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arxiv: 1907.03469 · v1 · pith:QXJ7HOKUnew · submitted 2019-07-08 · 🌊 nlin.CD

Suppression of Chaos in Mutually Coupled Synchronized Generalized Lorenz Systems

V. Ramiya Gowse , B. Palanivel , S.V.M.Satyanarayana , S.Sivaprakasam This is my paper

Pith reviewed 2026-05-25 00:54 UTC · model grok-4.3

classification 🌊 nlin.CD
keywords chaos suppressionmutual couplingsynchronizationGeneralized Lorenz systemanti-synchronizationLyapunov exponentpermutation entropypower spectra
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The pith

Controlling a parameter in the mutual coupling between two Generalized Lorenz systems suppresses chaos while keeping the systems synchronized.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper studies two Generalized Lorenz systems whose state variables are linked by nonlinear coupling controls. Synchronization between the oscillators is first established. Adjusting one parameter that scales the coupling strength then eliminates the chaotic fluctuations, replacing them with regular oscillations, all while the synchronized state remains intact. The change is confirmed by power spectra losing their broadband character, permutation entropy dropping, and Lyapunov exponents becoming negative. The same coupled setup also permits a switch from synchronization to anti-synchronization when the systems remain in the chaotic regime.

Core claim

By suitably controlling a parameter having a bearing on the coupling coefficient between the two Lorenz oscillators, the GLS, while preserving synchronization is rendered to a state wherein chaotic nature of state variables is suppressed and state variables exhibit oscillatory character. The suppression of chaos is verified by power spectra, permutation entropy and Lyapunov exponent calculations. When operated in chaotic domain, we show the possibility of transition from the state of synchronization to the state of anti-synchronization.

What carries the argument

Nonlinear mutual coupling controls whose strength is scaled by a single tunable parameter, acting on the synchronization manifold of the two Generalized Lorenz systems.

If this is right

  • State variables change from chaotic to periodic oscillatory behavior while synchronization holds.
  • Power spectra, permutation entropy, and Lyapunov exponents all indicate loss of chaos.
  • The same coupling arrangement permits a transition from synchronization to anti-synchronization inside the chaotic regime.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same coupling adjustment might be tried on other pairs of chaotic oscillators to test whether chaos can be removed without desynchronization.
  • The observed route from synchronized chaos to synchronized order could be examined for use in switching between signal types in coupled dynamical systems.
  • Varying the coupling parameter while staying on the synchronization manifold may reveal additional ordered states not reported in the present calculations.

Load-bearing premise

The nonlinear coupling and chosen parameter values suppress chaos without breaking the synchronization manifold or creating new instabilities.

What would settle it

A positive largest Lyapunov exponent or a broadband power spectrum persisting after the parameter is tuned would show that chaos was not suppressed.

Figures

Figures reproduced from arXiv: 1907.03469 by B. Palanivel, S.Sivaprakasam, S.V.M.Satyanarayana, V. Ramiya Gowse.

Figure 5
Figure 5. Figure 5: figure 5. It is observed, from Figure 5, that, as t [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
read the original abstract

In this work, the dynamics of a system of mutually coupled Generalized Lorenz systems (GLS) is investigated. The state variables of two Lorenz oscillators are coupled mutually via non-linear controls and synchronization is achieved between the state variables. We find that by suitably controlling a parameter having a bearing on the coupling coefficient between the two Lorenz oscillators, the GLS, while preserving synchronization is rendered to a state wherein chaotic nature of state variables is suppressed and state variables exhibit oscillatory character. The suppression of chaos is verified by power spectra, permutation entropy and Lyapunov exponent calculations. When operated in chaotic domain, we show the possibility of transition from the state of synchronization to the state of anti-synchronization.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript examines mutually coupled Generalized Lorenz systems (GLS) with nonlinear controls. It claims that synchronization between the two oscillators is achieved, and that tuning a parameter related to the coupling coefficient suppresses chaos on the synchronized trajectory (converting it to oscillatory behavior) while preserving synchronization. Suppression is verified via power spectra, permutation entropy, and Lyapunov exponent calculations. The work also reports a transition from synchronization to anti-synchronization when the system is operated in the chaotic regime.

