Attractor Behavior Research Articles

Dopamine builds and reveals reward-associated latent behavioral attractors.

Phasic variations in dopamine levels are interpreted as a teaching signal reinforcing rewarded behaviors. However, behavior also depends on the motivational, neuromodulatory effect of phasic dopamine. In this study, we reveal a neurodynamical principle that unifies these roles in a recurrent network-based decision architecture embodied through an action-perception loop with the task space, the MAGNet model. Dopamine optogenetic conditioning in mice was accounted for by an embodied network model in which attractors encode internal goals. Dopamine-dependent synaptic plasticity created "latent" attractors, to which dynamics converged, but only locally. Attractor basins were widened by dopamine-modulated synaptic excitability, rendering goals accessible globally, i.e. from distal positions. We validated these predictions optogenetically in mice: dopamine neuromodulation suddenly and specifically attracted animals toward rewarded locations, without off-target motor effects. We thus propose that motivational dopamine reveals dopamine-built attractors representing potential goals in a behavioral landscape.

Far-from-equilibrium attractors with full relativistic Boltzmann approach in boost-invariant and non-boost-invariant systems

We study the universal behavior associated with a Relativistic Boltzmann Transport (RBT) approach with the full collision integral in 0+1D conformal systems. We show that all momentum moments of the distribution function exhibit universal behavior. Furthermore, the RBT approach allows to calculate the full distribution function, showing that an attractor behavior is present in both the longitudinal and transverse momentum dependence. We compare our results to the far-from-equilibrium attractors determined with other approaches, such as kinetic theory in Relaxation Time Approximation (RTA) and relativistic hydrodynamic theories, both in their viscous (DNMR) an anisotropic (aHydro) formulations, finding a very similar evolution, but an even faster thermalization in RBT for higher order moments. For the first time, we extended this analysis also to study the attractor behavior under a temperature-dependent viscosity η/s(T)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\eta /s(T)$$\\end{document}, accounting also for the rapid increase toward the hadronic phase. We find that a partial breaking of the scaling behavior with respect to τ/τeq\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ au /\ au _{eq}$$\\end{document} emerges only at T≈Tc\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$T \\approx T_c$$\\end{document} generating a transient deviation from attractors; interestingly this in realistic finite systems may occur around the freeze-out dynamics. Finally, we investigate for the first time results beyond the boost-invariant picture, finding that also in such a case the system evolves toward the universal attractor. In particular, we present the forward and pull-back attractors at different space-time rapidities including rapidity regions where initially the distribution function is even vanishing.

Dynamic behaviors of far and near memristive electromagnetic induction in spoon neural network.

In this paper, a special spoon neural network is proposed, which is composed of four neurons with direct connection and indirect connection. On this basis, the far induction network and the near induction network (NINN) are constructed by using hyperbolic tangent memristors to explore the influence of electromagnetic induction between neurons at different positions on the dynamic behavior of attractors. NINN exhibits more complex attractor structures and wider chaotic parameters, and also displays a heterogeneous coexisting attractor of limit cycles and chaos under network parameter control. By varying the parameters, coexisting chaotic attractors can be synthesized into a double scrolls attractor, and their oscillation amplitude can be controlled without changing the chaotic characteristics. The type of attractors in human brain determines the clarity of memory. These complex dynamic behaviors demonstrate that near induction has a more pronounced effect on the forgetting and disappearance of memory compared to far induction. Finally, a circuit using switches to change the type of electromagnetic induction is constructed and the results are verified.

On fractional discrete financial system: Bifurcation, chaos, and control

Abstract The dynamic analysis of financial systems is a developing field that combines mathematics and economics to understand and explain fluctuations in financial markets. This paper introduces a new three-dimensional (3D) fractional financial map and we dissect its nonlinear dynamics system under commensurate and incommensurate orders. As such, we evaluate when the equilibrium points are stable or unstable at various fractional orders. We use many numerical methods, phase plots in 2D and 3D projections, bifurcation diagrams and the maximum Lyapunov exponent. These techniques reveal that financial maps exhibit chaotic attractor behavior. This study is grounded on the Caputo-like discrete operator, which is specifically influenced by the variance of the commensurate and incommensurate orders. Furthermore, we confirm the presence and measure the complexity of chaos in financial maps by the 0–1 test and the approximate entropy algorithm. Additionally, we offer nonlinear-type controllers to stabilize the fractional financial map. The numerical results of this study are obtained using MATLAB.

