Submitted Successfully!
To reward your contribution, here is a gift for you: A free trial for our video production service.
Thank you for your contribution! You can also upload a video entry or images related to this topic.
Version Summary Created by Modification Content Size Created at Operation
1 -- 1169 2023-06-15 01:25:58 |
2 format correction -26 word(s) 1143 2023-06-15 02:32:14 |

Video Upload Options

Do you have a full video?


Are you sure to Delete?
If you have any further questions, please contact Encyclopedia Editorial Office.
Elaskar, S.; Del Río, E. Chaotic Intermittency. Encyclopedia. Available online: (accessed on 24 June 2024).
Elaskar S, Del Río E. Chaotic Intermittency. Encyclopedia. Available at: Accessed June 24, 2024.
Elaskar, Sergio, Ezequiel Del Río. "Chaotic Intermittency" Encyclopedia, (accessed June 24, 2024).
Elaskar, S., & Del Río, E. (2023, June 15). Chaotic Intermittency. In Encyclopedia.
Elaskar, Sergio and Ezequiel Del Río. "Chaotic Intermittency." Encyclopedia. Web. 15 June, 2023.
Chaotic Intermittency

Chaotic intermittency is characterized by a signal that alternates aleatory between long regular (pseudo-laminar) phases and irregular bursts (pseudo-turbulent or chaotic phases).

chaotic intermittency types maps RPD noise

1. Introduction

Chaos theory has many fields of investigation, such as the different routes to chaos. One route is chaotic intermittency, which has been observed in several subjects. Batchelor and Townsend, in the middle of the previous century, used the word intermittency to give an account of the fluctuating velocity in turbulent flows [1]. Subsequently, intermittency has been found in many and varied physical phenomena such as the nonlinear behavior of transient periodic plasma and conducting fluids [2][3][4][5], fluid mechanics and turbulent flows [6][7][8][9][10], Rayleigh–Benard convection [11], electronic digital oscillator [12], logistic map [13], Alfven wave-fronts and derivative nonlinear Schrodinger equation [14][15], premixed combustion [16], Lorenz system [17], coupled oscillators [18][19][20][21], catalytic reactors [22], Ginzburg Landau equation [23], solar cycles [24], spatiotemporal chaos [25][26], thermoacoustic instability [27], control chaos [28], etc. Furthermore, chaotic intermittency is also observed in systems in economics [29][30], medicine, [31][32], neuroscience [33][34], genetics [35], and marine biology [36][37]. Therefore, a more suitable understanding of chaotic intermittency can collaborate to describe these phenomena accurately. In addition, the correct description of chaotic intermittency possesses significance for systems whose precise equations are partially or totally unknown.
There are three classical routes by which continuous or discrete dynamical systems can evolve from regular functioning to chaotic behaviors: quasi-periodic route, period-doubling scenario, and intermittency [38]. In chaotic intermittency, the dynamical system solutions display alternation between regular or pseudo-laminar phases and chaotic bursts or non-regular phases. The laminar phases correspond to pseudo-equilibrium, pseudo-periodic solutions, and quasi-invariant objects close to them that the system may consume for a long time. On the other hand, the burst ones are consistent with chaotic evolution [39].
Approximately 50 years ago, chaotic intermittency was categorized into three different types, known as I, II, and III [17][40][41]. This classification was according to the fixed point eigenvalues in the local Poincaré map or the Floquet multipliers of the continuous-system periodic solution [38][39][42]. Later works introduced other types of chaotic intermittency such as on–off, eyelet, ring, and in–out, type-X, and type-V [43][44][45][46][47][48].
