Attractor Regime Research Articles

First Passage Times of Long Transient Dynamics in Ecology.

Long transient dynamics in ecological models are characterized by extended periods in one state or regime before an eventual, and often abrupt, transition. One mechanism leading to long transient dynamics is the presence of ghost attractors, states where system dynamics slow down and the system lingers before eventually transitioning to the true attractor. This transition results solely from system dynamics rather than external factors. This paper investigates the dynamics of a classical herbivore-grazer model with the potential for ghost attractors or alternative stable states. We propose an intuitive threshold for first passage time analysis applicable to both bistable and ghost attractor regimes. By formulating the first passage time problem as a backward Kolmogorov equation, we examine how the mean first passage time changes as parameters are varied from the ghost attractor regime to the bistable one, through a saddle-node bifurcation. Our results reveal that the mean and variance of first passage times vary smoothly across the bifurcation threshold, eliminating the deterministic distinction between ghost attractors and bistable regimes. This work suggests that first passage time analysis can be an informative way to classify the length of a long transient. A better understanding of the duration of long transients may contribute to greater ecological understanding and more effective environmental management.

When tadpoles matter: one-loop corrections for spectator Higgs in inflation

We consider the classical attractor regime of the spectator Abelian Higgs model in power-law inflation, and compute the one-loop corrections to its evolution. For computations we utilize dimensional regularization and the propagators in the unitary gauge. The corrections to both the scalar condensate and the energy-momentum tensor exhibit secular ultraviolet contributions, that tend to slow down the rolling of the scalar down its potential, and drive it away from the classical attractor. These corrections need not be suppressed if the U(1) charge is much larger than the scalar self-coupling, which is seen already in flat space. In addition, at late times the secular corrections necessarily invalidate the perturbative loop expansion. We find the late time secular corrections to be captured by the renormalization group, which opens up the possibility to resum them past the breakdown of perturbativity.

Inflationary gravitational waves in consistent D → 4 Einstein-Gauss-Bonnet gravity

We study the slow-roll single field inflation in the context of the consistent D → 4 Einstein-Gauss-Bonnet gravity that was recently proposed in [1]. In addition to the standard attractor regime, we find a new attractor regime which we call the Gauss-Bonnet attractor as the dominant contribution comes from the Gauss-Bonnet term. Around this attractor solution, we find power spectra and spectral tilts for the curvature perturbations and gravitational waves (GWs) and also a model-independent consistency relation among observable quantities. The Gauss-Bonnet term provides a nonlinear k4 term to the GWs dispersion relation which has the same order as the standard linear k2 term at the time of horizon crossing around the Gauss-Bonnet attractor. The Gauss-Bonnet attractor regime thus provides a new scenario for the primordial GWs which can be tested by observations. Finally, we study non-Gaussianity of GWs in this model and estimate the nonlinear parameters fs1s2s3NL, sq and fs1s2s3NL, eq by fitting the computed GWs bispectra with the local-type and equilateral-type templates respectively at the squeezed limit and at the equilateral shape. For helicities (+++) and (−−−), fs1s2s3NL, sq is larger while fs1s2s3NL, eq is larger for helicities (++−) and (−−+).

Abelian Higgs model in power-law inflation: the propagators in the unitary gauge

We consider the Abelian Higgs model in the broken phase as a spectator in cosmological spaces of general D space-time dimensions, and allow for the condensate to be time-dependent. We fix the unitary gauge using Dirac’s formalism for constrained systems, and then quantize the gauge-fixed system. Vector and scalar perturbations develop time­dependent masses. We work out their propagators assuming the cosmological background is that of power-law inflation, characterized by a constant principal slow-roll parameter, and that the scalar condensate is in the attractor regime, scaling as the Hubble rate. Our propagators correctly reduce to known results in the Minkowski and de Sitter space limits. We use the vector propagator to compute the equal-time correlators of electric and magnetic fields and find that at super-Rubble separations the former is enhanced, while the latter is suppressed compared to the vacuum fluctuations of the massless vector field. These correlators satisfy the hierarchy governed by Faraday’s law.

Charged vector inflation

We present a model of inflation in which the inflaton field is charged under a triplet of $U(1)$ gauge fields. The model enjoys an internal $O(3)$ symmetry supporting the isotropic FRW solution. With an appropriate coupling between the gauge fields and the inflaton field, the system reaches an attractor regime in which the gauge fields furnish a small constant fraction of the total energy density. We decompose the scalar perturbations into the adiabatic and entropy modes and calculate the contributions of the gauge fields into the curvature perturbations power spectrum. We also calculate the entropy power spectrum and the adiabatic-entropy cross correlation. In addition to the metric tensor perturbations, there are tensor perturbations associated with the gauge field perturbations which are coupled to metric tensor perturbations. We show that the correction in primordial gravitational tensor power spectrum induced from the matter tensor perturbation is a sensitive function of the gauge coupling.