Stability analysis for the phytoplankton-zooplankton model with depletion of dissolved oxygen and strong Allee effects

The photosynthetic activity of phytoplankton in the seas is responsible for an estimated 50–80 % of the world's oxygen generation. Both phytoplankton and zooplankton require some of this synthesized oxygen for cellular respiration. This study aims to better understand how the oxygen-phytoplankton dynamics are altered due to the Allee effect in phytoplankton development, particularly when considering the time-dependent oxygen generation rate. The dynamic analysis of the model is dedicated to finding the possible equilibrium points. The analysis reveals that three equilibrium points can be obtained. The stability study demonstrates that one of the equilibrium points is always stable. The remaining equilibrium points are stable under specific conditions. We also identify bifurcations originating from these equilibrium points, including transcritical, pitchfork, and Hopf bifurcation. We derive conditions for stable limit cycles (supercritical Hopf bifurcation) and, in some cases, establish the non-existence of solutions. Numerical simulations are performed to validate our theoretical findings. Furthermore, it is noted that the Allee threshold for the phytoplankton population (k0) significantly influences the overall dynamics of the system. When k0≤0.001, the population of plankton is at risk of extinction. On the other hand, when 0.001<k0≤0.01, the population of zooplankton is at risk of extinction. When 0.01<k0≤2, the solution reaches a stable condition of coexistence. Conversely, when k0≥2.1, the solution exhibits periodic attractor behaviour.

Dynamical Analysis of a Quadratic Megastable Chaotic Oscillator and Its Application in Biometric Fingerprint Image Encryption

This investigation centers on megastable systems, distinguished by their countable infinite attractors, with a particular emphasis on the Quadratic Megastable Oscillator (QMO). Unlike traditional megastable oscillators reliant on external excitation, our proposed QMO operates autonomously, contributing to its distinctiveness. Through a comprehensive exploration of the QMO, we elucidate various dynamical behaviors, enriching the understanding of its complex system dynamics. In contrast to conventional megastable oscillators, the QMO yields nested types of multiple attractors for diverse initial conditions, elegantly depicted in phase portraits. To gauge the sustainability of chaotic oscillation, we employ influential parameter bifurcation plots, providing a nuanced insight into the system’s dynamical evolution. The complexity of the proposed system is further underscored by its intricate basins of attraction, accommodating an infinite number of coexisting attractors. Exploring trajectory dynamics, we observe that certain initial conditions lead trajectories to distant destinations, evading the influence of local attractors. This behavior underscores the uniqueness of the QMO and highlights its potential applications in scenarios requiring nonlocalized attractor behaviors. Taking a practical turn, the QMO is applied to biometric fingerprint image encryption, demonstrating its efficacy in real-world applications. Rigorous statistical analyses and vulnerability assessments confirm the success of the QMO in providing secure encryption within chaotic system-based frameworks. This research contributes not only to the theoretical understanding of megastable systems but also establishes the QMO as a valuable tool in encryption applications, emphasizing its robustness and versatility in complex dynamical scenarios.

Attractor behavior of [formula omitted] modified gravity and the cosmic acceleration

In this paper, we have performed the dynamical system analysis of f(T) gravity cosmological models at both background and perturbation levels. We have presented three models pertaining to three distinct functional forms of f(T). The first form is that of the logarithmic form of the torsion scalar T, the second one is in the power law form, and the third one is the combination of the first two forms. For all these three forms of f(T), we have derived the corresponding cosmological parameters in terms of the dynamical variables. Subsequently, the critical points are obtained and the condition(s) of its existence has been derived. Critical points of each model have been analyzed individually and the corresponding cosmology are derived. The stability behavior of these critical points are discussed from the behavior of the eigenvalues and the phase portraits. At least one stable node has been obtained in each of these models. Further from the evolution plots of the cosmological parameters, the accelerating behavior of the cosmological models are also verified.

Asymptotic Behavior of Attractors of the Two-Dimensional Navier–Stokes System in a Domain with Small Periodic Obstacles

Asymptotic Behavior of Attractors of the Two-Dimensional Navier–Stokes System in a Domain with Small Periodic Obstacles

A Conservative Memristive Chaotic System with Extreme Multistability and Its Application in Image Encryption.