Significance. If the central claim holds, the result illustrates parameter-based chaos suppression in a synchronized pair of chaotic oscillators without loss of the synchronization manifold. The use of multiple independent diagnostics (spectra, entropy, LE) for the on-manifold dynamics is a positive feature of the numerical exploration.

major comments (2)
  1. [chaos suppression verification / Lyapunov exponent results] The abstract and results sections claim that synchronization is preserved after the coupling-related parameter is tuned to suppress chaos. However, the reported Lyapunov exponents, power spectra, and permutation entropy characterize the on-manifold dynamics; no explicit computation or plot of the transverse Lyapunov exponents (or equivalent error dynamics transverse to the manifold) is shown after the parameter change. Because the tuned parameter directly modifies the coupling, transverse stability must be re-verified to support the preservation claim.
  2. [numerical results on synchronization preservation] The weakest assumption in the central claim (that the nonlinear coupling controls and chosen parameter range suppress on-manifold chaos without violating the synchronization manifold) is not directly tested. The manuscript should include at least one figure or table showing that the largest transverse Lyapunov exponent remains negative across the reported parameter values that convert the synchronized state from chaotic to periodic.
minor comments (2)
  1. [Abstract] The abstract refers to 'a parameter having a bearing on the coupling coefficient' without naming the parameter or giving its range; this should be stated explicitly (with equation reference) for reproducibility.
  2. [model equations] Notation for the coupling terms and the controlled parameter should be introduced once in the model equations section and used consistently thereafter.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments on our manuscript. The points raised highlight the need for explicit verification of transverse stability, which we will address in the revision.

read point-by-point responses

  1. Referee: [chaos suppression verification / Lyapunov exponent results] The abstract and results sections claim that synchronization is preserved after the coupling-related parameter is tuned to suppress chaos. However, the reported Lyapunov exponents, power spectra, and permutation entropy characterize the on-manifold dynamics; no explicit computation or plot of the transverse Lyapunov exponents (or equivalent error dynamics transverse to the manifold) is shown after the parameter change. Because the tuned parameter directly modifies the coupling, transverse stability must be re-verified to support the preservation claim.

    Authors: We agree that the current manuscript lacks explicit post-tuning verification of transverse stability. In the revised manuscript we will add calculations of the transverse Lyapunov exponents (including the largest one) for the tuned values of the coupling-related parameter, confirming that it remains negative and thereby supporting preservation of the synchronization manifold. revision: yes

  2. Referee: [numerical results on synchronization preservation] The weakest assumption in the central claim (that the nonlinear coupling controls and chosen parameter range suppress on-manifold chaos without violating the synchronization manifold) is not directly tested. The manuscript should include at least one figure or table showing that the largest transverse Lyapunov exponent remains negative across the reported parameter values that convert the synchronized state from chaotic to periodic.

    Authors: We acknowledge that the manuscript does not directly test transverse stability after the parameter change. The revised version will incorporate a new figure or table displaying the largest transverse Lyapunov exponent versus the coupling-related parameter over the range where on-manifold chaos is suppressed, verifying that it stays negative. revision: yes

Circularity Check

0 steps flagged

No circularity: numerical exploration of coupled system dynamics

full rationale

The paper consists of numerical investigation of mutually coupled Generalized Lorenz systems. Synchronization is achieved via nonlinear coupling, a parameter related to the coupling coefficient is tuned to convert on-manifold dynamics from chaotic to oscillatory, and suppression is verified directly via power spectra, permutation entropy, and Lyapunov exponents. No derivation chain, fitted parameter, self-citation, ansatz, or uniqueness theorem is invoked that reduces any claimed result to its own inputs by construction. The central claims rest on explicit simulation outputs rather than self-referential definitions or predictions.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities are introduced; the study numerically explores the dynamics of an established system class under a described coupling scheme.

pith-pipeline@v0.9.0 · 5656 in / 1096 out tokens · 23589 ms · 2026-05-25T00:54:16.871386+00:00 · methodology

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Reference graph

Works this paper leans on

45 extracted references · 45 canonical work pages · 1 internal anchor

  1. [1]

    Introduction A great deal of research had been carried out on ch aos synchronization and control of chaos in various systems due to their ap plicability in multiple disciplines of research and applications [1-13]. Th e dynamical behavior of any chaotic system is known to be dependent sensitively on the initial values and the attractors and phase portraits...

    work page

  2. [2]

    System Description We consider two Lorenz oscillators and they are mut ually coupled via non-linear controls [26]. Lorenz oscillator -1 (LO-1) is descr ibed by the system of following equations: ( ) 33213 231212 1121 ucx xxx uxxdx bx x uxxax ++= +−+= +−= & & & (1) and the second Lorenz oscillator (LO-2) is described as follows : ( ) 63213 31212 4121 5 ucy...

    work page

  3. [3]

    1( E E E c cdab baa EEEV σσ σ & ( ) ( )TEEEQEEE 321321−= (12) Where,           − −+−+ −+ = c dcba aba Q 00

    work page

  4. [4]

