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We prove that in systems undergoing Hopf bifurcations, the effects of periodic forcing can be amplified by the shearing in the system to create sustained chaotic behavior. Specifically, strange attractors with SRB measures are shown to exist. The analysis is carried out for infinite dimensional systems, and the results are applicable to partial differential equations. Application of the general results to a concrete equation, namely the Brusselator, is given.

We examine spectral operator-theoretic properties of linear and nonlinear dynamical systems with globally stable attractors. Using the Kato decomposition, we develop a spectral expansion for general linear autonomous dynamical systems with analytic observables and define the notion of generalized eigenfunctions of the associated Koopman operator. We interpret stable, unstable and center subspaces in terms of zero-level sets of generalized eigenfunctions. We then utilize conjugacy properties of Koopman eigenfunctions and the new notion of open eigenfunctions—defined on subsets of state space—to extend these results to nonlinear dynamical systems with an equilibrium. We provide a characterization of (global) center manifolds, center-stable, and center-unstable manifolds in terms of joint zero-level sets of families of Koopman operator eigenfunctions associated with the nonlinear system. After defining a new class of Hilbert spaces, that capture the on- and off-attractor properties of dissipative dynamics, and introducing the concept of modulated Fock spaces, we develop spectral expansions for a class of dynamical systems possessing globally stable limit cycles and limit tori, with observables that are square-integrable in on-attractor variables and analytic in off-attractor variables. We discuss definitions of stable, unstable, and global center manifolds in such nonlinear systems with (quasi)-periodic attractors in terms of zero-level sets of Koopman operator eigenfunctions. We define the notion of isostables for a general class of nonlinear systems. In contrast with the systems that have discrete Koopman operator spectrum, we provide a simple example of a measure-preserving system that is not chaotic but has continuous spectrum, and discuss experimental observations of spectrum on such systems. We also provide a brief characterization of the data types corresponding to the obtained theoretical results and define the coherent principal dimension for a class of datasets based on the lattice-type principal spectrum of the associated Koopman operator.

It is believed that local activation is the origin of all complexities, and the locally active memristive synaptic neural network can generate complex chaotic dynamic behaviors, such as hyperchaotic, multi-scroll, multi-stability and hidden dynamical behaviors. However, there are few studies on the simultaneous occurrence of multiple complex dynamic behaviors in neural networks. No chaotic system of multi-scroll hyperchaotic hidden attractors based on neural network has been found yet. To solve the problem, in this paper, we propose a new locally active memristive Hopfield neural network (HNN) model based on a multi-segment function, which is affected by electromagnetic radiation and external current. The multi-scroll hyperchaotic hidden attractors are found in the memristive HNN for the first time. Theoretical analysis and numerical simulation results show that the memristive HNN model has no equilibrium point, and the number of multi-scroll attractors is controlled by the state equation parameters of the memristive synapse. In addition, the structures and number of scrolls are also affected by electromagnetic radiation and external current. At the same time, under the appropriate parameter conditions, by modifying the initial value of the system, the memristive HNN has a controllable number of coexisting attractors, showing extreme multi-stability. Finally, a memristive HNN analog circuit is designed. The hardware experiment results reproduce the multi-scroll dynamics phenomenon, which verifies the correctness of the theoretical analysis and numerical simulation.

We study the continuity set (the set of all continuous points) of pullback random attractors from a parametric space into the space of all compact subsets of the state space with Hausdorff metric. We find a general theorem that the continuity set is an IOD-type (a countable intersection of open dense sets) with the local similarity under appropriate conditions of random dynamical systems, and we further show that any IOD-type set in the parametric space has the continuous cardinality, which affirmatively answers the unsolved question about the cardinality of the continuity set of attractors in the literature. Applying to the random nonautonomous nonlocal parabolic equations on an unbounded domain driven by colored noise, we establish the existence and IOD-type continuity of pullback random attractors in time, sample-translation and noise-size, moreover, we prove that the continuity set of the pullback random attractor on the plane of time and sample-translation is composed of diagonal rays whose number of bars is the continuous cardinality.

Flexible computation is a hallmark of intelligent behavior. However, little is known about how neural networks contextually reconfigure for different computations. In the present work, we identified an algorithmic neural substrate for modular computation through the study of multitasking artificial recurrent neural networks. Dynamical systems analyses revealed learned computational strategies mirroring the modular subtask structure of the training task set. Dynamical motifs, which are recurring patterns of neural activity that implement specific computations through dynamics, such as attractors, decision boundaries and rotations, were reused across tasks. For example, tasks requiring memory of a continuous circular variable repurposed the same ring attractor. We showed that dynamical motifs were implemented by clusters of units when the unit activation function was restricted to be positive. Cluster lesions caused modular performance deficits. Motifs were reconfigured for fast transfer learning after an initial phase of learning. This work establishes dynamical motifs as a fundamental unit of compositional computation, intermediate between neuron and network. As whole-brain studies simultaneously record activity from multiple specialized systems, the dynamical motif framework will guide questions about specialization and generalization. The authors identify reusable ‘dynamical motifs’ in artificial neural networks. These motifs enable flexible recombination of previously learned capabilities, promoting modular, compositional computation and rapid transfer learning. This discovery sheds light on the fundamental building blocks of intelligent behavior.