In this work, a novel conservative memristive chaotic system is constructed based on a smooth memristor. In addition to generating multiple types of quasi-periodic trajectories within a small range of a single parameter, the amplitude of the system can be controlled by changing the initial values. Moreover, the proposed system exhibits nonlinear dynamic characteristics, involving extreme multistability behavior of isomorphic and isomeric attractors. Finally, the proposed system is implemented using STMicroelectronics 32 and applied to image encryption. The excellent encryption performance of the conservative chaotic system is proven by an average correlation coefficient of 0.0083 and an information entropy of 7.9993, which provides a reference for further research on conservative memristive chaotic systems in the field of image encryption.

Predicting Tipping Points in a Family of PWL Systems: Detecting Multistability via Linear Operators Properties

The study of dynamical systems is based on the solution of differential equations that may exhibit various behaviors, such as fixed points, limit cycles, periodic, quasi-periodic attractors, chaotic behavior, and coexistence of attractors, to name a few. In this paper, we present a simple and novel method for predicting the occurrence of tipping points in a family of Piece-Wise Linear systems (PWL) that exhibit a transition from monostability to multistability with the variation of a single parameter, without the need to compute time series, i.e., without solving the differential equations of the system. The linearized system of the model is analyzed, the stable and unstable manifolds are taken to be real vectors in space, and the changes suffered by these vectors as a result of the modification of the parameter are examined using such simple metrics as the magnitude of a vector or the angle between two vectors in space. The results obtained with the linear analysis of the system agree well with those obtained with the numerical resolution of the dynamical system itself. The work presented here is an extension of previous results on this topic and contributes to the understanding of the mechanisms by which a system changes its stability by fragmenting its basin of attraction. This, in turn, enriches the field by providing an alternative to numerical resolution to identify quantitative changes in the dynamics of complex systems without having to solve the differential equation system.

Cosmographic and phase-space analysis of dynamical Chern–Simons modified and fractal gravities

We examine the cosmic scenario of interacting Kaniadakis holographic dark energy in dynamical Chern–Simons modified gravity and the fractal universe. For this purpose, the Hubble, deceleration, coincidence, equation-of-state and jerk parameters have been evaluated in view of the redshift parameter. It is observed that deceleration parameter [Formula: see text] evaluates the accelerated expansion of the universe in both gravities. The coincidence parameter [Formula: see text] exhibits the transition of behavior from dark matter to dark energy era of the universe. The behavior of the equation-of-state parameter [Formula: see text] describes the quintessence and vacuum regions of the universe in both gravities for maximum choices of the interacting parameters. The jerk parameter shows the correspondence of the given model with [Formula: see text]CDM model and other standard models in both gravities. All the parameters exhibit consistent behavior with the Planck 2018 data. We also analyze the dynamical stability in both frameworks by formulating dynamical models in the form of a system of differential equations and evaluating their corresponding critical points. Critical points of both models are stable and phase plots indicate attractor behavior that implies stability of the models. Further, stability conditions of both models signify the accelerating expansion of the universe. In addition, we discuss the thermodynamics of this model with the generalized second law and find its validity in both frameworks.

Coexistence behavior of asymmetric attractors in hyperbolic-type memristive Hopfield neural network and its application in image encryption

The neuron model has been widely employed in neural-morphic computing systems and chaotic circuits. This study aims to develop a novel circuit simulation of a three-neuron Hopfield neural network (HNN) with coupled hyperbolic memristors through the modification of a single coupling connection weight. The bistable mode of the hyperbolic memristive HNN (mHNN), characterized by the coexistence of asymmetric chaos and periodic attractors, is effectively demonstrated through the utilization of conventional nonlinear analysis techniques. These techniques include bifurcation diagrams, two-parameter maximum Lyapunov exponent plots, local attractor basins, and phase trajectory diagrams. Moreover, an encryption technique for color images is devised by leveraging the mHNN model and asymmetric structural attractors. This method demonstrates significant benefits in correlation, information entropy, and resistance to differential attacks, providing strong evidence for its effectiveness in encryption. Additionally, an improved modular circuit design method is employed to create the analog equivalent circuit of the memristive HNN. The correctness of the circuit design is confirmed through Multisim simulations, which align with numerical simulations conducted in Matlab.