    This is the case if and only if the following three conditions hold: 0 00

    1( σσ σ To ensure that the origin of the error system is as ymptotically stable, we let the matrix Q be positive definite. This is the case if and only if the following three conditions hold: 0 00

    work page

  5. [5]

    In the next sec tion we present our numerical findings

    1(2) 1( ) 1()( 0.)( > − −+−+ −+ >−+−+ −+ > c dcba aba iii dcba abaii ai σσ σ σσ σ (13) All the three cases are verified for specific value s of σ and hence for the system under study, the condition 0>V& is satisfied, which implies that the error function 5 is asymptotically stable at origin. In the next sec tion we present our numerical findings. The four...

    work page

  6. [6]

    The parameters are chosen su ch that the system is in the chaotic regime

    SUPPRESSION OF CHAOS Lorenz oscillators LO-1 and LO-2 are coupled to eac h other as defined in equations (1) and (2). The parameters are chosen su ch that the system is in the chaotic regime. The strength of coupling between th e two oscillators LO-1 and LO-2 are controlled by introducing a scale factor ( sf) for the control functions u i (i = 1 to 6). Th...

    work page

  7. [7]

    We vary the parameter σ of the control functions between 0 and 1 and seek for possible states of synchronization

    Co-existence of Synchronization and anti- synchronization The dynamics of all the three state variables are c onsidered for this study. We vary the parameter σ of the control functions between 0 and 1 and seek for possible states of synchronization. In Figure 6 (a- f), we present the temporal evolution of error dynamics (defined by equation 4) along with ...

    work page

  8. [8]

    The temporal evolution of the state variables is studie d and their synchronization properties are investigated

    CONCLUSION Two Lorenz oscillators are coupled mutually via non -linear control functions. The temporal evolution of the state variables is studie d and their synchronization properties are investigated. The strength of coupli ng, as controlled by a scale factor, is varied (increased) which is found to res ult in a reduction in the complexity of chaotic dy...

    work page

  9. [9]

    Pecora L M , Carroll T L, Synchronization in chaoti c systems, Phys Rev Lett 1990; 64:821-25

    REFERENCE: 1. Pecora L M , Carroll T L, Synchronization in chaoti c systems, Phys Rev Lett 1990; 64:821-25

    work page 1990

  10. [10]

    Pecora L M, Carroll T L, Johnson G A, Mar D J, Fund amentals of synchronization in chaotic systems, concepts, and a pplications, Chaos 1997;7: 520-43

    work page 1997

  11. [11]

    Gonzalez-Miranda J M, Synchronization and control o f chaos: An introduction for scientists and engineers, Imperial College press 2004. 14

    work page 2004

  12. [12]

    Hayes S, Grebogi C and Yorke J A: Communicating wit h chaos, Phys Rev Lett 1993;70: 3031-34

    work page 1993

  13. [13]

    Annovazzi-Lodi V, Donati S , Scire A, Synchronizati on of chaotic lasers by optical feedback for cryptographic applications, IEEE J. Quant. Electron 1997;33: 1449-54

    work page 1997

  14. [14]

    Shinbrot T, Ott E, Grebogi C, Yorke J A, Using chao s to direct trajectories to target, Phys .Rev.Lett 1990;65: 3215-18

    work page 1990

  15. [15]

    Medvedev G S, Synchronization of Coupled limit cycl es, J Nonlin Sci 2011;21:441-64

    work page 2011

  16. [16]

    Ujjwal S R, Punetha N, Prasad A, Ramaswamy R, Emerg ence of chimeras through induced multistability, Phys. Rev. E,2007;95:032203

    work page 2007

  17. [17]

    Rosenblum M G, Pikovsky A S, Kurths J, Phase synchr onization of chaotic oscillators, Phys Rev Lett 1996 ;76: 1804-07

    work page 1996

  18. [18]

    Goluskin D, Bounding averages rigorously using semi definite programming: Mean moments of the Lorenz system, J N onlin sci 2018; 28:621-51

    work page 2018

  19. [19]

    Wang L, Peng J, Jin Y, Ma J, Synchronous dynamics a nd bifurcation analysis in two delay coupled oscillators with recurrent inhibitory loops, J Nonlin Sci 2013;23:283-302

    work page 2013

  20. [20]

    Ataeiand M, Haghighatdar F, Adaptive nonlinear feed back design for chaos suppression and regulation in uncertain unifi ed chaotic system, Int J Innov Comput I 2011; 7(5(A)):2051-62

    work page 2011

  21. [21]

    Pyragas K (1992) Continuous control of chaos by sel f-controlling feedback, Phys Lett A 1992; 170:421-28

    work page 1992

  22. [22]

    Ott E, GrebogiC, Yorke J A, Controlling Chaos, Phys . Rev. Lett. 1990; 641196-99

    work page 1990

  23. [23]

