--70

EN

A new discrete time-reversible map of a unit square onto itself is proposed. The map comprises of piecewise linear two-dimensional operations, and is able to represent the macroscopic features of both equilibrium and nonequilibrium dynamical systems. Our operations are analogous to sinusoidally driven shear in the two dimensions, and a radial compression/ expansion of a point lying outside/inside a circle centred around origin. Depending upon the radius, the map transitions from being ergodic and non-dissipative (like in equilibrium situations) to a limit cycle through intermediate multifractal situations (like in nonequilibrium situations). All dissipative cases of the proposed map suggest that the Kaplan-Yorke dimension is smaller than the embedding dimension, a feature typically arising in nonequilibrium steady-states. The proposed map differs from the existing maps like the Baker map and Arnold’s cat map in the sense that (i) it is reversible, and (ii) it generates an intricate multifractal phase-space portrait.

15

Hyperchaos, adaptive control and synchronization of a novel 4-D hyperchaotic system with two quadratic nonlinearities

Vaidyanathan S.

Archives of Control Sciences

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2016

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Vol. 26, no. 4

471--495

EN

This research work announces an eleven-term novel 4-D hyperchaotic system with two quadratic nonlinearities. We describe the qualitative properties of the novel 4-D hyperchaotic system and illustrate their phase portraits. We show that the novel 4-D hyperchaotic system has two unstable equilibrium points. The novel 4-D hyperchaotic system has the Lyapunov exponents L1 = 3.1575, L2 = 0.3035, L3 = 0 and L4 = −33.4180. The Kaplan-Yorke dimension of this novel hyperchaotic system is found as DKY = 3.1026. Since the sum of the Lyapunov exponents of the novel hyperchaotic system is negative, we deduce that the novel hyperchaotic system is dissipative. Next, an adaptive controller is designed to stabilize the novel 4-D hyperchaotic system with unknown system parameters. Moreover, an adaptive controller is designed to achieve global hyperchaos synchronization of the identical novel 4-D hyperchaotic systems with unknown system parameters. The adaptive control results are established using Lyapunov stability theory. MATLAB simulations are depicted to illustrate all the main results derived in this research work

16

The dynamics of a parametrically driven damped pendulum

Das A., Kumar K.

International Journal of Applied Mechanics and Engineering

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2015

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Vol. 20, no. 2

257--266

EN

Ordered and chaotic states of a parametrically driven planar pendulum with viscous damping are numerically investigated. The damping makes the number of chaotic windows fewer but with larger width. Stroboscopic maps of the chaotic motion of the pendulum, driven either subharmonically or harmonically, show strange attractors with inversion symmetry in the phase plane.

17

Analysis and implementation of dynamical system with periodical discrete jumps

Gotthans T., Petrzela J.

Przegląd Elektrotechniczny

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2013

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R. 89, nr 1a

156--163

EN

The novel method for dynamical motion quantification is discussed in this paper. The proposed approach is verified on the cyclically symmetrical vector field with jump functions generating the large state space attractors recently discovered by the authors. The core part of the calculation engine involves continuous Fourier transform and Cartesian to spherical conversion. It turns out that the existing methods give the incorrect results, have huge demands on the computer performance or completely fail to converge in the finite time. The fully analog as well as hybrid circuitry implementation using off-the-shelf components is also presented and validated by means of the network simulator Orcad Pspice.

PL

W artykule opisano nową metodę dynamicznego kwantowania ruchu. Zasadniczą część obliczeń stanowi transformata Fouriera i sferyczna konwersja Kartezjańska. Założenia potwierdzają symulacje Pspice.

18

On a continuity of characteristic exponents of linear discrete time-varying systems

Czornik A., Mokry P., Niezabitowski M.

Archives of Control Sciences

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2012

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Vol. 22, no. 1

17-27

EN

In this paper we present a sufficient condition for continuity of Lyapunov exponents of discrete time-varying linear system. Basing on this result we show that Lyapunov exponents of time-invariant systems depend continuously on the time-varying perturbations.

19

Simple environment for developing methods of controlling chaos in spatially distributed systems

Korus Ł.

International Journal of Applied Mathematics and Computer Science

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2011

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Vol. 21, no. 1

149-159

EN

The paper presents a simple mathematical model called a coupled map lattice (CML). For some range of its parameters, this model generates complex, spatiotemporal behavior which seems to be chaotic. The main purpose of the paper is to provide results of stability analysis and compare them with those obtained from numerical simulation. The indirect Lyapunov method and Lyapunov exponents are used to examine the dependence on initial conditions. The net direction phase is introduced to measure the symmetry of the system state trajectory. In addition, a real system, which can be modeled by the CML, is presented. In general, this article describes basic elements of environment, which can be used for creating and examining methods of chaos controlling in systems with spatiotemporal dynamics.

20

Energometric Approach to Testing Chaoticity of Systems

Pánek D., Lahoda J., Hrušák J., Štork M.

Przegląd Elektrotechniczny

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2010

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R. 86, nr 1

166-168

EN

The paper deals with a relatively new approach to testing of chaoticity of non-linear systems. The proposed approach is based on computing of a measure of auto covariance of the abstract state space energy – covariance analysis. The results obtained by the covariance analysis are compared with standard methods: computing of Lyapunov exponents and bifurcation analysis.

PL

Artykuł porusza zagadnienie określania miary chaotyczności nieliniowych układów poprzez odpowiednie testy. Zaproponowano nowe podejście bazujące na pomiarze kowariancji abstrakcyjnego pojęcia energii przestrzeni stanu. Wyniki symulacji numerycznych uzyskane przy zastosowaniu analizy kowariancji porównano z konwencjonalnymi metodami bazującymi na estymacji wykładników Lapunowa oraz analizy bifurkacyjnej.