Limiting Behavior of Random Attractors of Stochastic Supercritical Wave Equations Driven by Multiplicative Noise

This paper deals with the limiting behavior of random attractors of stochastic wave equations with supercritical drift driven by linear multiplicative white noise defined on unbounded domains. We first establish the uniform Strichartz estimates of the solutions with respect to noise intensity, and then prove the convergence of the solutions of the stochastic equations with respect to initial data as well as noise intensity. To overcome the non-compactness of Sobolev embeddings on unbounded domains, we first utilize the uniform tail-ends estimates to truncate the solutions in a bounded domain and then employ a spectral decomposition to establish the pre-compactness of the collection of all random attractors. We finally prove the upper semicontinuity of random attractor as noise intensity approaches zero.

Attractors' Analysis and Bifurcation Diagrams for an Impacting Inverted Pendulum in the Presence of a Two-Terms Harmonic Excitation

Abstract In this work, we examine the nonlinear dynamics of an inverted pendulum between lateral rebounding barriers. We continue the numerical investigation started in (Demeio et al., 2006, “Response Scenario and Non-Smooth Features in the Nonlinear Dynamics of an Impacting Inverted Pendulum,” ASME J. Comput. Nonlin. Dyn., 1(1), pp. 56–64) by adding the contribution of the second harmonic in the external forcing term. We investigate the behavior of the periodic attractors by bifurcation diagrams with respect to each amplitude and by behavior charts of single attractors in the amplitude parameters plane for fixed frequency. We study the effects of the second harmonic term on the existence domain of each attractor, on local bifurcations, and on the changes in the basins of attraction. The behavior of some robust chaotic attractor is also considered. In the evolution of the periodic attractors we have observed that the addition of the second harmonic generates a rich variety of behaviors, such as loss of stability and formation of isolas of periodic orbits. In the case of chaotic attractors, we have studied one attractor at high frequency, ω = 18, and one at low frequency, ω = 3. In the high frequency case we detect a transition from a scattered to a confined attractor, whereas at the lower frequency the chaotic attractor is present over a wide range of the second harmonic's amplitude. Finally, we extend the investigation of the chaotic attractors by bifurcation diagrams with respect to the frequency.

A Multistable Discrete Memristor and Its Application to Discrete-Time FitzHugh–Nagumo Model

This paper presents a multistable discrete memristor that is based on the discretization of a continuous-time model. It has been observed that the discrete memristor model is capable of preserving the characteristics of the continuous memristor model. Furthermore, a three-dimensional memristor discrete-time FitzHugh–Nagumo model is constructed by integrating the discrete memristor into a two-dimensional FitzHugh–Nagumo (FN) neuron model. Subsequently, the dynamic behavior of the proposed neuron model is analyzed through Lyapunov exponents, phase portraits, and bifurcation diagrams. The results show multiple kinds of coexisting hidden attractor behaviors generated by this neuron model. The proposed approach is expected to have significant implications for the design of advanced neural networks and other computational systems, with potential applications in various fields, including robotics, control, and optimization.

Unitary Evolutions Sourced By Interacting Quantum Memories: Closed Quantum Systems Directing Themselves Using Their State Histories

We propose, formulate and examine novel quantum systems and behavioral phases in which momentary choices of the system&amp;apos;s memories interact in order to source the internal interactions and unitary time evolutions of the system. In a closed system of the kind, the unitary evolution operator is updated, moment by moment, by being remade out of the system&amp;apos;s `experience&amp;apos;, that is, its quantum state history. The `Quantum Memory Made&amp;apos; Hamiltonians (QMM-Hs) which generate these unitary evolutions are Hermitian nonlocal-in-time operators composed of arbitrarily-chosen past-until-present density operators of the closed system or its arbitrary subsystems. The time evolutions of the kind are described by novel nonlocal nonlinear von Neumann and Schrödinger equations. We establish that nontrivial Purely-QMM unitary evolutions are `Robustly Non-Markovian&amp;apos;, meaning that the maximum temporal distances between the chosen quantum memories must exceed finite lower bounds which are set by the interaction couplings. After general formulation and considerations, we focus on the sufficiently-involved task of obtaining and classifying behavioral phases of one-qubit pure-state evolutions generated by first-to-third order polynomial QMM-Hs made out of one, two and three quantum memories. The behavioral attractors resulted from QMM-Hs are characterized and classified using QMM two-point-function observables as the natural probes, upon combining analytical methods with extensive numerical analyses. The QMM phase diagrams are shown to be outstandingly rich, having diverse classes of unprecedented unitary evolutions with physically remarkable behaviors. Moreover, we show that QMM interactions cause novel purely-internal dynamical phase transitions. Finally, we suggest independent fundamental and applied domains where the proposed `Experience Centric&amp;apos; Unitary Evolutions can be applied natuarlly and advantageously.

