International Research Journal of Engineering and Technology (IRJET) e-ISSN:2395-0056
Volume: 12 Issue: 05 | May 2025 www.irjet.net p-ISSN:2395-0072
Spreadsheet-Based Coupling for Synchronization of the Henon and Logistic Maps
Aditya Krishan Goel
The Shri Ram School Aravali, Haryana. India
Abstract - This paper presents a coupling design to give the targeted synchronization in parameter mismatched chaotic discrete dynamical systems via spreadsheet software. We propose an open-plus-closed-loop (OPCL) type design for which a suitablestabilitycriterionisderived.Wedemonstratetheproposedcouplingdesignusingthe1Dlogisticmapand2DHenon map. Detailed spreadsheet implementation with formulas and tables is provided to facilitate practical implementation of chaoticsynchronizationwithoutrequiringspecializedhardwareorsoftwaretools.
Keywords – Dynamical Systems Theory, Difference and Functional Equations, Numerical Analysis, Optimization, Mathematical Applications
Introduction
Coupledchaoticoscillatorscanoftenmodelmanydynamicalsystemsandhelpunderstandcoherentbehaviorsassociated withthem.Dependingupon thenatureoftheproblemunderconsideration,eithercontinuous ordiscretedynamical systems are used as individual oscillators for investigation. In this paper, we focus on discrete maps as they are straightforward to implementinspreadsheetsoftware.
Research on synchronization in coupled chaotic dynamical systems has increased significantly in the last few years because of its significance in several fields such as encryption, information theory, signal processing, ecology, climatology, sociology, and power systems. Various types of synchronization are complete synchronization (CS), phase synchronization (PS), anti-phase synchronization (APS), lag synchronization (LS), generalized synchronization (GS), and anti-synchronization (AS).Designingcouplingmethodshavebeensuggestedforachievingaspecifictypeofsynchronization.
Theprimarydistinctionofsuchcouplingdesignisthatitbeginswithnoaprioriinformationaboutthecouplingfunction. The coupling is formulated on a general stability criterion to aim towards a desired coherent state in target dynamical systems.
We use spreadsheet software here to implement synchronization among coupled chaotic maps based solely on the coupling theory. We adopt an open-plus-closed-loop (OPCL) type coupling strategy as an approach toward the target synchronization.
THEORY OF COUPLING
Wedescribethetheoryusingann-dimensionalmapasgivenby
whichdrivesanotheridenticalmap,
toachieveagoal ,where isarealmatrix, standsforparameter(s),and denotestheiteration number.Notethatthereismismatchintheparametersofthesystems(1)and(2),whichisdenotedbytheterm .The responsesystemwiththecouplingtermisgivenby
International Research Journal of Engineering and Technology (IRJET) e-ISSN:2395-0056
Volume: 12 Issue: 05 | May 2025 www.irjet.net p-ISSN:2395-0072
TheunidirectionalOPCLcouplingisdefinedby
where istheJacobianof and isanarbitraryrealmatrixwhoseeigenvaluesmustlieinsidetheunit circleonthecomplexplaneforastablesynchronization.
Spreadsheet Implementation of Chaotic Maps Synchronization
3.1 Logistic Map
We start with the one-dimensional logistic map as a driver. The map is given by
The coupled response map in this case takes the form (6)
We can implement these equations in a spreadsheet as follows:
Table 1: Logistic map implementation
In Excel, the formulas for implementing this would be:
Constants (placed in named cells)
-mu: 3.8
-Delta_mu: 0.1
-h11: 0.65
-a11: 1 (for complete synchronization)
Cell Formulae
Assuming the constants are defined as named ranges and the initial values are in row 2:
- For driver (column B starting from B3):
International Research Journal of Engineering and Technology (IRJET) e-ISSN:2395-0056
Volume: 12 Issue: 05 | May 2025 www.irjet.net p-ISSN:2395-0072
- For response (column C starting from C3): - For error (column D starting from D3):
For anti-synchronization (AS), set a11 = -1 and h11 = 0.65, and the formula remains the same. For amplification (with factor 3), set a11 = 3 and h11 = 0.65, and the formula remains the same.