The monodromy operator multipliers determine the type of intermittency [49]. Type-I intermittency happens if the periodic solution loses its stability by a cyclic-fold bifurcation [50], then a multiplier goes away from the unit circle by the real axis across +1. Type-II intermittency is born in a sub-critical Hopf bifurcation or a Neimark–Sacker bifurcation [38][51]. Accordingly, two complex-conjugate Floquet multipliers get out of the unit circle. Therefore, it is a consequence of a bifurcation scenario for 𝑇2 torus breakdown. A sub-critical period-doubling bifurcation generates type-III intermittency. In this bifurcation, an unstable period-2 orbit encounters and destabilizes a stable period-1 orbit. Type-III intermittency presents a progressive increase throughout the laminar phase of a period-2 component in the motion [52]. In addition, a one-dimensional map 𝐹(𝑥) showing a sub-critical period-doubling bifurcation possesses a positive Schwartzian derivative, 𝑆𝐹(𝑥):
S F ( x ) = F ( x ) / F ( x ) 1.5 F ( x ) / F ( x ) 2 > 0 .
Chaotic intermittency may be investigated using one-dimensional maps (Poincaré maps) [38][39][42]. These maps have two main characteristics: a particular local map and a reinjection process or reinjection mechanism. The specific local map depicts the type of intermittency. The reinjection mechanism returns the trajectories from the chaotic regime to the laminar one. Therefore, the reinjection mechanism determines the reinjection probability density function (RPD). It considers only points in the laminar region, but in the preceding period they have not been there. Note that the RPD is used to describe the reinjection process and is defined by the chaotic dynamics of the system itself. The probability of the reinjection points being in a particular sub-interval inside the laminar interval equals the integral of the RPD in the sub-interval. The RPD function specifies the probability that trajectories are returned (reinjected) into the laminar zone around to the unstable or even the vanished fixed point, and together with the local map outline all the dynamical features of the system.
The precise determination of the RPD function is extremely significant in correctly describing the chaotic intermittency phenomenon. In addition, the evaluation of the RPD function from experimental or numerical is a hard task due to the large amount of data involved, and it is difficult to estimate the statistical fluctuations generated in the numerical result and experimental data. Several strategies were utilized to calculate the RPD function. Most results in the classical theory of chaotic intermittency were obtained considering uniform reinjection inside the laminar interval [41][42][53]. Other implemented RPD functions were constructed using specific characteristics of the nonlinear processes. For example, in type-I intermittency the reinjection was restricted to the fixed point [54], and for the intermittency of type-III the RPD function was assumed proportional to (𝑥𝑎)1/2, with 𝑎= constant [55]. Notwithstanding, these RPD functions cannot be generalized to other nonlinear systems.
Two new methodologies together with their theoretical background to obtain the RPD function were introduced in recent years. One of them is the M function methodology [56][57][58][59][60][61][62][63][64]. This methodology introduces a generalized power law for the RPD. It has been shown to be very accurate for a broad class of one-dimensional maps showing type-I, II, III, and V intermittencies. In addition, the M function methodology includes the classical approximation because it contains uniform reinjection as a particular case. The second one is called the continuity technique. It utilizes the Perron–Frobenius operator to compute the reinjection probability density function. In the same way as the methodology of the M function, the continuity technique has been shown to be accurate for several maps displaying different types of intermittency [65][66][67].
To more precisely describe the intermittency phenomenon, other statistical functions are used, such as the probability density of the laminar lengths (𝜓(𝑙)), the average laminar length (𝑙̲ ¯), and the characteristic relation (𝑙̲ =𝑙̲ (𝜀). Nonetheless, these functions depend on the RPD. To calculate them, researchers previously have to know the reinjection probability density function [39][42].
The RPD and the other statistical functions used to describe the chaotic intermittency are affected by the noise and the lower boundary of reinjection (LBR). The M function methodology has been extended to incorporate both phenomena [68][69][70].

