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We extend the recently developed generalized Floquet theory [Phys. Rev. Lett. 110, 170602 (2013)] to systems with infinite memory, i.e., a dependence on the whole previous history. In particular, we show that a lower asymptotic bound exists for the Floquet exponents associated to such cases. As examples, we analyze the cases of an ideal 1D system, a Brownian particle, and a circuit resonator with an ideal transmission line. All these examples show the usefulness of this new approach to the study of dynamical systems with memory, which are ubiquitous in science and technology.In this paper, we present an explicit Darboux transformation of the generalized mixed nonlinear Schrödinger (GMNLS) equation. The compact determinant representation of the n-fold Darboux transformation of the GMNLS equation is constructed and the nth-order solution is built. We further prove that only the even-fold Darboux transformation and the even-order solution of the GMNLS equation can, respectively, be reduced to the Darboux transformation and solution of the Kundu-Eckhaus equation. Furthermore, two different kinds of explicit one-soliton solutions of the GMNLS equation are constructed and discussed.It has been demonstrated that cellular automata had the highest computational capacity at the edge of chaos [N. H. Packard, in Dynamic Patterns in Complex Systems, edited by J. A. S. Kelso, A. Avasimibe nmr J. Mandell, and M. F. Shlesinger (World Scientific, Singapore, 1988), pp. 293-301; C. G. Langton, Physica D 42(1), 12-37 (1990); J. P. Crutchfield and K. Young, in Complexity, Entropy, and the Physics of Information, edited by W. H. Zurek (Addison-Wesley, Redwood City, CA, 1990), pp. 223-269], the parameter at which their behavior transitioned from ordered to chaotic. This same concept has been applied to reservoir computers; a number of researchers have stated that the highest computational capacity for a reservoir computer is at the edge of chaos, although others have suggested that this rule is not universally true. Because many reservoir computers do not show chaotic behavior but merely become unstable, it is felt that a more accurate term for this instability transition is the “edge of stability.” Here, I find two examples where the computational capacity of a reservoir computer decreases as the edge of stability is approached in one case because generalized synchronization breaks down and in the other case because the reservoir computer is a poor match to the problem being solved. The edge of stability as an optimal operating point for a reservoir computer is not in general true, although it may be true in some cases.The multiwavelet scale multidimensional recurrence quantification analysis (MWMRQA) method is proposed in this paper, which is a combination of multidimensional recurrence quantification analysis and wavelet packet decomposition. It allows us to quantify the recurrence properties of a single multidimensional time series under different wavelet scales. We apply the MWMRQA method to the Lorenz system and the Chinese stock market, respectively, and show the feasibility of this method as well as the dynamic variation of the Lorenz system and the Chinese stock market under different wavelet scales. This provides another perspective for other disciplines that need to study the recurrence properties of different scales in the future.By developing quasi-discrete multiple-scale method combined with tight-binding approximation, a novel quadratic Riccati differential equation is first derived for the soliton dynamics of the condensed bosons trapped in the optical lattices. For a lack of exact solutions, the trial solutions of the Riccati equation have been analytically explored for the condensed bosons with various scattering length as. When the lattice depth is rather shallow, the results of sub-fundamental gap solitons are in qualitative agreement with the experimental observation. For the deeper lattice potentials, we predict that in the case of as>0, some novel intrinsically localized modes of symmetrical envelope, topological (kink) envelope, and anti-kink envelope solitons can be observed within the bandgap in the system, of which the amplitude increases with the increasing lattice spacing and (or) depth. In the case of as less then 0, the bandgap brings out intrinsically localized gray or black soliton. This well provides experimental protocols to realize transformation between the gray and black solitons by reducing light intensity of the laser beams forming optical lattice.The inverse mechano-electrical problem in cardiac electrophysiology is the attempt to reconstruct electrical excitation or action potential wave patterns from the heart’s mechanical deformation that occurs in response to electrical excitation. Because heart muscle cells contract upon electrical excitation due to the excitation-contraction coupling mechanism, the resulting deformation of the heart should reflect macroscopic action potential wave phenomena. However, whether the relationship between macroscopic electrical and mechanical phenomena is well-defined and unique enough to be utilized for an inverse imaging technique in which mechanical activation mapping is used as a surrogate for electrical mapping has yet to be determined. Here, we provide a numerical proof-of-principle that deep learning can be used to solve the inverse mechano-electrical problem in phenomenological two- and three-dimensional computer simulations of the contracting heart wall, or in elastic excitable media, with muscle fiber anisotropy. We trained a convolutional autoencoder neural network to learn the complex relationship between electrical excitation, active stress, and tissue deformation during both focal or reentrant chaotic wave activity and, consequently, used the network to successfully estimate or reconstruct electrical excitation wave patterns from mechanical deformation in sheets and bulk-shaped tissues, even in the presence of noise and at low spatial resolutions. We demonstrate that even complicated three-dimensional electrical excitation wave phenomena, such as scroll waves and their vortex filaments, can be computed with very high reconstruction accuracies of about 95% from mechanical deformation using autoencoder neural networks, and we provide a comparison with results that were obtained previously with a physics- or knowledge-based approach.