Bifurcation memory
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Bifurcation memory is a generalized name for some specific features of the behaviour of the
dynamical system In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space. Examples include the mathematical models that describe the swinging of a ...
near the
bifurcation Bifurcation or bifurcated may refer to: Science and technology * Bifurcation theory, the study of sudden changes in dynamical systems ** Bifurcation, of an incompressible flow, modeled by squeeze mapping the fluid flow * River bifurcation, the ...
.


General information

The phenomenon is known also under the names of "''stability loss delay for dynamical bifurcations''" and "''ghost attractor''". The essence of the effect of bifurcation memory lies in the appearance of a special type of
transition process Transition or transitional may refer to: Mathematics, science, and technology Biology * Transition (genetics), a point mutation that changes a purine nucleotide to another purine (A ↔ G) or a pyrimidine nucleotide to another pyrimidine (C ↔ ...
. 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", 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. In the literature, 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; 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":


History of studying

The earliest of those described on this subject in the scientific literature should be recognized, perhaps, the result presented in 1973, which was obtained under the guidance of , 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 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
ship A ship is a large watercraft that travels the world's oceans and other sufficiently deep waterways, carrying cargo or passengers, or in support of specialized missions, such as defense, research, and fishing. Ships are generally distinguished ...
s. Subsequently, similar phenomena were also found in biological systems — in the system of blood coagulation and in one of the mathematical models of
myocardium Cardiac muscle (also called heart muscle, myocardium, cardiomyocytes and cardiac myocytes) is one of three types of vertebrate muscle tissues, with the other two being skeletal muscle and smooth muscle. It is an involuntary, striated muscle that ...
.


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. In addition, the special sort of
tachycardia Tachycardia, also called tachyarrhythmia, is a heart rate that exceeds the normal resting rate. In general, a resting heart rate over 100 beats per minute is accepted as tachycardia in adults. Heart rates above the resting rate may be normal (su ...
s connected with the effects of bifurcation memory are considered in
cardiophysics Cardiophysics is an interdisciplinary science that stands at the junction of cardiology and medical physics, with researchers using the methods of, and theories from, physics to study cardiovascular system at different levels of its organisation ...
.


See also

* Bifurcation (disambiguation) *
Bifurcation diagram In mathematics, particularly in dynamical systems, a bifurcation diagram shows the values visited or approached asymptotically (fixed points, periodic orbits, or chaotic attractors) of a system as a function of a bifurcation parameter in the syst ...
*
Bifurcation theory Bifurcation theory is the mathematical study of changes in the qualitative or topological structure of a given family of curves, such as the integral curves of a family of vector fields, and the solutions of a family of differential equations. Mo ...
*
Phase portrait A phase portrait is a geometric representation of the trajectories of a dynamical system in the phase plane. Each set of initial conditions is represented by a different curve, or point. Phase portraits are an invaluable tool in studying dyn ...