2. Types of Chaotic Intermittency

Let reserachers analyze a periodic solution of an autonomous continuous-time system. It is stable for some values of the control parameters. At that time, if a control parameter is modified until the periodic solution loses stability, the development of the solution shall depend on how the Floquet multipliers go away from the unit circle in the complex plane [38][42][53]. Note that the metamorphoses of a family of solutions around a closed orbit is a complex problem of bifurcation theory [71].
The existence of type-I, II, and III intermittency depends on the monodromy operator multiplier. Type-I intermittency occurs by a cyclic-fold bifurcation, then a multiplier goes away from the unit circle across +1. Type-II intermittency is born in a sub-critical Hopf bifurcation (or a Neimark–Sacker bifurcation), in which two complex-conjugate multipliers move away from the unit circle. Finally, type-III intermittency appears if a multiplier gets out of the unit circle by −1, and a sub-critical period-doubling bifurcation happens [38][49][51].


  1. Batchelor, G.; Townsend, C. The nature of turbulent motion at large wave-number. Proc. R. Soc. London Ser. A 1949, 199, 238–255.
  2. Irimiciuc, S.; Saviuc, A.; Tudose-Sandu-Ville, F.; Toma, S.; Nedeff, F.; Marcela Rusu, C.; Agop, M. Non-Linear Behaviors of Transient Periodic Plasma Dynamics in a Multifractal Paradigm. Symmetry 2020, 12, 1356.
  3. Chertovskih, R.; Rempel, E; Chimanski, E. Magnetic field generation by intermittent convection. Phys. Lett. 2017, 381, 3300–3306.
  4. Belyaev, I.; Biryukov, D.; Gerasimov, D.; Yurin, E. On-off intermittency and hard turbulence in the flow of fluid in the magnetic field. Chaos Interdiscip. J. Nonlinear Sci. 2019, 29, 083119.
  5. Goldman, M. Plasma Wave Turbulence and Electromagnetic Radiation Caused by Electron Beams; Grant 84-0007. AFOSR-TR-86-2062; Air Office Scientific Research: Arlington, Virginia, 1986.
  6. Schmiegel, J.; Pons, F. Stochastic Intermittency Fields in a von Kármán Experiment. Symmetry 2021, 13, 1752.
  7. Manasseh, R. Breakdown regimes of inertia waves in a precessing cylinder. J. Fluid Mech. 1992, 243, 261–296.
  8. Loiseau, J.; Robinet, J.; Leriche, E. Intermittency and transition to chaos in the cubical lid-driven cavity flow. Fluid Dyn. Res. 2016, 48, 061421.
  9. Gao, J.; Zheng, Z.; Ma, J. Controlling turbulence via target waves generated by local phase space compression. Int. J. Mod. Phys. B 2008, 22, 3855–3863.
  10. Malm, A.; Waigh, T. Elastic turbulence in entangled semi-dilute DNA solutions measured with optical coherence tomography velocimetry. Sci. Rep. 2017, 7, 1186.
  11. Malasoma, J.; Werny, P.; Boiron, M. Multichannel type-I intermittency in two models of Rayleigh-Benard convection. Phys. Rev. Lett. 2004, 51, 487–500.
  12. Stavrinides, S.; Miliou, A.; Laopoulos, T.; Anagnostopoulos, A. The intermittency route to chaos of an electronic digital oscillator. Int. J. Bifurc. Chaos 2008, 18, 1561–1566.
  13. Elaskar, S.; del Rio, E.; Elaskar, S. Intermittency Reinjection in the Logistic Map. Symmetry 2022, 14, 281.
  14. Sanmartín, J.; López-Rebollal, O.; del Río, E.; Elaskar, S. Hard transition to chaotic dynamics in Alfven wave-fronts. Phys. Plasmas 2004, 11, 2026–2035.
  15. Sánchez-Arriaga, G.; Sanmartín, J.; Elaskar, S. Damping models in the truncated derivative nonlinear Schrödinger equation. Phys. Plasmas 2007, 14, 082108.
  16. Pizza, G.; Frouzakis, C.; Mantzaras, J. Chaotic dynamics in premixed Hydrogen/air channel flow combustion. Combust. Theor. Model 2012, 16, 275–299.
  17. Manneville, P.; Pomeau, Y. Intermittency and Lorenz model. Phys. Lett. A 1979, 75, 1–2.
  18. Casagrande, V.; Mikhailov, A. Birhythmicity, synchronization, and turbulence in an oscillatory system with nonlocal inertial coupling. Phys. D Nonlinear Phenom. 2005, 205, 154–169.
  19. Saha, A.; Feudel, U. Characteristics of in-out intermittency in delay-coupled FitzHugh–Nagumo oscillators. Eur. Phys. J. Spec. Top. 2018, 227, 1205–1219.
  20. Gil, S.; Mikhailov, A. Networks on the edge of chaos: Global feedback control of turbulence in oscillator networks. Phys. Rev. E 2009, 79, 026219.
