Showing 1–17 of 17 results for author: Torrent, M

Preconditioning Magnetic Systems in Kohn-Sham Density Functional Theory

Authors: Clémentine Barat, Antoine Levitt, Marc Torrent

Abstract: The convergence of the self-consistent field iterations in Kohn-Sham density functional theory can be significantly hindered by the presence of small eigenvalues in the dielectric matrix, which are often associated with electronic phase transitions in magnetic systems. In this work, we study this type of convergence issues and propose a new preconditioning scheme to mitigate them. Our precondition… ▽ More The convergence of the self-consistent field iterations in Kohn-Sham density functional theory can be significantly hindered by the presence of small eigenvalues in the dielectric matrix, which are often associated with electronic phase transitions in magnetic systems. In this work, we study this type of convergence issues and propose a new preconditioning scheme to mitigate them. Our preconditioning scheme is inspired by the Stoner model and based on a non-interacting susceptibility that neglects orbital variations. We demonstrate the effectiveness of our approach on a range of ferromagnetic systems, showing that it can significantly reduce the number of iterations required to achieve convergence in the vicinity of magnetic phase transitions. △ Less

Submitted 25 June, 2026; originally announced June 2026.

GPU acceleration of plane-wave density functional theory calculations in Abinit

Authors: Ioanna-Maria Lygatsika, Marc Sarraute, Lucas Baguet, Pierre Kestener, Marc Torrent

Abstract: We report on the GPU port of the Abinit high-performance simulation code for plane-wave DFT calculations. Large-scale electronic structure calculations require computing the electronic wave function by solving the Kohn-Sham equations discretized over a large number of plane waves. Porting such calculations to GPU nodes relies not only on extensive usage of vendor libraries from a development persp… ▽ More We report on the GPU port of the Abinit high-performance simulation code for plane-wave DFT calculations. Large-scale electronic structure calculations require computing the electronic wave function by solving the Kohn-Sham equations discretized over a large number of plane waves. Porting such calculations to GPU nodes relies not only on extensive usage of vendor libraries from a development perspective, but also on algorithmic revisions of the iterative diagonalization procedure in the resolution of the Kohn-Sham equations to identify GPU-efficient mathematical operations (linear algebra, FFTs) applied to the wave function distributed in memory. The present contribution discusses the Abinit implementation on multi-GPU architectures, providing detailed performance results for heterogeneous CPU-GPU nodes versus CPU nodes. Particular attention is given to comparing two diagonalization algorithms -- Locally Optimal Block Preconditioned Conjugate Gradient and Chebyshev polynomial filtering -- in terms of GPU efficiency. △ Less

Submitted 27 May, 2026; v1 submitted 13 April, 2026; originally announced April 2026.

Comments: 14 pages, 7 figures

Abinit 2025: New Capabilities for the Predictive Modeling of Solids and Nanomaterials

Authors: Matthieu J. Verstraete, Joao Abreu, Guillaume E. Allemand, Bernard Amadon, Gabriel Antonius, Maryam Azizi, Lucas Baguet, Clementine Barat, Louis Bastogne, Romuald Bejaud, Jean-Michel Beuken, Jordan Bieder, Augustin Blanchet, Francois Bottin, Johann Bouchet, Julien Bouquiaux, Eric Bousquet, James Boust, Fabien Brieuc, Veronique Brousseau-Couture, Nils Brouwer Fabien Bruneval, Alois Castellano, Emmanuel Castiel, Jean-Baptiste Charraud, Jean Clerouin , et al. (49 additional authors not shown)

Abstract: Abinit is a widely used scientific software package implementing density functional theory and many related functionalities for excited states and response properties. This paper presents the novel features and capabilities, both technical and scientific, which have been implemented over the past 5 years. This evolution occurred in the context of evolving hardware platforms, high-throughput calcul… ▽ More Abinit is a widely used scientific software package implementing density functional theory and many related functionalities for excited states and response properties. This paper presents the novel features and capabilities, both technical and scientific, which have been implemented over the past 5 years. This evolution occurred in the context of evolving hardware platforms, high-throughput calculation campaigns, and the growing use of machine learning to predict properties based on databases of first principles results. We present new methodologies for ground states with constrained charge, spin or temperature; for density functional perturbation theory extensions to flexoelectricity and polarons; and for excited states in many-body frameworks including GW, dynamical mean field theory, and coupled cluster. Technical advances have extended abinit high-performance execution to graphical processing units and intensive parallelism. Second principles methods build effective models on top of first principles results to scale up in length and time scales. Finally, workflows have been developed in different community frameworks to automate \abinit calculations and enable users to simulate hundreds or thousands of materials in controlled and reproducible conditions. △ Less

Submitted 11 July, 2025; originally announced July 2025.