Traditional neural network models of associative memories were used to store and retrieve static patterns. We develop reservoir-computing based memories for complex dynamical attractors, under two common recalling scenarios in neuropsychology: location-addressable with an index channel and content-addressable without such a channel. We demonstrate that, for location-addressable retrieval, a single reservoir computing machine can memorize a large number of periodic and chaotic attractors, each retrievable with a specific index value. We articulate control strategies to achieve successful switching among the attractors, unveil the mechanism behind failed switching, and uncover various scaling behaviors between the number of stored attractors and the reservoir network size. For content-addressable retrieval, we exploit multistability with cue signals, where the stored attractors coexist in the high-dimensional phase space of the reservoir network. As the length of the cue signal increases through a critical value, a high success rate can be achieved. The work provides foundational insights into developing long-term memories and itinerancy for complex dynamical patterns. Artificial associative memories in neural network models have shown ability to store and retrieve static patterns of complex systems, however analysis of dynamic patterns remains challenging. The authors develop a reservoir computing based memory approach for complex multistable dynamical systems.

After the discovery in early 1960s by E. Lorenz and Y. Ueda of the first example of a chaotic attractor in numerical simulation of a real physical process, a new scientific direction of analysis of chaotic behavior in dynamical systems arose. Despite the key role of this first discovery, later on a number of works have appeared supposing that chaotic attractors of the considered dynamical models are rather artificial, computer-induced objects, i.e., they are generated not due to the physical nature of the process, but only by errors arising from the application of approximate numerical methods and finite-precision computations. Further justification for the possibility of a real existence of chaos in the study of a physical system developed in two directions. Within the first direction, effective analytic-numerical methods were invented providing the so-called computer-assisted proof of the existence of a chaotic attractor. In the framework of the second direction, attempts were made to detect chaotic behavior directly in a physical experiment, by designing a proper experimental setup. The first remarkable result in this direction is the experiment of L. Chua, in which he designed a simple RLC circuit (Chua circuit) containing a nonlinear element (Chua diode), and managed to demonstrate the real evidence of chaotic behavior in this circuit on the screen of oscilloscope. The mathematical model of the Chua circuit (further, Chua system) is also known to be the first example of a system in which the existence of a chaotic hidden attractor was discovered and the bifurcation scenario of its birth was described. Despite the nontriviality of this discovery and cogency of the procedure for hidden attractor localization, the question of detecting this type of attractor in a physical experiment remained open. This article aims to give an exhaustive answer to this question, demonstrating both a detailed formulation of a radiophysical experiment on the localization of a hidden attractor in the Chua circuit, as well as a thorough description of the relationship between a physical experiment, mathematical modeling, and computer simulation.

In the field of artificial neural networks, researchers often use the hyperbolic tangent function as an activation function to imitate the firing rules of biological neurons and to add nonlinear characteristics to neural networks. However, prior studies have neglected to consider the effect of the bias of the activation function, which represents the firing threshold of biological neurons, on the dynamical behaviors of neural networks. In this paper, we aim to study the influence of neuronal thresholds on dynamics of the Hopfield neural network (HNN). The bias of the activation function is used as the control parameter in this investigation. The proposed HNN model is analyzed through various methods, including phase portraits, 0-1 tests, Lyapunov exponent spectra, bifurcation diagrams, and bi-parameter dynamic maps. The results of the analysis illustrate that the firing threshold could induce a range of complex dynamical phenomena in the HNN, such as periodic attractors, chaotic attractors, and forward and reverse period doubling bifurcations. Furthermore, the hardware implementation of the proposed HNN model is successfully demonstrated through circuit simulations. These experiments confirm the consistency of the results obtained through numerical simulations. Finally, the potential application of the proposed HNN model is further explored by constructing an image encryption system. The results demonstrate that the chaotic attractor has good randomness properties and that its application in image encryption has a high level of security. This study may provide valuable insights into dynamics of the HNN model influenced by the neuron firing threshold and highlight the potential for practical applications of these models in engineering.

We study a rough difference equation on a discrete time set, where the driving Hölder rough path is a realization of a stochastic process. Using a modification of Davie’s approach (Cong et al. in J. Dyn. Differ. Equ. 34:605–636, 2022) and the discrete sewing lemma, we derive norm estimates for the discrete solution. In particular, when the discrete time set is regular, the system generates a discrete random dynamical system. We also generalize a recent result in Duc and Kloeden (Numerical attractors for rough differential equations, 2021) on the existence and upper semi-continuity of a global random pullback attractor under the dissipativity and the linear growth condition for the drift.