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Dec 2002

Volume 12, Issue 4, pp. 985-1076

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Critical points and transitions in an electric power transmission model for cascading failure blackouts

B. A. Carreras, V. E. Lynch, I. Dobson, and D. E. Newman

Chaos 12, 985 (2002); http://dx.doi.org/10.1063/1.1505810 (10 pages) | Cited 62 times

Online Publication Date: 9 September 2002

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Cascading failures in large-scale electric power transmission systems are an important cause of blackouts. Analysis of North American blackout data has revealed power law (algebraic) tails in the blackout size probability distribution which suggests a dynamical origin. With this observation as motivation, we examine cascading failure in a simplified transmission system model as load power demand is increased. The model represents generators, loads, the transmission line network, and the operating limits on these components. Two types of critical points are identified and are characterized by transmission line flow limits and generator capability limits, respectively. Results are obtained for tree networks of a regular form and a more realistic 118-node network. It is found that operation near critical points can produce power law tails in the blackout size probability distribution similar to those observed. The complex nature of the solution space due to the interaction of the two critical points is examined.© 2002 American Institute of Physics.
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84.70.+p High-current and high-voltage technology: power systems; power transmission lines and cables
02.50.Cw Probability theory

Targeting in dissipative chaotic systems: A survey

Serdar Iplikci and Yagmur Denizhan

Chaos 12, 995 (2002); http://dx.doi.org/10.1063/1.1505809 (11 pages) | Cited 1 time

Online Publication Date: 10 September 2002

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The large number of unstable equilibrium modes embedded in the strange attractor of dissipative chaotic systems usually presents a sufficiently rich repertoire for the choice of the desirable motion as a target. Once the system is close enough to the chosen target local stabilization techniques can be employed to capture the system within the desired motion. The ergodic behavior of chaotic systems on their strange attractors guarantees that the system will eventually visit a close neighborhood of the target. However, for arbitrary initial conditions within the basin of attraction of the strange attractor the waiting time for such a visit may be intolerably long. In order to reduce the long waiting time it usually becomes indispensable to employ an appropriate method of targeting, which refers to the task of steering the system toward the close neighborhood of the target. This paper provides a survey of targeting methods proposed in the literature for dissipative chaotic systems. © 2002 American Institute of Physics.
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05.45.-a Nonlinear dynamics and chaos

Multifractal chaotic attractors in a system of delay-differential equations modeling road traffic

Leonid A. Safonov, Elad Tomer, Vadim V. Strygin, Yosef Ashkenazy, and Shlomo Havlin

Chaos 12, 1006 (2002); http://dx.doi.org/10.1063/1.1507903 (9 pages) | Cited 21 times

Online Publication Date: 13 September 2002

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We study a system of delay-differential equations modeling single-lane road traffic. The cars move in a closed circuit and the system’s variables are each car’s velocity and the distance to the car ahead. For low and high values of traffic density the system has a stable equilibrium solution, corresponding to the uniform flow. Gradually decreasing the density from high to intermediate values we observe a sequence of supercritical Hopf bifurcations forming multistable limit cycles, corresponding to flow regimes with periodically moving traffic jams. Using an asymptotic technique we find approximately small limit cycles born at Hopf bifurcations and numerically preform their global continuations with decreasing density. For sufficiently large delay the system passes to chaos following the Ruelle–Takens–Newhouse scenario (limit cycles–two-tori–three-tori–chaotic attractors). We find that chaotic and nonchaotic attractors coexist for the same parameter values and that chaotic attractors have a broad multifractal spectrum. © 2002 American Institute of Physics.
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45.70.Vn Granular models of complex systems; traffic flow
05.45.-a Nonlinear dynamics and chaos
02.30.Oz Bifurcation theory
02.30.Hq Ordinary differential equations
02.10.Ud Linear algebra

Generation of undular bores in the shelves of slowly-varying solitary waves

G. A. El and R. H. J. Grimshaw

Chaos 12, 1015 (2002); http://dx.doi.org/10.1063/1.1507381 (12 pages) | Cited 17 times

Online Publication Date: 16 September 2002

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We study the long-time evolution of the trailing shelves that form behind solitary waves moving through an inhomogeneous medium, within the framework of the variable-coefficient Korteweg–de Vries equation. We show that the nonlinear evolution of the shelf leads typically to the generation of an undular bore and an expansion fan, which form apart but start to overlap and nonlinearly interact after a certain time interval. The interaction zone expands with time and asymptotically as time goes to infinity occupies the whole perturbed region. Its oscillatory structure strongly depends on the sign of the inhomogeneity gradient of the variable background medium. We describe the nonlinear evolution of the shelves in terms of exact solutions to the KdV–Whitham equations with natural boundary conditions for the Riemann invariants. These analytic solutions, in particular, describe the generation of small “secondary” solitary waves in the trailing shelves, a process observed earlier in various numerical simulations. © 2002 American Institute of Physics.
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47.35.-i Hydrodynamic waves

An extension to chaos control via Lie derivatives: Fully linearizable systems

Ricardo Femat

Chaos 12, 1027 (2002); http://dx.doi.org/10.1063/1.1510041 (7 pages) | Cited 6 times