Roadmap on Electronic Structure Codes in the Exascale Era

Authors: Vikram Gavini, Stefano Baroni, Volker Blum, David R. Bowler, Alexander Buccheri, James R. Chelikowsky, Sambit Das, William Dawson, Pietro Delugas, Mehmet Dogan, Claudia Draxl, Giulia Galli, Luigi Genovese, Paolo Giannozzi, Matteo Giantomassi, Xavier Gonze, Marco Govoni, Andris Gulans, François Gygi, John M. Herbert, Sebastian Kokott, Thomas D. Kühne, Kai-Hsin Liou, Tsuyoshi Miyazaki, Phani Motamarri , et al. (16 additional authors not shown)

Abstract: Electronic structure calculations have been instrumental in providing many important insights into a range of physical and chemical properties of various molecular and solid-state systems. Their importance to various fields, including materials science, chemical sciences, computational chemistry and device physics, is underscored by the large fraction of available public supercomputing resources d… ▽ More Electronic structure calculations have been instrumental in providing many important insights into a range of physical and chemical properties of various molecular and solid-state systems. Their importance to various fields, including materials science, chemical sciences, computational chemistry and device physics, is underscored by the large fraction of available public supercomputing resources devoted to these calculations. As we enter the exascale era, exciting new opportunities to increase simulation numbers, sizes, and accuracies present themselves. In order to realize these promises, the community of electronic structure software developers will however first have to tackle a number of challenges pertaining to the efficient use of new architectures that will rely heavily on massive parallelism and hardware accelerators. This roadmap provides a broad overview of the state-of-the-art in electronic structure calculations and of the various new directions being pursued by the community. It covers 14 electronic structure codes, presenting their current status, their development priorities over the next five years, and their plans towards tackling the challenges and leveraging the opportunities presented by the advent of exascale computing. △ Less

Submitted 26 September, 2022; originally announced September 2022.

Comments: Submitted as a roadmap article to Modelling and Simulation in Materials Science and Engineering; Address any correspondence to Vikram Gavini (vikramg@umich.edu) and Danny Perez (danny_perez@lanl.gov)

The CECAM Electronic Structure Library and the modular software development paradigm

Authors: Micael J. T. Oliveira, Nick Papior, Yann Pouillon, Volker Blum, Emilio Artacho, Damien Caliste, Fabiano Corsetti, Stefano de Gironcoli, Alin M. Elena, Alberto Garcia, Victor M. Garcia-Suarez, Luigi Genovese, William P. Huhn, Georg Huhs, Sebastian Kokott, Emine Kucukbenli, Ask H. Larsen, Alfio Lazzaro, Irina V. Lebedeva, Yingzhou Li, David Lopez-Duran, Pablo Lopez-Tarifa, Martin Luders, Miguel A. L. Marques, Jan Minar , et al. (12 additional authors not shown)

Abstract: First-principles electronic structure calculations are very widely used thanks to the many successful software packages available. Their traditional coding paradigm is monolithic, i.e., regardless of how modular its internal structure may be, the code is built independently from others, from the compiler up, with the exception of linear-algebra and message-passing libraries. This model has been qu… ▽ More First-principles electronic structure calculations are very widely used thanks to the many successful software packages available. Their traditional coding paradigm is monolithic, i.e., regardless of how modular its internal structure may be, the code is built independently from others, from the compiler up, with the exception of linear-algebra and message-passing libraries. This model has been quite successful for decades. The rapid progress in methodology, however, has resulted in an ever increasing complexity of those programs, which implies a growing amount of replication in coding and in the recurrent re-engineering needed to adapt to evolving hardware architecture. The Electronic Structure Library (\esl) was initiated by CECAM (European Centre for Atomic and Molecular Calculations) to catalyze a paradigm shift away from the monolithic model and promote modularization, with the ambition to extract common tasks from electronic structure programs and redesign them as free, open-source libraries. They include "heavy-duty" ones with a high degree of parallelisation, and potential for adaptation to novel hardware within them, thereby separating the sophisticated computer science aspects of performance optimization and re-engineering from the computational science done by scientists when implementing new ideas. It is a community effort, undertaken by developers of various successful codes, now facing the challenges arising in the new model. This modular paradigm will improve overall coding efficiency and enable specialists (computer scientists or computational scientists) to use their skills more effectively. It will lead to a more sustainable and dynamic evolution of software as well as lower barriers to entry for new developers. △ Less

Submitted 24 June, 2020; v1 submitted 11 May, 2020; originally announced May 2020.

