Mean Li-Yorke chaos in Banach spaces

We investigate the notion of mean Li-Yorke chaos for operators on Banach spaces. We show that it differs from the notion of distributional chaos of type 2, contrary to what happens in the context of topological dynamics on compact metric spaces. We prove that an operator is mean Li-Yorke chaotic if and only if it has an absolutely mean irregular vector. As a consequence, absolutely Ces\`aro bounded operators are never mean Li-Yorke chaotic. Dense mean Li-Yorke chaos is shown to be equivalent to the existence of a dense (or residual) set of absolutely mean irregular vectors. As a consequence, every mean Li-Yorke chaotic operator is densely mean Li-Yorke chaotic on some infinite-dimensional closed invariant subspace. A (Dense) Mean Li-Yorke Chaos Criterion and a sufficient condition for the existence of a dense absolutely mean irregular manifold are also obtained. Moreover, we construct an example of an invertible hypercyclic operator $T$ such that every nonzero vector is absolutely mean irregular for both $T$ and $T^{-1}$. Several other examples are also presented. Finally, mean Li-Yorke chaos is also investigated for $C_0$-semigroups of operators on Banach spaces.Comment: 26 page

Nonequilibrium free energy, H theorem and self-sustained oscillations for Boltzmann-BGK descriptions of semiconductor superlattices

Semiconductor superlattices (SL) may be described by a Boltzmann-Poisson kinetic equation with a Bhatnagar-Gross-Krook (BGK) collision term which preserves charge, but not momentum or energy. Under appropriate boundary and voltage bias conditions, these equations exhibit time-periodic oscillations of the current caused by repeated nucleation and motion of charge dipole waves. Despite this clear nonequilibrium behavior, if we `close' the system by attaching insulated contacts to the superlattice and keeping its voltage bias to zero volts, we can prove the H theorem, namely that a free energy $\Phi(t)$ of the kinetic equations is a Lyapunov functional ($\Phi\geq 0$, $d\Phi/dt\leq 0$). Numerical simulations confirm that the free energy decays to its equilibrium value for a closed SL, whereas for an `open' SL under appropriate dc voltage bias and contact conductivity $\Phi(t)$ oscillates in time with the same frequency as the current self-sustained oscillations.Comment: 15 pages, 3 figures, minor revision of latex fil

A moment based approach to the dynamical solution of the Kuramoto model

We examine the dynamics of the Kuramoto model with a new analytical approach. By defining an appropriate set of moments the dynamical equations can be exactly closed. We discuss some applications of the formalism like the existence of an effective Hamiltonian for the dynamics. We also show how this approach can be used to numerically investigate the dynamical behavior of the model without finite size effects.Comment: 6 pages, 5 figures, Revtex file, to appear in J. Phys.

Chaos in resonant-tunneling superlattices

Spatio-temporal chaos is predicted to occur in n-doped semiconductor superlattices with sequential resonant tunneling as their main charge transport mechanism. Under dc voltage bias, undamped time-dependent oscillations of the current (due to the motion and recycling of electric field domain walls) have been observed in recent experiments. Chaos is the result of forcing this natural oscillation by means of an appropriate external microwave signal.Comment: 3 pages, LaTex, RevTex, 3 uuencoded figures (1.2M) are available upon request from [email protected], to appear in Phys.Rev.

Chapman-Enskog method and synchronization of globally coupled oscillators

The Chapman-Enskog method of kinetic theory is applied to two problems of synchronization of globally coupled phase oscillators. First, a modified Kuramoto model is obtained in the limit of small inertia from a more general model which includes ``inertial'' effects. Second, a modified Chapman-Enskog method is used to derive the amplitude equation for an O(2) Takens-Bogdanov bifurcation corresponding to the tricritical point of the Kuramoto model with a bimodal distribution of oscillator natural frequencies. This latter calculation shows that the Chapman-Enskog method is a convenient alternative to normal form calculations.Comment: 7 pages, 2-column Revtex, no figures, minor change

Universality of the Gunn effect: self-sustained oscillations mediated by solitary waves

The Gunn effect consists of time-periodic oscillations of the current flowing through an external purely resistive circuit mediated by solitary wave dynamics of the electric field on an attached appropriate semiconductor. By means of a new asymptotic analysis, it is argued that Gunn-like behavior occurs in specific classes of model equations. As an illustration, an example related to the constrained Cahn-Allen equation is analyzed.Comment: 4 pages,3 Post-Script figure