    Fronzoni L, Giocondo M, Experimental evidence of su ppression of chaos by resonant parametric perturbations, Phys. Rev. A 1991; 43:6483-87

    work page 1991

  24. [24]

    Chaurasia S S, Sinha S , Suppression of chaos throu gh coupling to an external chaotic system, Nonlin Dyn 2017;87:159–167

    work page 2017

  25. [25]

    Aguirre A, Billings S A, Closed-Loop suppression of chaos in nonlinear driven oscillators, J Nonlin Sci 1995; 5: 189-206

    work page 1995

  26. [26]

    Chil M K, Rim S H, Kye W H, Ryu J W, Park Y J, Ant isynchronization of chaotic oscillators, Phys Lett A 2003;320: 39–46. 15

    work page 2003

  27. [27]

    Rajesh S, Nandakumaran V M, Suppression of chaos in a directly modulated semiconductor laser with delayed feedback , Phys Lett A 2003; 319:340-47

    work page 2003

  28. [28]

    Chen J, Lu J, Wu X, Bidirectionally coupled synchro nization of the generalized Lorenz systems, JSSC 2011; 24:433-448

    work page 2011

  29. [29]

    Liu B, Li J P, Zheng W, Synchronization and adaptiv e anti- synchronization control for Lorenz systems under ch annel noise with applications, Asian J Control 2013;15( 4):1-11

    work page 2013

  30. [30]

    Guo S, Ma J, Alsaed A , Suppression of chaos via co ntrol of energy flow, Pramana – J. Phys. 2018;90(39):1-7

    work page 2018

  31. [31]

    Chacon R, Suppression of chaos by external parametr ic perturbation, Phys. Rev. E 1995;51(1): 761-764

    work page 1995

  32. [32]

    Rev.E 2005;72:036206-1-6

    Choe C -U, Höhne K, Benner H, Kivshar Y S, Chaos su ppression in the parametrically driven Lorenz system, Phys. Rev.E 2005;72:036206-1-6

    work page 2005

  33. [33]

    Lima R, Pettini M, Suppression of chaos by resonant parametric perturbations, Phys. Rev. A 1990; 41 (2):726-33

    work page 1990

  34. [34]

    Chai Y, Chen L,Wu R, Inverse projective synchroniza tion between two different hyperchaotic systems with fractional order, J Appl Math 2012;1- 18

    work page 2012

  35. [35]

    Rosenstein’ M T, Collins J J, De Luca C J, A practi cal method for calculating largest Lyapunov exponents from small d ata sets, Physica D 1993;65: 117-134

    work page 1993

  36. [36]

    Wolf A, Swift J B, Swinney H L, Vastano J A, Determ ining lyapunov exponents from a time series, Physica D 1985; 16:285-317

    work page 1985

  37. [37]

    Abarbanel H D I, Brown R, Kennel M B, Local Lyapuno v exponents computed from observed data, J Nonlin Sci1992;2:343-65

    work page

  38. [38]

    Bandt C, Pompe B, Permutation entropy: a natural co mplexity measure for time series, Phys Rev Lett 2002; 88: 174102

    work page 2002

  39. [39]

    Riedl M, Muller A, Wessel N, Practical consideratio ns of permutation entropy, Eur. Phys. J. Special Topics 2013:249-262

    work page 2013

  40. [40]

    Liu L, Miao S , Cheng M , Gao X, Permutation entrop y for random binary sequences, Entropy 2015;17:8207-16. 16

    work page 2015

  41. [41]

    Li C, Chen Q, Huang T: Coexistence of anti-phase an d complete synchronization in coupled chen system via a single variable, Chaos, Sol and Frac 2008 ;38: 461-464

    work page 2008

  42. [42]

    Sivaprakasam S, Pierce I, Rees P, Spencer P S, Shor e K.A, Inverse synchronization in semiconductor laser diodes, Phys . Rev. A 2001;64:013805

    work page 2001

  43. [43]

    RamiyaGowse V, Palanivel B, Sivaprakasam S, Co-exis tence of synchronization and anti-synchronization in general ized Lorenz system with application to secure communications, arXiv:1805.01122

    work page internal anchor Pith review Pith/arXiv arXiv

  44. [44]

    Bhowmick S K, Hens C, Ghosh D, Dana S K, Mixed sync hronization in chaotic oscillators using scalar coupling, Phys Lett A 2012 ; 376:2490-95

    work page 2012

  45. [45]

    Ren L, Guo R, Synchronization and Antisynchronizati on for a class of chaotic systems by a simple adaptive controller, Ma th Problems in Engineering 2015;1-7

    work page 2015