Comments: Revised version as finally accepted by J. Chem. Phys. to appear within the Special Topic in Electronic Structure Software (version prior to JCP's typesetting and proofs)

Density functional perturbation theory within non-collinear magnetism

Authors: Fabio Ricci, Sergei Prokhorenko, Marc Torrent, Matthieu J. Verstraete, Eric Bousquet

Abstract: We extend the density functional perturbation theory formalism to the case of non-collinear magnetism. The main problem comes with the exchange-correlation (XC) potential derivatives, which are the only ones that are affected by the non-collinearity of the system. Most of the present XC functionals are constructed at the collinear level, such that the off-diagonal (containing magnetization densiti… ▽ More We extend the density functional perturbation theory formalism to the case of non-collinear magnetism. The main problem comes with the exchange-correlation (XC) potential derivatives, which are the only ones that are affected by the non-collinearity of the system. Most of the present XC functionals are constructed at the collinear level, such that the off-diagonal (containing magnetization densities along $x$ and $y$ directions) derivatives cannot be calculated simply in the non-collinear framework. To solve this problem, we consider here possibilities to transform the non-collinear XC derivatives to a local collinear basis, where the $z$ axis is aligned with the local magnetization at each point. The two methods we explore are i) expanding the spin rotation matrix as a Taylor series, ii) evaluating explicitly the XC for the local density approximation through an analytical expression of the expansion terms. We compare the two methods and describe their practical implementation. We show their application for atomic displacement and electric field perturbations at the second order, within the norm-conserving pseudopotential methods. △ Less

Submitted 14 January, 2019; originally announced January 2019.

Journal ref: Phys. Rev. B 99, 184404 (2019)

First-principle study of paraelectric and ferroelectric CsH$_2$PO$_4$ including dispersion forces: stability and related vibrational, dielectric and elastic properties

Authors: Benoit Van Troeye, Michiel Jan van Setten, Matteo Giantomassi, Marc Torrent, Gian-Marco Rignanese, Xavier Gonze

Abstract: Using density functional theory (DFT) and density functional perturbation theory (DFPT), we investigate the stability and response functions of CsH$_2$PO$_4$, a ferroelectric material at low temperature. This material cannot be described properly by the usual (semi-)local approximations within DFT. The long-range e$^-$-e$^-$ correlation needs to be properly taken into account, using, for instance,… ▽ More Using density functional theory (DFT) and density functional perturbation theory (DFPT), we investigate the stability and response functions of CsH$_2$PO$_4$, a ferroelectric material at low temperature. This material cannot be described properly by the usual (semi-)local approximations within DFT. The long-range e$^-$-e$^-$ correlation needs to be properly taken into account, using, for instance, Grimme's DFT-D methods, as investigated in this work. We find that DFT-D3(BJ) performs the best for the members of the dihydrogenated alkali phosphate family (KH$_2$PO$_4$, RbH$_2$PO$_4$, CsH$_2$PO$_4$), leading to experimental lattice parameters reproduced with an average deviation of 0.5 %. With these DFT-D methods, the structural, dielectric, vibrational and mechanical properties of CsH$_2$PO$_4$ are globally in excellent agreement with the available experiments ($<$ 2% MAPE for Raman-active phonons). Our study suggests the possible existence of a new low-temperature phase for CsH$_2$PO$_4$, not yet reported experimentally. Finally, we report the implementation of DFT-D contributions to elastic constants within DFPT. △ Less

Submitted 26 January, 2018; originally announced January 2018.