Tracking collective cell motion by topological data analysis

By modifying and calibrating an active vertex model to experiments, we have simulated numerically a confluent cellular monolayer spreading on an empty space and the collision of two monolayers of different cells in an antagonistic migration assay. Cells are subject to inertial forces and to active forces that try to align their velocities with those of neighboring ones. In agreement with experiments, spreading tests exhibit finger formation in the moving interfaces, swirls in the velocity field, and the polar order parameter and correlation and swirl lengths increase with time. Cells inside the tissue have smaller area than those at the interface, as observed in recent experiments. In antagonistic migration assays, a population of fluidlike Ras cells invades a population of wild type solidlike cells having shape parameters above and below the geometric critical value, respectively. Cell mixing or segregation depends on the junction tensions between different cells. We reproduce experimentally observed antagonistic migration assays by assuming that a fraction of cells favor mixing, the others segregation, and that these cells are randomly distributed in space. To characterize and compare the structure of interfaces between cell types or of interfaces of spreading cellular monolayers in an automatic manner, we apply topological data analysis to experimental data and to numerical simulations. We use time series of numerical simulation data to automatically group, track and classify advancing interfaces of cellular aggregates by means of bottleneck or Wasserstein distances of persistent homologies. These topological data analysis techniques are scalable and could be used in studies involving large amounts of data. Besides applications to wound healing and metastatic cancer, these studies are relevant for tissue engineering, biological effects of materials, tissue and organ regeneration.Comment: 34 pages, 25 figures, the final version will appear in PLoS Computational Biolog

Energy and Momentum Distributions of a (2+1)-dimensional black hole background

Using Einstein, Landau-Lifshitz, Papapetrou and Weinberg energy-momentum complexes we explicitly evaluate the energy and momentum distributions associated with a non-static and circularly symmetric three-dimensional spacetime. The gravitational background under study is an exact solution of the Einstein's equations in the presence of a cosmological constant and a null fluid. It can be regarded as the three-dimensional analogue of the Vaidya metric and represents a non-static spinless (2+1)-dimensional black hole with an outflux of null radiation. All four above-mentioned prescriptions give exactly the same energy and momentum distributions for the specific black hole background. Therefore, the results obtained here provide evidence in support of the claim that for a given gravitational background, different energy-momentum complexes can give identical results in three dimensions. Furthermore, in the limit of zero cosmological constant the results presented here reproduce the results obtained by Virbhadra who utilized the Landau-Lifshitz energy-momentum complex for the same (2+1)-dimensional black hole background in the absence of a cosmological constant.Comment: 19 pages, LaTeX, v3: references added, to appear in Int.J.Mod.Phys.

