There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

다중혈관 관상동맥 환자에서 y-문합을 이용하여 양쪽 내흉동맥만을 사용한 우회술의 조기 성적

抗原变异可使得多种致病微生物易于逃避宿主免疫应答。表达在感染红细胞表面的恶性疟原虫红细胞表面蛋白1（PfPMP1）与感染红细胞、内皮细胞、树突状细胞以及胎盘的单个或多个受体作用，在黏附及免疫逃避中起关键的作用。每个单倍体基因组var基因家族编码约60种成员，通过启动转录不同的var基因变异体为抗原变异提供了分子基础。

疟原虫var基因转换速率变化导致抗原变异[英]／Paul H, Robert P, Christodoulou Z, et al//Proc Natl Acad Sci U S A

Contents: Introduction: Differential Equations and Dynamical Systems.- An Introduction to Chaos: Four Examples.- Local Bifurcations.- Averaging and Perturbation from a Geometric Viewpoint.- Hyperbolic Sets, Sympolic Dynamics, and Strange Attractors.- Global Bifurcations.- Local Codimension Two Bifurcations of Flows.- Appendix: Suggestions for Further Reading. Postscript Added at Second Printing. Glossary. References. Index.

A Reflection on Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields

Certain subsystems of nonlinear, chaotic systems can be made to synchronize by linking them with common signals. The criterion for this is the sign of the sub-Lyapunov exponents. We apply these ideas to a real set of synchronizing chaotic circuits.

Synchronization in chaotic systems

Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields

One important aspect of dynamical systems is the study of the long-term behavior of a set of ordinary differential equations (ODEs) In recent years many systems that are simple to write down have been discovered whose solutions are chaotic They oscillate irregularly, never settling down to a regular pattern Two trajectories which start close together will separate quickly Systems whose time evolution is governed by a parameter p can undergo intriguing variations in the behavior of trajectories In many cases, there are values p* such that the long-term behavior of typical trajectories of p p*  For example, the system may go from stable periodic behavior for p p*  Such sudden, discontinuous changes or "bifurcations" are quite common Research in chaos and bifurcations in dynamical processes has advanced at a rapid pace during the past decade, acquiring an extraordinary breadth of applications in fields as diverse as fluid mechanics, electrical engineering and neurophysiology The new results interest a wide spectrum of the scientific community, many of whose members, however, lack the mathematical background necessary to decipher the literature Accordingly, Guckenheimer and Holmes have written their book as a "user's guide" to an extensive and rapidly growing field The book surveys the theory and techniques needed to understand chaotic behavior of ODEs The first chapter contains a brief introduction of the theory of ODEs; it is a review of topics usually found in a standard text like Hirsch and Smale (1) The second chapter considers four examples of chaotic systems: the forced van der Pol oscillator, Duffing's equation, the celebrated Lorenz equations, and Holmes' "bouncing ball map" (perhaps more familiar as the map which describes the motion of a periodically forced, damped planar pendulum in the absence of gravity) These examples

Book Review: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields

Cheetahs and beetles run, dolphins and salmon swim, and bees and birds fly with grace and economy surpassing our technology. Evolution has shaped the breathtaking abilities of animals, leaving us the challenge of reconstructing their targets of control and mechanisms of dexterity. In this review we explore a corner of this fascinating world. We describe mathematical models for legged animal locomotion, focusing on rapidly running insects and highlighting past achievements and challenges that remain. Newtonian body--limb dynamics are most naturally formulated as piecewise-holonomic rigid body mechanical systems, whose constraints change as legs touch down or lift off. Central pattern generators and proprioceptive sensing require models of spiking neurons and simplified phase oscillator descriptions of ensembles of them. A full neuromechanical model of a running animal requires integration of these elements, along with proprioceptive feedback and models of goal-oriented sensing, planning, and learning. We outline relevant background material from biomechanics and neurobiology, explain key properties of the hybrid dynamical systems that underlie legged locomotion models, and provide numerous examples of such models, from the simplest, completely soluble "peg-leg walker" to complex neuromuscular subsystems that are yet to be assembled into models of behaving animals. This final integration in a tractable and illuminating model is an outstanding challenge.

/pdf/the-dynamics-of-legged-locomotion-models-analyses-and-4e8knboeub.pdf

The Dynamics of Legged Locomotion: Models, Analyses, and Challenges

© Publications mathématiques de l’I.H.É.S., 1979, tous droits réservés. L’accès aux archives de la revue « Publications mathématiques de l’I.H.É.S. » (http:// www.ihes.fr/IHES/Publications/Publications.html) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

/pdf/structural-stability-of-lorenz-attractors-3jifvrj2c3.pdf

Structural stability of Lorenz attractors

Mixed-mode oscillations (MMOs) are trajectories of a dynamical system in which there is an alternation between oscillations of distinct large and small amplitudes. MMOs have been observed and studied for over thirty years in chemical, physical, and biological systems. Few attempts have been made thus far to classify different patterns of MMOs, in contrast to the classification of the related phenomena of bursting oscillations. This paper gives a survey of different types of MMOs, concentrating its analysis on MMOs whose small-amplitude oscillations are produced by a local, multiple-time-scale “mechanism.” Recent work gives substantially improved insight into the mathematical properties of these mechanisms. In this survey, we unify diverse observations about MMOs and establish a systematic framework for studying their properties. Numerical methods for computing different types of invariant manifolds and their intersections are an important aspect of the analysis described in this paper.

/pdf/mixed-mode-oscillations-with-multiple-time-scales-2gobomm5x9.pdf

John Guckenheimer

Papers

Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields

Book Review: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields

The Dynamics of Legged Locomotion: Models, Analyses, and Challenges

Structural stability of Lorenz attractors

Mixed-Mode Oscillations with Multiple Time Scales