Comments: This paper was published in Physical Review B the 25 January 2017 (21 pages, 4 figures)

Journal ref: B. Van Troeye, M. J. van Setten, M. Giantomassi, M. Torrent, G.-M. Rignanese, and X. Gonze, Phys. Rev. B 95, 024112 (2017)

Interatomic force constants including the DFT-D dispersion contribution

Authors: Benoit Van Troeye, Marc Torrent, Xavier Gonze

Abstract: Grimme's DFT-D dispersion contribution to interatomic forces constants, required for the computation of the phonon band structures in density-functional perturbation theory, has been derived analytically. The implementation has then been validated with respect to frozen phonons, and applied on materials where weak cohesive forces play a major role i.e. argon, graphite, benzene. We show that these… ▽ More Grimme's DFT-D dispersion contribution to interatomic forces constants, required for the computation of the phonon band structures in density-functional perturbation theory, has been derived analytically. The implementation has then been validated with respect to frozen phonons, and applied on materials where weak cohesive forces play a major role i.e. argon, graphite, benzene. We show that these dispersive contributions have to be considered to properly reproduce the experimental vibrational properties of these materials, although the lattice parameter change, coming from the ground-state relaxation with the proper functional, induces the most important change with respect to a treatment without dispersion corrections. In the current implementation, the contribution of these dispersion corrections to the dynamical matrices (with a number of elements that is proportional to the square of the number of atoms) has only a cubic scaling with the number of atoms. In practice the overload with respect to density-functional calculations is small, making this methodology promising to study vibrational properties of large dispersive systems. △ Less

Submitted 26 January, 2018; originally announced January 2018.

Comments: This paper was published in Physical Review B the 20 April 2016 (22 pages, 7 figures)

Journal ref: Phys. Rev. B 93, 144304 (2016)

Precise effective masses from density functional perturbation theory

Authors: Jonathan Laflamme Janssen, Yannick Gillet, Samuel Poncé, Alexandre Martin, Marc Torrent, Xavier Gonze

Abstract: The knowledge of effective masses is a key ingredient to analyze numerous properties of semiconductors, like carrier mobilities, (magneto-)transport properties, or band extrema characteristics yielding carrier densities and density of states. Currently, these masses are usually calculated using finite-difference estimation of density functional theory (DFT) electronic band curvatures. However, fin… ▽ More The knowledge of effective masses is a key ingredient to analyze numerous properties of semiconductors, like carrier mobilities, (magneto-)transport properties, or band extrema characteristics yielding carrier densities and density of states. Currently, these masses are usually calculated using finite-difference estimation of density functional theory (DFT) electronic band curvatures. However, finite differences require an additional convergence study and are prone to numerical noise. Moreover, the concept of effective mass breaks down at degenerate band extrema. We assess the former limitation by developing a method that allows to obtain the Hessian of DFT bands directly, using density functional perturbation theory (DFPT). Then, we solve the latter issue by adapting the concept of `transport equivalent effective mass' to the $\vec{k} \cdot \hat{\vec{p}}$ framework. The numerical noise inherent to finite-difference methods is thus eliminated, along with the associated convergence study. The resulting method is therefore more general, more robust and simpler to use, which makes it especially appropriate for high-throughput computing. After validating the developed techniques, we apply them to the study of silicon, graphane, and arsenic. The formalism is implemented into the ABINIT software and supports the norm-conserving pseudopotential approach, the projector augmented-wave method, and the inclusion of spin-orbit coupling. The derived expressions also apply to the ultrasoft pseudopotential method. △ Less

Submitted 19 August, 2017; originally announced August 2017.

Comments: 22 pages, 6 figures

Journal ref: Phys. Rev. B 93, 205147 (2016)

A wavelet-based Projector Augmented-Wave (PAW) method: reaching frozen-core all-electron precision with a systematic, adaptive and localized wavelet basis set

Authors: Tonatiuh Rangel, Damien Caliste, Luigi Genovese, Marc Torrent

Abstract: We present a Projector Augmented-Wave~(PAW) method based on a wavelet basis set. We implemented our wavelet-PAW method as a PAW library in the ABINIT package [http://www.abinit.org] and into BigDFT [http://www.bigdft.org]. We test our implementation in prototypical systems to illustrate the potential usage of our code. By using the wavelet-PAW method, we can simulate charged and special boundary c… ▽ More We present a Projector Augmented-Wave~(PAW) method based on a wavelet basis set. We implemented our wavelet-PAW method as a PAW library in the ABINIT package [http://www.abinit.org] and into BigDFT [http://www.bigdft.org]. We test our implementation in prototypical systems to illustrate the potential usage of our code. By using the wavelet-PAW method, we can simulate charged and special boundary condition systems with frozen-core all-electron precision. Furthermore, our work paves the way to large-scale and potentially order-N simulations within a PAW method. △ Less

Submitted 14 May, 2016; originally announced May 2016.