2D Henon Map
Now we consider the 2D Henon map,
as a driver system. The coupled response system is:
Implementation in a spreadsheet:
2: Henon map implementation
Table
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In Excel, the formulas for implementing this would be:
Constants (placed in named cells)
- mu1: 1.8
- Delta_mu1: 0.1
- mu2: -0.005
- Delta_mu2: 0.0005
- h11: 0.95
- h12: 0
- h21: 0
- h22: 0.95
For Complete Synchronization
- a11: 1
- a12: 0
- a21: 0
- a22: 1
Cell Formulae
Assuming constants are defined as named ranges and initial values are in row 2:
- For driver (column B starting from B3):
- For driver (column C starting from C3):
- For response (column D starting from D3):
- For response (column E starting from E3):
- For error (column F starting from F3):
- For error (column G starting from G3):
For Anti-Synchronization: change a11 = -1, a12 = 0, a21 = 0, a22 = -1 while keeping the formulas the same.
For Generalized Synchronization: change a11 = 2, a12 = -1, a21 = 2, a22 = 1 while keeping the formulas the same.
International Research Journal of Engineering and Technology (IRJET) e-ISSN:2395-0056
Volume: 12 Issue: 05 | May 2025 www.irjet.net p-ISSN:2395-0072
Results and Discussion
4.1. Logistic Map Synchronization Results
Usingthespreadsheetimplementation,varioussynchronizationstatescanbeachievedwiththelogisticmap.Forcomplete synchronization(CS),afterapproximately10iterations,theerrorbetweenthedriverandresponsesystemsapproacheszero. ThisconfirmsthattheOPCLcouplingsuccessfullyachievesthegoalofCS.
Foranti-synchronization(AS),theresponsesystemconvergestothenegativevalueofthedriversystem,asevidencedby theerrorapproachingzerowhencomparing to .
Inthecaseofamplification,theresponsesystemstabilizestoasignalthatisascaledversionofthedriversignal,withthe scalingfactordeterminedbytheparameter .
4.2
Henon Map Synchronization Results
Forthe2DHenonmap,completesynchronization(CS)isachievedwhenbothcomponents( and )oftheresponse systemconvergetotherespectivecomponentsofthedriversystem.Asshowninthetable,botherrors( and ) approachzeroafterabout10iterations.
Anti-synchronization(AS)isachievedwhenweset [ ].Inthiscase,theresponsesystemconvergestothe negativevaluesofthedriversystem.
Forgeneralizedsynchronization(GS)with [ ],theresponsevariables and convergetoalinear combinationofthedrivervariables and .Specifically, approaches andYi approaches
Fig. 1 Logistic Map Synchronization
International Research Journal of Engineering and Technology (IRJET) e-ISSN:2395-0056
Volume: 12 Issue: 05 | May 2025 www.irjet.net p-ISSN:2395-0072
4.3 Discussion
The spreadsheet realization of chaotic map synchronization has a number of benefits compared to hardware realizations:
Accessibility:Spreadsheetsoftwareisubiquitousanddoesn'tneedspecifichardwareorprogrammingexpertise.
Visualization:Spreadsheets allowinstant visualization ofthesynchronization process,facilitating easierobservation andanalysisofthedynamics.
Flexibility: Parameters can be readily changed to investigate various synchronization regimes without the need for physicalreconfiguration.
Educational value: The spreadsheet implementation is a very good educational resource to grasp chaotic dynamics andsynchronization.
Reproducibility:Theresultsarereproduciblewitheaseandwithouttheneedforspecializedhardware.
The design of the OPCL coupling turns out to be successful in realizing different forms of synchronization, such as completesynchronization,anti-synchronization,andgeneralizedsynchronization.Thestabilityconditionobtainedensuresthe synchronizationtobestableandrobustoveriterations.
One weakness of the spreadsheet method is computational efficiency. For extremely large numbers of iterations or systems of higher dimensionality, special software or hardware implementations are likely to be more practical. Yet, for pedagogical purposes and smaller-scale investigations, the spreadsheet method provides a fine balance between capability andaccessibility.
Fig. 2 Henon Map Synchronization
International Research Journal of Engineering and Technology (IRJET) e-ISSN:2395-0056
Volume: 12 Issue: 05 | May 2025 www.irjet.net p-ISSN:2395-0072
Conclusion
Wehavedemonstratedacouplingdesigntogivethetargetedsynchronizationinparametermismatchedchaoticdiscrete dynamicalsystemsviaspreadsheetsoftware.TheOPCL-basedcouplingdesignisabletoaccomplishdifferentsynchronization statessuchascompletesynchronization,anti-synchronization,andgeneralizedsynchronization.
The spreadsheet solution brings synchronization within reach of more users, without the requirement for particular hardware or programming expertise. It can be especially useful in classroom and exploratory studies of chaotic dynamic.
References
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