Bifurcation memory

Bifurcation memory is a generalized name for some specific features of the behaviour of the dynamical system near the bifurcation.

General information

The phenomenon is known also under the names of "stability loss delay for dynamical bifurcations"[A: 1] and "ghost attractor".[A: 2]

The essence of the effect of bifurcation memory lies in the appearance of a special type of transition process. An ordinary transition process is characterized by asymptotic approach of the dynamical system from the state defined by its initial conditions to the state corresponding to its stable stationary regime in the basin of attraction of which the system found itself. However, near the bifurcation boundary can be observed two types of transition processes: passing through the place of the vanished stationary regime, the dynamic system slows down its asymptotic motion temporarily, "as if recollecting the defunct orbit",[A: 3] with the number of revolutions of the phase trajectory in this area of bifurcation memory depending on proximity of the corresponding parameter of the system to its bifurcation value, — and only then the phase trajectory rushes to the state that corresponds to stable stationary regime of the system.

Bifurcation situations generate in state space bifurcation tracks that isolate regions of unusual transition processes (phase spots). The transition process in the phase spot is estimated qualitatively as a universal dependence of the index of loss of controllability on the control parameter.

— Feigin, 2004, [A: 1]

In the literature,[A: 3][A: 4] the effect of bifurcation memory is associated with a dangerous "bifurcation of merging".

The twice repeated bifurcation memory effects in dynamical systems were also described in literature;[A: 5] they were observed, when parameters of the dynamical system under consideration were chosen in the area of either crossing two different bifurcation boundaries, or their close neighbourhood.

The known definitions

It is claimed that the term "bifurcation memory":

...was proposed in Ref.[A: 6] to describe the fact that solutions of a system of differential equations (when the boundary of the region in which they exist is crossed in the parameter space) retain similarity with the already nonexistent type of solutions as long as the variable parameter values insignificantly differ from the limit value.
In mathematical models describing processes in time, this fact is known as a corollary of the theorem on continuous dependence of solutions of differential equations (on a finite time interval) on their parameters; from this standpoint, it is not fundamentally new.[note 1]

— Ataullakhanov etc., 2007, [A: 4]

History of studying

The earliest of those described on this subject in the scientific literature should be recognized, perhaps, the result presented in 1973,[A: 7] which was obtained under the guidance of L. S. Pontryagin, a Soviet academician, and which initiated then a number of foreign studies of the mathematical problem known as "stability loss delay for dynamical bifurcations".[A: 1]

A new wave of interest in the study of the strange behaviour of dynamic systems in a certain region of the state space has been caused by the desire to explain the non-linear effects revealed during the getting out of controllability of ships.[A: 3][A: 1]

Subsequently, similar phenomena were also found in biological systems — in the system of blood coagulation[A: 8][A: 4] and in one of the mathematical models of myocardium.[A: 9][A: 10]

Topicality

The topicality of scientific studies of the bifurcation memory is obviously driven by the desire to prevent conditions of reduced controllability of the vehicle.[A: 3][A: 1]

In addition, the special sort of tachycardias connected with the effects of bifurcation memory are considered in cardiophysics.[B: 1]

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See also

Notes
  1. The theorem on the continuous dependence of solutions of differential equations has not yet been proven for the general case of infinite systems of differential equations. In this sense, the thought stated in the quotation above should be still understood, hence, only as a believable hypothesis.

References
  • Books
  1. Elkin, Yu. E.; Moskalenko, A. V. (2009). "Базовые механизмы аритмий сердца" [Basic mechanisms of cardiac arrhythmias]. In Ardashev, A. V. (ed.). Клиническая аритмология [Clinical arrhythmology] (in Russian). Moscow: MedPraktika. pp. 45–74. ISBN 978-5-98803-198-7.
  • Papers
  1. Feigin, M; Kagan, M (2004). "Emergencies as a manifestation of effect of bifurcation memory in controlled unstable systems". International Journal of Bifurcation and Chaos (journal). 14 (7): 2439–2447. Bibcode:2004IJBC...14.2439F. doi:10.1142/S0218127404010746. ISSN 0218-1274.
  2. Deco, G; Jirsa, VK (2012). "Ongoing cortical activity at rest: criticality, multistability, and ghost attractors". J Neurosci (journal). 32 (10): 3366–75. doi:10.1523/JNEUROSCI.2523-11.2012. PMC 6621046. PMID 22399758.
  3. Feigin, M I (2001). Проявление эффектов бифуркационной памяти в поведении динамической системы [Manifestation of the bifurcation memory effect in behaviour of dynamic system]. Soros Educational Journal (journal) (in Russian). 7 (3): 121–127. Archived from the original on November 30, 2007.
  4. Ataullakhanov, F I; Lobanova, E S; Morozova, O L; Shnol’, E E; Ermakova, E A; Butylin, A A; Zaikin, A N (2007). "Intricate regimes of propagation of an excitation and self-organization in the blood clotting model". Phys. Usp. (journal). 50: 79–94. doi:10.1070/PU2007v050n01ABEH006156. ISSN 0042-1294.
  5. Feigin, M I (2008). О двукратных проявлениях эффекта бифуркационной памяти в динамических системах [On twice repeated manifestation of the bifurcation memory effect in dynamical systems]. Вестник научно-технического развития (journal) (in Russian). 3 (7): 21–25. ISSN 2070-6847.
  6. Nishiura, Y; Ueyama, D (1999). "A skeleton structure of self-replicating dynamics". Physica D (journal). 130 (1–2): 73–104. Bibcode:1999PhyD..130...73N. doi:10.1016/S0167-2789(99)00010-X. hdl:2115/69146. ISSN 0167-2789.
  7. Shishkova, M A (1973). "Studies of a system of differential equations with a small parameter at the highest derivative". Soviet Math. Dokl. (journal). 14: 384–387.
  8. Ataullakhanov, F I; Zarnitsyna, V I; Kondratovich, A Yu; Lobanova, E S; Sarbash, V I (2002). "A new class of stopping self-sustained waves: a factor determining the spatial dynamics of blood coagulation". Phys. Usp. (journal). 45 (6): 619–636. doi:10.1070/PU2002v045n06ABEH001090. ISSN 0042-1294.
  9. Elkin, Yu. E.; Moskalenko, A.V.; Starmer, Ch.F. (2007). "Spontaneous halt of spiral wave drift in homogeneous excitable media". Mathematical Biology & Bioinformatics (journal). 2 (1): 1–9. ISSN 1994-6538.
  10. Moskalenko, A. V.; Elkin, Yu. E. (2009). "The lacet: a new type of the spiral wave behavior". Chaos, Solitons and Fractals (journal). 40 (1): 426–431. Bibcode:2009CSF....40..426M. doi:10.1016/j.chaos.2007.07.081. ISSN 0960-0779.
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