Parallel eigensolvers in plane-wave Density Functional Theory

Authors: Antoine Levitt, Marc Torrent

Abstract: We consider the problem of parallelizing electronic structure computations in plane-wave Density Functional Theory. Because of the limited scalability of Fourier transforms, parallelism has to be found at the eigensolver level. We show how a recently proposed algorithm based on Chebyshev polynomials can scale into the tens of thousands of processors, outperforming block conjugate gradient algorith… ▽ More We consider the problem of parallelizing electronic structure computations in plane-wave Density Functional Theory. Because of the limited scalability of Fourier transforms, parallelism has to be found at the eigensolver level. We show how a recently proposed algorithm based on Chebyshev polynomials can scale into the tens of thousands of processors, outperforming block conjugate gradient algorithms for large computations. △ Less

Submitted 7 October, 2014; v1 submitted 17 June, 2014; originally announced June 2014.

Generation of Projector Augmented-Wave atomic data: a 71 elements validated table in the XML format

Authors: François Jollet, Marc Torrent, Natalie Holzwarth

Abstract: A Projector Augmented Wave (PAW) atomic data file is needed to be generated for each element, and plays in the PAW method the role of the pseudopotential file for norm-conserving (NC) or ultra-soft (US) plane wave calculations. In this paper, we present a review on how to obtain these data as well as results concerning their accuracy, their transferability and their efficiency for bulk solids. Fol… ▽ More A Projector Augmented Wave (PAW) atomic data file is needed to be generated for each element, and plays in the PAW method the role of the pseudopotential file for norm-conserving (NC) or ultra-soft (US) plane wave calculations. In this paper, we present a review on how to obtain these data as well as results concerning their accuracy, their transferability and their efficiency for bulk solids. Following \cite{Lejaeghere}, we propose a new criterium to test PAW atomic data and we provide a new table written in a XML format potentially readable by every PAW electronic structure code. △ Less

Submitted 1 October, 2013; v1 submitted 27 September, 2013; originally announced September 2013.

Hydrogen diffusion in the proton conductor Gd-doped barium cerate

Authors: Jessica Hermet, Marc Torrent, François Bottin, Guilhem Dezanneau, Gregory Geneste

Abstract: The energy landscape and diffusion barriers of protonic defects in Gd-doped BaCeO3, a compound candidate as electrolyte for protonic ceramic fuel cells, have been investigated by density functional theory calculations, starting from a previously computed energy landscape consisting of 16 kinds of stable sites (8 close to dopants and 8 far from them). The simplified string method has been used to d… ▽ More The energy landscape and diffusion barriers of protonic defects in Gd-doped BaCeO3, a compound candidate as electrolyte for protonic ceramic fuel cells, have been investigated by density functional theory calculations, starting from a previously computed energy landscape consisting of 16 kinds of stable sites (8 close to dopants and 8 far from them). The simplified string method has been used to determine accurately the Minimum Energy Paths between those sites, that might imply either proton reorientations, intra-octahedral or inter-octahedral hopping mechanisms. At contrast with simple cubic perovskites such as barium stannate or barium zirconate, very different values for energy barriers (from 0.02 eV to 0.58 eV) are found in this highly distorted orthorhombic perovskite, and no specific process appears to be clearly rate-limiting. Some inter-octahedral hoppings (when possible) are found to be more favourable than the intra-octahedral ones, while reorientations exhibit a wide range of energy barriers. △ Less

Submitted 1 January, 2013; originally announced January 2013.