Notes


References

* Books * Papers {{Reflist , group="A:" , refs = {{cite journal , last1 = Ataullakhanov , first1 = F I , last2 = Zarnitsyna , first2 = V I , last3 = Kondratovich , first3 = A Yu , last4 = Lobanova , first4 = E S , last5 = Sarbash , first5 = V I , title = A new class of stopping self-sustained waves: a factor determining the spatial dynamics of blood coagulation , url = http://ufn.ru/ru/articles/2002/6/c/ , journal = Phys. Usp. , type = journal , year = 2002 , volume = 45 , issue = 6 , pages = 619–636 , doi = 10.1070/PU2002v045n06ABEH001090 , s2cid = 250754001 , issn = 0042-1294 {{cite journal , last1 = Ataullakhanov , first1 = F I , last2 = Lobanova , first2 = E S , last3 = Morozova , first3 = O L , last4 = Shnol’ , first4 = E E , last5 = Ermakova , first5 = E A , last6 = Butylin , first6 = A A , last7 = Zaikin , first7 = A N , title = Intricate regimes of propagation of an excitation and self-organization in the blood clotting model , url = http://ufn.ru/ru/articles/2007/1/d/ , journal = Phys. Usp. , type = journal , year = 2007 , volume = 50 , pages = 79–94 , doi = 10.1070/PU2007v050n01ABEH006156 , s2cid = 53344915 , issn = 0042-1294 {{cite journal , last1 = Elkin , first1 = Yu. E. , last2 = Moskalenko , first2 = A.V. , last3 = Starmer , first3 = Ch.F. , title = Spontaneous halt of spiral wave drift in homogeneous excitable media , url = http://mi.mathnet.ru/eng/mbb/v2/i1/p73 , journal = Mathematical Biology & Bioinformatics , type = journal , year = 2007 , volume = 2 , issue = 1 , pages = 1–9 , issn = 1994-6538 {{cite journal , last1 = Moskalenko , first1 = A. V. , last2 = Elkin , first2 = Yu. E. , title = The lacet: a new type of the spiral wave behavior , journal = Chaos, Solitons and Fractals , type = journal , year = 2009 , volume = 40 , issue = 1 , pages = 426–431 , doi = 10.1016/j.chaos.2007.07.081 , bibcode = 2009CSF....40..426M , issn = 0960-0779 {{cite journal , last1 = Deco , first1 = G , last2 = Jirsa , first2 = VK , title = Ongoing cortical activity at rest: criticality, multistability, and ghost attractors , pmid = 22399758 , pmc = 6621046 , journal = J Neurosci , type = journal , year = 2012 , volume = 32 , issue = 10 , pages = 3366–75 , doi = 10.1523/JNEUROSCI.2523-11.2012 {{cite journal, last1=Feigin , first1=M I , script-title=ru:Проявление эффектов бифуркационной памяти в поведении динамической системы , trans-title=Manifestation of the bifurcation memory effect in behaviour of dynamic system , url=http://journal.issep.rssi.ru/ , journal=Soros Educational Journal , type=journal , year=2001 , volume=7 , issue=3 , pages=121–127 , language=ru , url-status=dead , archive-url=https://web.archive.org/web/20071130004824/http://journal.issep.rssi.ru/ , archive-date=November 30, 2007 {{cite journal , last1 = Feigin , first1 = M I , script-title=ru:О двукратных проявлениях эффекта бифуркационной памяти в динамических системах , trans-title=On twice repeated manifestation of the bifurcation memory effect in dynamical systems , url = http://www.vntr.ru/ftpgetfile.php?id=133 , journal = Вестник научно-технического развития , type = journal , year = 2008 , volume = 3 , issue = 7 , pages = 21–25 , language = ru , issn = 2070-6847 {{cite journal , last1 = Feigin , first1 = M , last2 = Kagan , first2 = M , title = Emergencies as a manifestation of effect of bifurcation memory in controlled unstable systems , journal = International Journal of Bifurcation and Chaos , type = journal , year = 2004 , volume = 14 , issue = 7 , pages = 2439–2447 , doi = 10.1142/S0218127404010746 , bibcode = 2004IJBC...14.2439F , issn = 0218-1274 {{cite journal , last1 = Nishiura , first1 = Y , last2 = Ueyama , first2 = D , title = A skeleton structure of self-replicating dynamics , journal = Physica D , type = journal , year = 1999 , volume = 130 , issue = 1–2 , pages = 73–104 , doi = 10.1016/S0167-2789(99)00010-X , bibcode = 1999PhyD..130...73N , issn = 0167-2789 , hdl = 2115/69146 , hdl-access= free {{cite journal , last1 = Shishkova , first1 = M A , title = Studies of a system of differential equations with a small parameter at the highest derivative , journal = Soviet Math. Dokl. , type = journal , year = 1973 , volume = 14 , pages = 384–387 Biophysics Nonlinear systems Dynamical systems Non-equilibrium thermodynamics