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Bifurcation memory is a generalized name for some specific features of the behaviour of the dynamical system near the bifurcation. An example is the recurrent neuron memory.
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.
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 et al., 2007, [A: 4]
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]
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]
Chaos theory is an interdisciplinary area of scientific study and branch of mathematics focused on underlying patterns and deterministic laws of dynamical systems that are highly sensitive to initial conditions, and were once thought to have completely random states of disorder and irregularities. Chaos theory states that within the apparent randomness of chaotic complex systems, there are underlying patterns, interconnection, constant feedback loops, repetition, self-similarity, fractals, and self-organization. The butterfly effect, an underlying principle of chaos, describes how a small change in one state of a deterministic nonlinear system can result in large differences in a later state. A metaphor for this behavior is that a butterfly flapping its wings in Texas can cause a tornado in Brazil.
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In mathematics, particularly in dynamical systems, a bifurcation diagram shows the values visited or approached asymptotically of a system as a function of a bifurcation parameter in the system. It is usual to represent stable values with a solid line and unstable values with a dotted line, although often the unstable points are omitted. Bifurcation diagrams enable the visualization of bifurcation theory. In the context of discrete-time dynamical systems, the diagram is also called orbit diagram.
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Gunduz Caginalp was a mathematician whose research has also contributed over 100 papers to physics, materials science and economics/finance journals, including two with Michael Fisher and nine with Nobel Laureate Vernon Smith. He began his studies at Cornell University in 1970 and received an AB in 1973 "Cum Laude with Honors in All Subjects" and Phi Beta Kappa. In 1976 he received a Master's degree, and in 1978 a PhD, both also at Cornell. He held positions at The Rockefeller University, Carnegie-Mellon University and the University of Pittsburgh, where he was a Professor of Mathematics until his death on December 7, 2021. He was born in Turkey, and spent his first seven years and ages 13–16 there, and the middle years in New York City.
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A cellular model is a mathematical model of aspects of a biological cell, for the purposes of in silico research.
Biological applications of bifurcation theory provide a framework for understanding the behavior of biological networks modeled as dynamical systems. In the context of a biological system, bifurcation theory describes how small changes in an input parameter can cause a bifurcation or qualitative change in the behavior of the system. The ability to make dramatic change in system output is often essential to organism function, and bifurcations are therefore ubiquitous in biological networks such as the switches of the cell cycle.
Institute of Mathematics of the National Academy of Sciences of Ukraine is a government-owned research institute in Ukraine that carries out basic research and trains highly qualified professionals in the field of mathematics. It was founded on 13 February 1934.
In mathematics, a periodic travelling wave is a periodic function of one-dimensional space that moves with constant speed. Consequently, it is a special type of spatiotemporal oscillation that is a periodic function of both space and time.
Autowaves are self-supporting non-linear waves in active media. The term is generally used in processes where the waves carry relatively low energy, which is necessary for synchronization or switching the active medium.
In the theory of autowave phenomena an autowave reverberator is an autowave vortex in a two-dimensional active medium.
In mathematics, the Kuramoto–Sivashinsky equation is a fourth-order nonlinear partial differential equation. It is named after Yoshiki Kuramoto and Gregory Sivashinsky, who derived the equation in the late 1970s to model the diffusive–thermal instabilities in a laminar flame front. The equation was independently derived by G. M. Homsy and A. A. Nepomnyashchii in 1974, in connection with the stability of liquid film on an inclined plane and by R. E. LaQuey et. al. in 1975 in connection with trapped-ion instability. The Kuramoto–Sivashinsky equation is known for its chaotic behavior.