Online Publication Date: 8 October 2002

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The technique of using Lie derivatives to control chaos introduced by Kocarev et al. [Chaos, Solitons Fractals 9, 1359–1366 (1998)] is extended in this contribution. Here, by using Lie derivatives in an extended space state, it is proved that chaos can be practically suppressed via feedback in spite of the Lie derivative being ill-posed at the reference. The main idea is to construct a dynamically equivalent system. In this way, the chaotic system can be practically stabilized around any point of singularity x0. The Lorenz equation is used as an illustrative example to show the application in the chaos control context. © 2002 American Institute of Physics.
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05.45.Gg Control of chaos, applications of chaos

Analysis of the Fenton–Karma model through an approximation by a one-dimensional map

E. G. Tolkacheva, D. G. Schaeffer, D. J. Gauthier, and C. C. Mitchell

Chaos 12, 1034 (2002); http://dx.doi.org/10.1063/1.1515170 (9 pages) | Cited 16 times

Online Publication Date: 29 October 2002

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The Fenton–Karma model is a simplification of complex ionic models of cardiac membrane that reproduces quantitatively many of the characteristics of heart cells; its behavior is simple enough to be understood analytically. In this paper, a map is derived that approximates the response of the Fenton–Karma model to stimulation in zero spatial dimensions. This map contains some amount of memory, describing the action potential duration as a function of the previous diastolic interval and the previous action potential duration. Results obtained from iteration of the map and numerical simulations of the Fenton–Karma model are in good agreement. In particular, the iterated map admits different types of solutions corresponding to various dynamical behavior of the cardiac cell, such as 1:1 and 2:1 patterns. © 2002 American Institute of Physics.
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87.17.-d Cell processes
87.19.L- Neuroscience
87.16.D- Membranes, bilayers, and vesicles
87.19.R- Mechanical and electrical properties of tissues and organs
02.60.-x Numerical approximation and analysis
05.45.Gg Control of chaos, applications of chaos

Transport in a slowly perturbed convective cell flow

A. P. Itin, R. de la Llave, A. I. Neishtadt, and A. A. Vasiliev

Chaos 12, 1043 (2002); http://dx.doi.org/10.1063/1.1520070 (11 pages) | Cited 6 times

Online Publication Date: 13 November 2002

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We study transport properties in a simple model of two-dimensional roll convection under a slow periodic (period of order 1/ϵ≫1) perturbation. The problem is considered in terms of conservation of the adiabatic invariant. It is shown that the adiabatic invariant is well conserved in the system. It results in almost regular dynamics on large time scales (of order ∼ ϵ−3 ln ϵ) and hence, fast transport. We study both generic systems and an example having some symmetry. © 2002 American Institute of Physics.
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47.27.T- Turbulent transport processes
47.20.Bp Buoyancy-driven instabilities (e.g., Rayleigh-Benard)
05.45.-a Nonlinear dynamics and chaos

Border collision bifurcations at the change of state-space dimension

Sukanya Parui and Soumitro Banerjee

Chaos 12, 1054 (2002); http://dx.doi.org/10.1063/1.1521390 (16 pages) | Cited 8 times

Online Publication Date: 19 November 2002

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We present the theory of border collision bifurcation for the special case where the state space is piecewise smooth, but two-dimensional in one side of the borderline, and one dimensional in the other side. This situation occurs in a class of switching circuits widely used in power electronic industry. We analyze this particular class of bifurcations in terms of the normal form, where the determinant of the Jacobian matrix at one side of the borderline is greater than unity in magnitude, and in the other side it is zero. © 2002 American Institute of Physics.
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05.45.Ra Coupled map lattices
05.45.Pq Numerical simulations of chaotic systems
02.30.Oz Bifurcation theory
02.10.Yn Matrix theory
02.30.Uu Integral transforms

Generation of large-amplitude solitons in the extended Korteweg–de Vries equation

Roger Grimshaw, Dmitry Pelinovsky, Efim Pelinovsky, and Alexey Slunyaev

Chaos 12, 1070 (2002); http://dx.doi.org/10.1063/1.1521391 (7 pages) | Cited 24 times

Online Publication Date: 19 November 2002

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We study the extended Korteweg–de Vries equation, that is, the usual Korteweg–de Vries equation but with the inclusion of an extra cubic nonlinear term, for the case when the coefficient of the cubic nonlinear term has an opposite polarity to that of the coefficient of the linear dispersive term. As this equation is integrable, the number and type of solitons formed can be determined from an appropriate spectral problem. For initial disturbances of small amplitude, the number and type of solitons generated is similar to the well-known situation for the Korteweg–de Vries equation. However, our interest here is in initial disturbances of larger amplitude, for which there is the possibility of the generation of large-amplitude “table-top” solitons as well as small-amplitude solitons similar to the solitons of the Korteweg–de Vries equation. For this case, and in contrast to some earlier results which assumed that an initial disturbance in the shape of a rectangular box would be typical, we show that the number and type of solitons formed depend crucially on the disturbance shape, and change drastically when the initial disturbance is changed from a rectangular box to a “sech”-profile. © 2002 American Institute of Physics.
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46.05.+b General theory of continuum mechanics of solids
02.30.Hq Ordinary differential equations
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