  21. Hu, B.; Zheng, Z. Phase synchronizations: Transitions from high-to low-dimensional tori through chaos. Int. J. Bifurc. Chaos 2000, 10, 2399–2414.
  22. Elnashaie, S.; Abashar, M.; Teymour, F. Bifurcation, instability and chaos in fluidized bed catalytic reactors with consecutive exothermic chemical reactions. Chaos Solitons Fractals 1993, 3, 1–33.
  23. Li, H.-H.; Xiao, J.-H.; Hu, G.; Hu, B. Intermittencies in complex Ginzburg–Landau equation by varying system size. Chin. Phys. B 2010, 19, 050516.
  24. Serre, T.; Nesme-Ribes, E. Nonlinear analysis of solar cycles. Astron. Astrophys. 2000, 360, 319–330.
  25. Coulibaly, S.; Clerc, M.; Selmi, F.; Barbay, S. Extreme events following bifurcation to spatiotemporal chaos in a spatially extended microcavity laser. Phys. Rev. A 2017, 95, 023816.
  26. Pavlos, G.; Iliopoulos, A.; Tsoutsouras, V.; Karakatsanis, L.; Pavlos, E. Spatiotemporal chaos in distributed systems: Theory and practice. In Chaotic Systems: Theory and Applications; World Scientific: Singapore, 2010; pp. 268–283.
  27. Sujith, R.; Unni, V. Complex system approach to investigate and mitigate thermoacoustic instability in turbulent combustors. Phys. Fluids 2020, 32, 061401.
  28. Zambrano, S.; Mariño, I.P.; Sanjuán, M. Controlling crisis-induced intermittency using its relation with a boundary crisis. New J. Phys. 2009, 11, 023025.
  29. Chian, A. Complex System Approach to Economic Dynamics. Lecture Notes in Economics and Mathematical Systems; Springer: Berlin, Germany, 2007.
  30. Bhansali, R.; Holland, M.; Kokoszka, P. Intermittency, long-memory and financial returns. In Long Memory in Economics; Springer: Berlin, Germany, 2007; pp. 39–68.
  31. Zebrowski, J.; Baranowski, R. Type-I intermittency in nonstationary systems: Models and human heart-rate variability. Physics A 2004, 336, 74–86.
  32. Velazquez, J.; Khosravani, H.; Lozano, A.; Bardakjian, B.; Carlen, P.; Wennberg, R. Type III intermittency in human partial epilepsy. Eur. J. Neurosci. 1999, 11, 2571–2576.
  33. Paradisi, P.; Allegrini, P.; Gemignani, A.; Laurino, M.; Menicucci, D.; Piarulli, A. Scaling and intermittency of brains events as a manifestation of consciousness. AIP Conf. Proc. 2012, 1510, 151–161.
  34. Bashkirtseva, I.; Nasyrova, V.; Ryashko, L. Scaling and intermittency of brains events as a manifestation of consciousness. Chaos Solitons Fractals 2018, 110, 76–81.
  35. Suzuki, Y.; Lu, M.; Ben-Jacob, E.; Onuchic, J. Periodic, quasi-periodic and chaotic dynamics in simple gene elements with time delays. Sci. Rep. 2016, 6, 21037.
  36. Gardiner, J.; Atema, J. The function of bilateral odor arrival time differences in olfactory orientation of sharks. Curr. Biol. 2010, 20, 1187–1191.
  37. Atema, J.; Brönmark, C.; Hansson, L. Aquatic odor dispersal fields: Opportunities and limits of detection, communication and navigation. In Chemical Ecology in Aquatic Systems; Oxford University Press: New York, NY, USA, 2012; pp. 1–18.
  38. Nayfeh, A.; Balachandran, B. Applied Nonlinear Dynamics; Wiley: New York, NY, USA, 1995.
  39. Elaskar, S.; del Rio, E. New Advances on Chaotic Intermittency and Applications; Springer: New York, NY, USA, 2017; ISBN 978–3-319-47836-4.
  40. Manneville, P. Intermittency, self-similarity and 1/f spectrum in dissipative dynamical systems. J. Phys. 1980, 41, 1235–1243.
  41. Hirsch, J.; Huberman, B.; Scalapino, D. Theory of intermittency. Phys. Rev. Lett. 1982, 25, 519–532.
  42. Schuster, H.; Just, W. Deterministic Chaos; Wiley VCH: Mörlenbach, Germany, 2005.
  43. Kaplan, H. Return to type-I intermittency. Phys. Rev. Lett. 1992, 68, 553–557.
  44. Price, T.; Mullin, P. An experimental observation of a new type of intermittency. Phys. D 1991, 48, 29–52.
  45. Platt, N.; Spiegel, E.; Tresser, C. On-off intermittency: A mechanism for bursting. Phys. Rev. Lett. 1993, 70, 279–282.
  46. Pikovsky, A.; Osipov, G.; Rosenblum, M.; Zaks, M.; Kurths, J. Attractor–repeller collision and eyelet intermittency at the transition to phase synchronization. Phys. Rev. Lett. 1997, 79, 47–50.
