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Abstract Forward attractors, especially invariant ones, of non-autonomous and random dynamical systems have not been as well studied as pullback attractors. This is mainly due to the different role that time plays in each case. Pullback attractors involve convergence at each instant of current time as the initial data is pulled back to distant past, whereas forward attractors are concerned with what happens in the distant future. Moreover, if they exist, they need not be unique. In this paper, ideas introduced in Kloeden [20] for asymptotically invariant forward attracting sets of non-autonomous ODEs are generalized and extended to study strictly invariant forward attractors of non-autonomous random dynamical systems on metric spaces. These are formulated as processes that include the effects of both noise and deterministic time variation. It is shown that the random forward attractors, when they exist, can be constructed by a pullback argument inside a positively invariant set. However, this is only a necessary condition, so additional sufficiency conditions are also given. Since forward attractors need not exist in dissipative systems, an alternative object, specifically the closed union of the forward omega limit sets for each initial time of a forward absorbing set is investigated. It is shown to be asymptotically invariant under certain conditions, so looks more and more like an invariant attractor as conventionally understood. It resembles the Vishik uniform attractor, but need not attract uniformly in the initial times and does not even require the system to be defined in the distant past. [ABSTRACT FROM AUTHOR]

q-deformations of functions and distributions have been used in the literature to explain several experimental observations. In this work, we study the dynamics of the Tinkerbell map under q -deformations. The system exhibits a rich variety of dynamical behavior as q varies, including occurrences of interior crises, paired cascades, simultaneous occurrence of Neimark-Sacker and reverse Neimark-Sacker bifurcations, and co-existence of attractors and multistability. Numerical analysis reveals the existence of 3 limit cycles occurring simultaneously in a certain parameter regime. An appropriate choice of initial conditions enables one to choose a desired attractor for the system among other co-existing ones, thus switching the system between different dynamical states. We demonstrate the possibility of secure encryption and decryption of messages with the q -deformed Tinkerbell map. The system's sensitivity to the initial conditions and to the deformation parameter makes the cryptic message secure, and decrypting the original message difficult. We propose the use of the q -deformed map as a novel method for transmission of messages securely. [ABSTRACT FROM AUTHOR]

Background  The assessment of pituitary tumor (PT) volume is important in the treatment and follow-up of patients with PT. Previously, PT volume estimation has been performed by conventional geometric equations (CGE) such as abc/2 (simplified ellipsoid volume equation) and 4πr 3 /3 (sphere), both presuming a symmetric tumor shape, which occurs uncommonly in patients with PT. In contrast, three-dimensional (3D) voxel-based software segmentation takes the irregular and asymmetric shapes that PTs often possess into account and might be a more accurate method for PT volume segmentation. The purpose of this study is twofold. (1) To compare 3D segmentation with CGE for PT volume estimation. (2) To assess inter-rater reliability in 3D segmentation of PTs. Methods  Nineteen high-resolution (1mm slice thickness) T1-weighted MRI examinations of patients with PT were independently analyzed and manually segmented, using the software ITK-SNAP, by two certified neuroradiologists. Concurrently, the volumes of the PTs were estimated with abc/2 and 4πr 3 /3 by a clinician, and the results were compared with the corresponding segmented volumes. Results  There was a significant decrease in PT volume attained from the segmentations compared with the calculations made with abc/2 (p  < 0.001, mean volume 18% higher than segmentation) and 4πr 3 /3 (p  < 0.001, mean volume 28% higher than segmentation). The intraclass correlation coefficient (ICC) for the two sets of segmented PTs was 0.99. Conclusion  CGE (abc/2 and 4πr 3 /3) significantly overestimates PT volume compared with 3D volumetric segmentation. The inter-rater agreement on manual 3D volumetric software segmentation is excellent. [ABSTRACT FROM AUTHOR]

Strange attractors have been extensively studied, but the same is not true for strange repellors. Some time-reversible systems have repellors that mirror their corresponding attractors and that exchange roles when time is reversed. In this paper, a conversion operator is introduced by which an easy transformation can be constructed between such a time-reversible system with an attractor/repellor pair and an irreversible one with a pair of attractors, or vice versa, thus expanding the list of such examples. [ABSTRACT FROM AUTHOR]

A finite-time version, based on implicit Lyapunov functions (ILFs), for the attractive ellipsoid method (AEM) is developed. Based on this, a robust control scheme is presented to ensure finite-time convergence of the solutions of a chain of integrators with bounded output perturbations to a minimal ellipsoidal set. The control parameters are obtained by solving a minimisation problem of the ‘size’ of the ellipsoid subject to a set of linear matrix inequalities (LMIs) constraints, and by applying the implicit function theorem. A numerical example is presented to support the implementability of these theoretical results. [ABSTRACT FROM AUTHOR]

In this paper we consider the analysis and design of an output feedback controller for a perturbed nonlinear system in which the output is sampled and quantised. Using the attractive ellipsoid method, which is based on Lyapunov analysis techniques, together with the relaxation of a nonlinear optimisation problem, sufficient conditions for the design of a robust control law are obtained. Since the original conditions result in nonlinear matrix inequalities, a numerical algorithm to obtain the solution is presented. The obtained control ensures that the trajectories of the closed-loop system will converge to a minimal (in a sense to be made specific) ellipsoidal region. Finally, numerical examples are presented in order to illustrate the applicability of the proposed design method. [ABSTRACT FROM PUBLISHER]

We introduce a numerical method to integrate the stochastic Landau-Lifshitz-Gilbert equation in spherical coordinates for generic discretization schemes. This method conserves the magnetization modulus and ensures the approach to equilibrium under the expected conditions. We test the algorithm on a benchmark problem: the dynamics of a uniformly magnetized ellipsoid. We investigate the influence of various parameters, and in particular, we analyze the efficiency of the numerical integration, in terms of the number of steps needed to reach a chosen long time with a given accuracy. [ABSTRACT FROM AUTHOR]

We study the dynamics of the Hénon map defined over complete, locally compact non-Archimedean fields of odd residue characteristic. We establish basic properties of its one-sided and two-sided filled Julia sets, and we determine, for each Hénon map, whether these sets are empty or nonempty, whether they are bounded or unbounded, and whether they are equal to the unit ball or not. On a certain region of the parameter space we show that the filled Julia set is an attractor. We prove that, for infinitely many distinct Hénon maps over Q3, this attractor is infinite and supports an SRB-type measure describing the distribution of all nearby forward orbits. We include some numerical calculations which suggest the existence of such infinite attractors over Q5 and Q7 as well. On a different region of the parameter space, we show that the Hénon map is topologically conjugate on its filled Julia set to the two-sided shift map on the space of bisequences in two symbols. [ABSTRACT FROM AUTHOR]