Mean Li-Yorke chaos in Banach spaces

[EN] We investigate the notion of mean Li-Yorke chaos for operators on Banach spaces. We show that it differs from the notion of distributional chaos of type 2, contrary to what happens in the context of topological dynamics on compact metric spaces. We prove that an operator is mean Li-Yorke chaotic if and only if it has an absolutely mean irregular vector. As a consequence, absolutely Cesaro bounded operators are never mean Li-Yorke chaotic. Dense mean Li-Yorke chaos is shown to be equivalent to the existence of a dense (or residual) set of absolutely mean irregular vectors. As a consequence, every mean Li-Yorke chaotic operator is densely mean Li-Yorke chaotic on some infinite-dimensional closed invariant subspace. A (Dense) Mean Li-Yorke Chaos Criterion and a sufficient condition for the existence of a dense absolutely mean irregular manifold are also obtained. Moreover, we construct an example of an invertible hypercyclic operator T such that every nonzero vector is absolutely mean irregular for both T and T-1. Several other examples are also presented. Finally, mean Li-Yorke chaos is also investigated for C-0-semigroups of operators on Banach spaces.This work was partially done on a visit of the first author to the Institut Universitari de Matematica Pura i Aplicada at Universitat Politecnica de Valencia, and he is very grateful for the hospitality and support. The first author was partially supported by project #304207/2018-7 of CNPq and by grant #2017/22588-0 of Sao Paulo Research Foundation (FAPESP). The second and third authors were supported by MINECO, Project MTM2016-75963-P. The third author was also supported by Generalitat Valenciana, Project PROMETEO/2017/102. We thank Frederic Bayart for providing us Theorem 27, which answers a previous question of us. We also thank the referee whose careful comments produced an improvement in the presentation of the article.Bernardes, NCJ.; Bonilla, A.; Peris Manguillot, A. (2020). Mean Li-Yorke chaos in Banach spaces. Journal of Functional Analysis. 278(3):1-31. https://doi.org/10.1016/j.jfa.2019.108343S1312783Albanese, A., Barrachina, X., Mangino, E. M., & Peris, A. (2013). Distributional chaos for strongly continuous semigroups of operators. Communications on Pure and Applied Analysis, 12(5), 2069-2082. doi:10.3934/cpaa.2013.12.2069Barrachina, X., & Conejero, J. A. (2012). Devaney Chaos and Distributional Chaos in the Solution of Certain Partial Differential Equations. Abstract and Applied Analysis, 2012, 1-11. doi:10.1155/2012/457019Barrachina, X., & Peris, A. (2012). Distributionally chaotic translation semigroups. Journal of Difference Equations and Applications, 18(4), 751-761. doi:10.1080/10236198.2011.625945Bayart, F., & Grivaux, S. (2006). Frequently hypercyclic operators. Transactions of the American Mathematical Society, 358(11), 5083-5117. doi:10.1090/s0002-9947-06-04019-0BAYART, F., & RUZSA, I. Z. (2013). Difference sets and frequently hypercyclic weighted shifts. Ergodic Theory and Dynamical Systems, 35(3), 691-709. doi:10.1017/etds.2013.77Bermúdez, T., Bonilla, A., Martínez-Giménez, F., & Peris, A. (2011). Li–Yorke and distributionally chaotic operators. Journal of Mathematical Analysis and Applications, 373(1), 83-93. doi:10.1016/j.jmaa.2010.06.011Bernal-González, L., & Bonilla, A. (2016). Order of growth of distributionally irregular entire functions for the differentiation operator. Complex Variables and Elliptic Equations, 61(8), 1176-1186. doi:10.1080/17476933.2016.1149820Bernardes, N. C., Bonilla, A., Müller, V., & Peris, A. (2013). Distributional chaos for linear operators. Journal of Functional Analysis, 265(9), 2143-2163. doi:10.1016/j.jfa.2013.06.019BERNARDES, N. C., BONILLA, A., MÜLLER, V., & PERIS, A. (2014). Li–Yorke chaos in linear dynamics. Ergodic Theory and Dynamical Systems, 35(6), 1723-1745. doi:10.1017/etds.2014.20Bernardes, N. C., Peris, A., & Rodenas, F. (2017). Set-Valued Chaos in Linear Dynamics. Integral Equations and Operator Theory, 88(4), 451-463. doi:10.1007/s00020-017-2394-6Bernardes, N. C., Bonilla, A., Peris, A., & Wu, X. (2018). Distributional chaos for operators on Banach spaces. Journal of Mathematical Analysis and Applications, 459(2), 797-821. doi:10.1016/j.jmaa.2017.11.005Bès, J., Menet, Q., Peris, A., & Puig, Y. (2015). Recurrence properties of hypercyclic operators. Mathematische Annalen, 366(1-2), 545-572. doi:10.1007/s00208-015-1336-3Conejero, J. A., Müller, V., & Peris, A. (2007). Hypercyclic behaviour of operators in a hypercyclic C0-semigroup. Journal of Functional Analysis, 244(1), 342-348. doi:10.1016/j.jfa.2006.12.008Alberto Conejero, J., Rodenas, F., & Trujillo, M. (2015). Chaos for the Hyperbolic Bioheat Equation. Discrete & Continuous Dynamical Systems - A, 35(2), 653-668. doi:10.3934/dcds.2015.35.653Downarowicz, T. (2013). Positive topological entropy implies chaos DC2. Proceedings of the American Mathematical Society, 142(1), 137-149. doi:10.1090/s0002-9939-2013-11717-xFeldman, N. S. (2002). Hypercyclicity and supercyclicity for invertible bilateral weighted shifts. Proceedings of the American Mathematical Society, 131(2), 479-485. doi:10.1090/s0002-9939-02-06537-1Foryś-Krawiec, M., Oprocha, P., & Štefánková, M. (2017). Distributionally chaotic systems of type 2 and rigidity. Journal of Mathematical Analysis and Applications, 452(1), 659-672. doi:10.1016/j.jmaa.2017.02.056Garcia-Ramos, F., & Jin, L. (2016). Mean proximality and mean Li-Yorke chaos. Proceedings of the American Mathematical Society, 145(7), 2959-2969. doi:10.1090/proc/13440Grivaux, S., & Matheron, É. (2014). Invariant measures for frequently hypercyclic operators. Advances in Mathematics, 265, 371-427. doi:10.1016/j.aim.2014.08.002Hou, B., Cui, P., & Cao, Y. (2009). Chaos for Cowen-Douglas operators. Proceedings of the American Mathematical Society, 138(3), 929-936. doi:10.1090/s0002-9939-09-10046-1Huang, W., Li, J., & Ye, X. (2014). Stable sets and mean Li–Yorke chaos in positive entropy systems. Journal of Functional Analysis, 266(6), 3377-3394. doi:10.1016/j.jfa.2014.01.005León-Saavedra, F. (2002). Operators with hypercyclic Cesaro means. Studia Mathematica, 152(3), 201-215. doi:10.4064/sm152-3-1LI, J., TU, S., & YE, X. (2014). Mean equicontinuity and mean sensitivity. Ergodic Theory and Dynamical Systems, 35(8), 2587-2612. doi:10.1017/etds.2014.41Martínez-Giménez, F., Oprocha, P., & Peris, A. (2009). Distributional chaos for backward shifts. Journal of Mathematical Analysis and Applications, 351(2), 607-615. doi:10.1016/j.jmaa.2008.10.049Martínez-Giménez, F., Oprocha, P., & Peris, A. (2012). Distributional chaos for operators with full scrambled sets. Mathematische Zeitschrift, 274(1-2), 603-612. doi:10.1007/s00209-012-1087-8Menet, Q. (2017). Linear chaos and frequent hypercyclicity. Transactions of the American Mathematical Society, 369(7), 4977-4994. doi:10.1090/tran/6808Müller, V., & Vrs˘ovský, J. (2009). Orbits of Linear Operators Tending to Infinity. Rocky Mountain Journal of Mathematics, 39(1). doi:10.1216/rmj-2009-39-1-219Wu, X. (2013). Li–Yorke chaos of translation semigroups. Journal of Difference Equations and Applications, 20(1), 49-57. doi:10.1080/10236198.2013.809712Wu, X., Oprocha, P., & Chen, G. (2016). On various definitions of shadowing with average error in tracing. Nonlinearity, 29(7), 1942-1972. doi:10.1088/0951-7715/29/7/1942Wu, X., Wang, L., & Chen, G. (2017). Weighted backward shift operators with invariant distributionally scrambled subsets. Annals of Functional Analysis, 8(2), 199-210. doi:10.1215/20088752-3802705Yin, Z., & Yang, Q. (2017). Distributionally n-Scrambled Set for Weighted Shift Operators. Journal of Dynamical and Control Systems, 23(4), 693-708. doi:10.1007/s10883-017-9359-6Yin, Z., & Yang, Q. (2017). Distributionally n-chaotic dynamics for linear operators. Revista Matemática Complutense, 31(1), 111-129. doi:10.1007/s13163-017-0226-

Dynamics of Electric Field Domains and Oscillations of the Photocurrent in a Simple Superlattice Model

A discrete model is introduced to account for the time-periodic oscillations of the photocurrent in a superlattice observed by Kwok et al, in an undoped 40 period AlAs/GaAs superlattice. Basic ingredients are an effective negative differential resistance due to the sequential resonant tunneling of the photoexcited carriers through the potential barriers, and a rate equation for the holes that incorporates photogeneration and recombination. The photoexciting laser acts as a damping factor ending the oscillations when its power is large enough. The model explains: (i) the known oscillatory static I-V characteristic curve through the formation of a domain wall connecting high and low electric field domains, and (ii) the photocurrent and photoluminescence time-dependent oscillations after the domain wall is formed. In our model, they arise from the combined motion of the wall and the shift of the values of the electric field at the domains. Up to a certain value of the photoexcitation, the non-uniform field profile with two domains turns out to be metastable: after the photocurrent oscillations have ceased, the field profile slowly relaxes toward the uniform stationary solution (which is reached on a much longer time scale). Multiple stability of stationary states and hysteresis are also found. An interpretation of the oscillations in the photoluminescence spectrum is also given.Comment: 34 pages, REVTeX 3.0, 10 figures upon request, MA/UC3M/07/9