  47. Lee, K.; Kwak, Y.; Lim, T. Phase jumps near a phase synchronization transition in systems of two coupled chaotic oscillators. Phys. Rev. Lett. 1998, 81, 321–324.
  48. Hramov, A.; Koronovskii, A.; Kurovskaya, M.; Boccaletti, S. Ring intermittency in coupled chaotic oscillators at the boundary of phase synchronization. Phys. Rev. Lett. 2006, 97, 114101.
  49. Guckenheimer, J.; Holmes, P. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Field; Springer: New York, NY, USA, 1983.
  50. Bai-lin, H. Elementary Simbolic Dynamics and Chaos in Dissipative Systems; World Scientific: Singapure, 1989.
  51. Wiggins, S. Introduction to Applied Nonlinear Dynamical Systems and Chaos; Springer: New York, NY, USA, 2003.
  52. Laugesen, J.; Carlsson, N.; Mosckilde, E.; Bountis, T. Anomalous statistics for type-III intermittency. Open Syst. Inf. Dyn. 1998, 4, 393–405.
  53. Marek, M.; Schreiber, I. Chaotic Behaviour of Deterministic Dissipative Systems; Cambridge University Press: Cambridge, UK, 1995.
  54. Kye, W.; Kim, C. Characteristic relations of type-I intermittency in presence of noise. Phys. Rev. E 2000, 62, 6304–6307.
  55. Kye, W.; Rim, S.; Kim, C.; Lee, J.; Ryu, J.; Yeom, B.; Park, Y. Experimental observation of characteristic relations of type-III intermittency in the presence of noise in a simple electronic circuit. Phys. Rev. E 2003, 68, 036203.
  56. del Rio, E.; Velarde, M.; Rodríguez-Lozano, A. Long time data series and difficulties with the characterization of chaotic attractors: A case with intermittency III. Chaos Solitons Fractals 1994, 4, 2169–2179.
  57. del Rio, E.; Elaskar, S. New characteristic relation in type-II intermittency. Int. J. Bifurc. Chaos 2010, 20, 1185–1191.
  58. Elaskar, S.; del Rio, E.; Donoso, J. Reinjection probability density in type-III intermittency. Phys. A 2011, 390, 2759–2768.
  59. del Rio, E.; Elaskar, S.; Makarov, S. Theory of intermittency applied to classical pathological cases. Chaos 2013, 23, 033112.
  60. del Rio, E.; Elaskar, S.; Donoso, J. Laminar length and characteristic relation in type-I intermittency. Commun. Numer. Simul. Nonlinear Sci. 2014, 19, 967–976.
  61. Krause, G.; Elaskar, S.; del Rio, E. Type-I intermittency with discontinuous reinjection probability density in a truncation model of the derivative nonlinear Schrödinger equation. Nonlinear Dyn. 2014, 77, 455–466.
  62. Krause, G.; Elaskar, S.; del Rio, E. Noise effect on statistical properties of type-I intermittency. Phys. A 2014, 402, 318–329.
  63. Elaskar, S.; del Rio, E.; Krause, G.; Costa, A. Effect of the lower boundary of reinjection and noise in type-II intermittency. Nonlinear Dyn. 2015, 79, 1411–1424.
  64. Elaskar, S.; del Rio, E.; Grioni, M. Chaotic intermittency with non-differentiable M(x) function. REDIM Rev. Fac. Ing. 2023, in press.
  65. Elaskar, S.; del Rio, E.; Zapico, E. Evaluation of the statistical properties for type-II intermittency using the Perron-Frobenius operator. Nonlinear Dyn. 2016, 86, 1107–1116.
  66. Elaskar, S.; del Rio, E.; Schulz, W. Analysis of the Type V Intermittency Using the Perron-Frobenius Operator. Symmetry 2022, 14, 2519.
  67. Elaskar, S.; del Rio, E.; Lorenzón, D. Calculation of the Statistical Properties in Intermittency Using the Natural Invariant Density. Symmetry 2021, 13, 935.
  68. del Rio, E.; Sanjuan, M.; Elaskar, S. Effect of noise on the reinjection probability density in intermittency. Commun. Numer. Simul. Nonlinear Sci. 2012, 17, 3587–3596.
  69. Elaskar, S.; del Rio, E.; Costa, A. Reinjection probability density for type-III intermittency with noise and lower boundary of reinjection. J. Comput. Nonlinear Dyn. ASME 2017, 12, 031020.
  70. Elaskar, S.; del Rio, E.; Gutiérrez Marcantoni, L. Non-uniform reinjection probability density function in type V intermittency. Nonlinear Dyn. 2018, 92, 683697.
  71. Arnold, V. Geometrical Methods in the Theory of Differential Equations; Springer: Berlin, Germany, 1988.
Contributors MDPI registered users' name will be linked to their SciProfiles pages. To register with us, please refer to : ,
View Times: 328
Revisions: 2 times (View History)
Update Date: 16 Jun 2023
Video Production Service