In the Lagrangian representation, the problem of advection of a passive marker particle by a prescribed flow defines a dynamical system. For two-dimensional incompressible flow this system is Hamiltonian and has just one degree of freedom. For unsteady flow the system is non-autonomous and one must in general expect to observe chaotic particle motion. These ideas are developed and subsequently corroborated through the study of a very simple model which provides an idealization of a stirred tank. In the model the fluid is assumed incompressible and inviscid and its motion wholly two-dimensional. The agitator is modelled as a point vortex, which, together with its image(s) in the bounding contour, provides a source of unsteady potential flow. The motion of a particle in this model device is computed numerically. It is shown that the deciding factor for integrable or chaotic particle motion is the nature of the motion of the agitator. With the agitator held at a fixed position, integrable marker motion ensues, and the model device does not stir very efficiently. If, on the other hand, the agitator is moved in such a way that the potential flow is unsteady, chaotic marker motion can be produced. This leads to efficient stirring. A certain case of the general model, for which the differential equations can be integrated for a finite time to produce an explicitly given, invertible, area-preserving mapping, is used for the calculations. The paper contains discussion of several issues that put this regime of chaotic advection in perspective relative to both the subject of turbulent advection and to recent work on critical points in the advection patterns of steady laminar flows. Extensions of the model, and the notion of chaotic advection, to more realistic flow situations are commented upon.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112084001233

Stirring by chaotic advection

18. R. C. Desrosiers, AIDS Pes. Horn. Retrovlr. 10, 331 fibrillation (6). Fuirthermi-ore, the width of 8. R. M. Ruprecht et al., J. AIDS 3, 591 (1990); P. M. (1994). thcse frcurc pcr a erltdt Ruprecht et al., Proc. Nat!. Acad. Sci. U.S.A. 87, 19. D. D. Ho and Y. Cao. N. Engl. 3. Med. 332. 1647reule spcrca beeatdo 5558 (1990); R. C. Horn, P. W. Finberg, S. Mullaney, (1995). the frequency of rotation of the rotor, thei H. M. Ruprecht, 3. Virol. 65, 220 (1991);R. M. RU20. F. Kirchhoff, H. W. Kostler, R. C. Desrosiers. 3. Virol. speed of its motion, and the wave speed prechit and R. Bronson, DNA Ce//Pie!O. 13, 59 (1994). 68. 2031 (1994). throuui h oplrpeninnicod 9 R. M. Huprecht et al.. unpublished data. 21.~Wethank L. Geronimo tor preparing this maniuscript. \"I~teDppe hnmeo cod 10. M. L. Marthas et a!., 3. Virol. 64, 3694 (1990). Supported in part by NIH grants 2RD1 A132330-05 MItL to the following relationship: 11 3. 5. Dibbs et a)., AIDS Res. Hum. Retrovir. 10, 607 and 2BD1 A135533 02, DAMD grant 1 7 94-J-4431 (1994). to H.M.R., and bytheCentertorAIDS Research core 1/1+ O, ./)<<f, < /(I >~5t 12, M. L. Marthas et a)., 3. Virol. 69, 4198 (1995). grant (P3

http://science.sciencemag.org/content/sci/270/5239/1222.full.pdf

Mechanisms of Cardiac Fibrillation

Chua's circuit is a simple electronic network which exhibits a variety of bifurcation phenomena and attractors. The circuit consists of two capacitors, an inductor, a linear resistor, and a nonlinear resistor. This paper describes the design methodology for a robust practical op amp implementation of Chua's circuit. In addition, we present experimental results and SPICE simulations for a working circuit using o -the-shelf components.

http://nonlinear.eecs.berkeley.edu/chaos/RobustOpAmpRealizationOfChuaCircuit.pdf

Robust OP Amp realization of Chua's circuit

In this introductory tutorial paper, we demonstrate the generality of Chua's oscillator in generating chaos and bifurcation phenomena by electronic laboratory experiments which illustrate the standard routes to chaos, and by giving a result which shows that Chua's oscillator can generate the same qualitative behavior as any member of a 21-parameter family C of continuous, odd-symmetric, piecewise-linear vector field in R/sup 3/. This result is of fundamental importance because it unifies many previously published papers on chaotic circuits and systems under one umbrella, thereby obviating the need to analyze these circuits and systems as separate and unrelated systems. Indeed, every bifurcation and chaotic phenomena exhibited by any member of the family C is also exhibited by this universal circuit. >

A universal circuit for studying and generating chaos. I. Routes to chaos

Following a non-technical personal reminiscence of the author's conception of Chua's circuit, this paper presents the systematic sequence of technical steps which the author used to design his chaotic circuit. The design procedure. though straightforward in nature. could not have been concocted without a working knowledge of several crucial properties of nonlinear circuits and their physical realizations. Die Entstehung der Chua-Schaltung

The Genesis of Chua's circuit

A new method to compute normal forms of vector-field singularities is proposed. Normal forms for some degenerate singularities of vector fields are computed. These normal forms are simpler than those known as Arnold-Takens normal form. Parameters in the normal forms are uniquely determined from the original singularity in the category of (jets of) coordinate transformations.

Normal forms for singularities of vector fields

Chaos in numerical analysis of ordinary differential equations

Central difference scheme and chaos

In this letter we report the numerical observation of the co-existence of three distinct chaotic attractors for at least one choice of parameters (α = 15.60, β = 28.58, m0 = −1/7, m1 = 2/7) in Chua's circuit. This new phenomenon may have escaped earlier detection because the three attractors are located very close to each other and requires the "zooming" ability of our confinor approach to uncover them.

Coexisting chaotic attractors in chua's circuit

We apply the new concept of confinors and anti-confinors, initially defined for ordinary differential equations constrained on a cusp manifold, to the equations governing the circuit dynamics of Chua’s circuit. We especially emphasize some properties of the confinors of Chua’s equation with respect to the patterns in the time waveforms. Some of these properties lead to a very accurate numerical method for the computation of the half-Poincare maps which reveal the precise structures of Chua’s strange attractors and the exact bifurcation diagrams with the help of a special sequence of change of coordinates. We also recall how such accurate methods allow the reliable numerical observation of the coexistence of three distinct chaotic attractors for at least one choice of the parameters. Chua’s equation seemssurprisingly rich in very new behaviors not yet reported even in other dynamical systems. The application of the theory of confinors to Chua’s equation and the use of sequences of Taylor’s coordinates could give new perspectives to the study of dynamical systems by uncovering very unusual behaviors not yet reported in the literature. The main paradox here is that the theory of confinors, which could appear as a theory of rough analysis of the phase portrait of Chua’s equation, leads instead to a very accurate analysis of this phase portrait.

Shigehiro Ushiki

Papers

Normal forms for singularities of vector fields

Chaos in numerical analysis of ordinary differential equations

Central difference scheme and chaos

Coexisting chaotic attractors in chua's circuit

The theory of confinors in chua’s circuit: accurate analysis of bifurcations and attractors