On the Minkowski content of self-similar random homogeneous iterated function systems

The Minkowski content of a compact set is a fine measure of its geometric scaling. For Lebesgue null sets it measures the decay of the Lebesgue measure of epsilon neighbourhoods of the set. It is well known that self-similar sets, satisfying reasonable separation conditions and non-log commensurable contraction ratios, have a well-defined Minkowski content. When dropping the contraction conditions, the more general notion of average Minkowski content still exists. For random recursive self-similar sets the Minkowski content also exists almost surely, whereas for random homogeneous self-similar sets it was recently shown by Zähle that the Minkowski content exists in expectation.&#x0D; In this short note we show that the upper Minkowski content, as well as the upper average Minkowski content of random homogeneous self-similar sets is infinite, almost surely, answering a conjecture posed by Zähle. Additionally, we show that in the random homogeneous equicontractive self-similar setting the lower Minkowski content is zero and the lower average Minkowski content is also infinite. These results are in stark contrast to the random recursive model or the mean behaviour of random homogeneous attractors.

Hybrid metric-Palatini Higgs inflation

We propose an extension of the Higgs inflation to the hybrid metric-Palatini gravity, where we introduce non-minimal couplings between Higgs and both the metric-type and the Palatini-type Ricci scalars. We study the inflationary phenomenology of our model and find that slow-roll inflation can be realized in the large-field regime, giving the observationally favored predictions. In particular, the scalar spectral index exhibits an attractor behavior to n s ∼ 0.964, while the tensor-to-scalar ratio can take an arbitrary value depending on the non-minimal coupling parameters, with the metric-Higgs limit r ∼ 10-3 being the maximum. We also investigate the unitarity property of our model. As the ultraviolet (UV) cutoff as a low-energy effective field theory (EFT) of this model is significantly lower than the Planck scale due to a strong curvature of field-space, we consider a possible candidate of UV-extended theories with an additional scalar field introduced so as to flatten the field-space in five-dimension. While the field-space can be flatten completely and this approach can lead to a weakly-coupled EFT, we gain an implication that Planck-scale EFT can be only realized in the limit of metric-Higgs inflation. We also discuss generalizations of the model up to mass-dimension four.

Effects of bias current and control of multistability in 3D hopfield neural network

This work studies the dynamics of a three dimensional Hopfield neural network focusing on the impact of bias terms. In the presence of bias terms, the models displays an odd symmetry and experiences typical behaviors including period doubling, spontaneous symmetry breaking, merging crisis, bursting oscillation, coexisting attractors and coexisting period-doubling reversals as well. Multistability control is investigated by employing the linear augmentation feedback strategy. We numerically prove that the multistable neural system can be adjusted to experience only a single attractor behavior when the coupling coefficient is gradually monitored. Experimental results from a microcontroller based realization of the underlined neural system are consistent with the theoretical analysis.

Implementation of the Simple Hyperchaotic Memristor Circuit with Attractor Evolution and Large-Scale Parameter Permission

A novel, simple, four-dimensional hyperchaotic memristor circuit consisting of two capacitors, an inductor and a magnetically controlled memristor is designed. Three parameters (a, b, c) are especially set as the research objects of the model through numerical simulation. It is found that the circuit not only exhibits a rich attractor evolution phenomenon, but also has large-scale parameter permission. At the same time, the spectral entropy complexity of the circuit is analyzed, and it is confirmed that the circuit contains a significant amount of dynamical behavior. By setting the internal parameters of the circuit to remain constant, a number of coexisting attractors are found under symmetric initial conditions. Then, the results of the attractor basin further confirm the coexisting attractor behavior and multiple stability. Finally, the simple memristor chaotic circuit is designed by the time-domain method with FPGA technology and the experimental results have the same phase trajectory as the numerical calculation results. Hyperchaos and broad parameter selection mean that the simple memristor model has more complex dynamic behavior, which can be widely used in the future, in areas such as secure communication, intelligent control and memory storage.