Comments: submitted to Phys. Rev. B

Strong Isotopic Effect in Phase II of Dense Solid Hydrogen and Deuterium

Authors: Grégory Geneste, Marc Torrent, François Bottin, Paul Loubeyre

Abstract: Quantum nuclear zero-point motions in solid H$_2$ and D$_2$ under pressure are investigated at 80 K up to 160 GPa by first-principles path-integral molecular dynamics calculations. Molecular orientations are well-defined in phase II of D$_2$, while solid H$_2$ exhibits large and very asymmetric angular quantum fluctuations in this phase, with possible rotation in the (bc) plane, making it difficul… ▽ More Quantum nuclear zero-point motions in solid H$_2$ and D$_2$ under pressure are investigated at 80 K up to 160 GPa by first-principles path-integral molecular dynamics calculations. Molecular orientations are well-defined in phase II of D$_2$, while solid H$_2$ exhibits large and very asymmetric angular quantum fluctuations in this phase, with possible rotation in the (bc) plane, making it difficult to associate a well-identified single classical structure. The mechanism for the transition to phase III is also described. Existing structural data support this microscopic interpretation. △ Less

Submitted 17 July, 2012; v1 submitted 10 May, 2012; originally announced May 2012.

Comments: 5 pages, 3 figures

Effect of External Noise Correlation in Optical Coherence Resonance

Authors: J. M. Buldu, J. Garcia-Ojalvo, Claudio R. Mirasso, M. C. Torrent, J. M. Sancho

Abstract: Coherence resonance occurring in semiconductor lasers with optical feedback is studied via the Lang-Kobayashi model with external non-white noise in the pumping current. The temporal correlation and the amplitude of the noise have a highly relevant influence in the system, leading to an optimal coherent response for suitable values of both the noise amplitude and correlation time. This phenomeno… ▽ More Coherence resonance occurring in semiconductor lasers with optical feedback is studied via the Lang-Kobayashi model with external non-white noise in the pumping current. The temporal correlation and the amplitude of the noise have a highly relevant influence in the system, leading to an optimal coherent response for suitable values of both the noise amplitude and correlation time. This phenomenon is quantitatively characterized by means of several statistical measures. △ Less

Submitted 20 April, 2001; originally announced April 2001.

Comments: RevTeX, 4 pages, 7 figures

Enhancements to the GW space-time method

Authors: L. Steinbeck, A. Rubio, L. Reining, M. Torrent, I. D. White, R. W. Godby

Abstract: We describe the following new features which significantly enhance the power of the recently developed real-space imaginary-time GW scheme (Rieger et al., Comp. Phys. Commun. 117, 211 (1999)) for the calculation of self-energies and related quantities of solids: (i) to fit the smoothly decaying time/energy tails of the dynamically screened Coulomb interaction and other quantities to model functi… ▽ More We describe the following new features which significantly enhance the power of the recently developed real-space imaginary-time GW scheme (Rieger et al., Comp. Phys. Commun. 117, 211 (1999)) for the calculation of self-energies and related quantities of solids: (i) to fit the smoothly decaying time/energy tails of the dynamically screened Coulomb interaction and other quantities to model functions, treating only the remaining time/energy region close to zero numerically and performing the Fourier transformation from time to energy and vice versa by a combination of analytic integration of the tails and Gauss-Legendre quadrature of the remaining part and (ii) to accelerate the convergence of the band sum in the calculation of the Green's function by replacing higher unoccupied eigenstates by free electron states (plane waves). These improvements make the calculation of larger systems (surfaces, clusters, defects etc.) accessible. △ Less

Submitted 26 August, 1999; originally announced August 1999.

Comments: 10 pages, 6 figures

Coherence and synchronization in diode-laser arrays with delayed global coupling

Authors: J. Garcia-Ojalvo, J. Casademont, C. R. Mirasso, M. C. Torrent, J. M. Sancho

Abstract: The dynamics of a semiconductor-laser array whose individual elements are coupled in a global way through an external mirror is numerically analysed. A coherent in-phase solution is seen to be preferred by the system at intermediate values of the feedback coupling strength. At low values of this parameter, a strong amplification of the spontaneous emission noise is observed. A tendency towards c… ▽ More The dynamics of a semiconductor-laser array whose individual elements are coupled in a global way through an external mirror is numerically analysed. A coherent in-phase solution is seen to be preferred by the system at intermediate values of the feedback coupling strength. At low values of this parameter, a strong amplification of the spontaneous emission noise is observed. A tendency towards chaos synchronization is also observed at large values of the feedback strength. △ Less

Submitted 29 October, 1998; originally announced October 1998.

Comments: 8 pages, LaTeX, 6 PS figures, to appear in International Journal of Bifurcation and Chaos