Self-Excited and Hidden Attractors in an Autonomous Josephson Jerk Oscillator: Analysis and Its Application to Text Encryption

By converting the resistive capacitive shunted junction model to a jerk oscillator, an autonomous chaotic Josephson jerk oscillator which can belong to oscillators with hidden and self-excited attractors is designed. The proposed autonomous Josephson jerk oscillator has two or no equilibrium points depending on DC bias current. The stability analysis of the two equilibrium points shows that one of the equilibrium points is unstable while for the other equilibrium point, the existence of a Hopf bifurcation is established. The dynamical behavior of autonomous Josephson jerk oscillator is analyzed by using standard tools of nonlinear analysis. For a suitable choice of the parameters, an autonomous Josephson jerk oscillator can generate antimonotonicity, periodic oscillations, self-excited chaotic attractors, hidden chaotic attractors, hidden chaotic bubble attractors, and coexistence between periodic and chaotic self-excited attractors. Finally, a text cryptographic encryption scheme with the help of generalized function projective synchronization of the proposed autonomous Josephson jerk oscillators in hidden chaotic attractor regime is illustrated through a numerical example, showing that a high-level security device can be produced using this system.

Application of chaos in a recurrent neural network to control in ill-posed problems: a novel autonomous robot arm.

Inspired by a viewpoint that complex/chaotic dynamics would play an important role in biological systems including the brain, chaotic dynamics introduced in a recurrent neural network was applied to robot control in ill-posed situations. By computer experiments we show that a model robot arm without an advanced visual processing function can catch a target object and bring it to a set position under ill-posed situations (e.g., in the presence of unknown obstacles). The key idea in these works is adaptive switching of a system parameter (connectivity) between a chaos regime and attractor regime in a neural network model, which generates, depending on environmental circumstances, either chaotic motions or definite motions corresponding to embedded attractors. The adaptive switching results in useful functional motions of the robot arm. These successful experiments indicate that chaotic dynamics is potentially useful for practical engineering control applications. In addition, this novel autonomous arm system is implemented in a hardware robot arm that can avoid obstacles and reach for a target in a situation where the robot can get only rough target information, including uncertainty, by means of a few sensors, as indicated in the appendix, A1 and A2.

Vanishing of local non-Gaussianity in canonical single field inflation

We study the production of observable primordial local non-Gaussianity in two opposite regimes of canonical single field inflation: attractor (standard single field slow-roll inflation) and non attractor (ultra slow-roll inflation). In the attractor regime, the standard derivation of the bispectrum's squeezed limit using co-moving coordinates gives the well known Maldacena's consistency relation fNL = 5 (1−ns) / 12. On the other hand, in the non-attractor regime, the squeezed limit offers a substantial violation of this relation given by fNL = 5/2. In this work we argue that, independently of whether inflation is attractor or non-attractor, the size of the observable primordial local non-Gaussianity is predicted to be fNLobs = 0 (a result that was already understood to hold in the case of attractor models). To show this, we follow the use of the so-called Conformal Fermi Coordinates (CFC), recently introduced in the literature. These coordinates parametrize the local environment of inertial observers in a perturbed FRW spacetime, allowing one to identify and compute gauge invariant quantities, such as n-point correlation functions. Concretely, we find that during inflation, after all the modes have exited the horizon, the squeezed limit of the 3-point correlation function of curvature perturbations vanishes in the CFC frame, regardless of the inflationary regime. We argue that such a cancellation should persist after inflation ends.

Overline{D3} induced geometric inflation

Effective supergravity inflationary models induced by anti-D3 brane interaction with the moduli fields in the bulk geometry have a geometric description. The Kähler function carries the complete geometric information on the theory. The non-vanishing bisectional curvature plays an important role in the construction. The new geometric formalism, with the nilpotent superfield representing the anti-D3 brane, allows a powerful generalization of the existing inflationary models based on supergravity. They can easily incorporate arbitrary values of the Hubble parameter, cosmological constant and gravitino mass. We illustrate it by providing generalized versions of polynomial chaotic inflation, T- and E-models of α-attractor type, disk merger. We also describe a multi-stage cosmological attractor regime, which we call cascade inflation.

Institutional Tipping Points in Organizational Climate Change Adaptation Processes

Despite increasing awareness of the urgency to respond to climate change through adaptation, progress with climate change adaptation differs considerably across social contexts, even within seemingly uniform institutional environments. Only a part of these differences in engaging in adaptation can be explained by differentiated exposure or sensitivity to climate change hazards. Institutions, and institutional change, play important roles in enabling or constraining adaptation at the social group scale. This paper borrows the concept of tipping points from the natural sciences (Lenton et al. 2008; Lenton 2013) and applies it to social processes of climate change adaptation by focusing on processes of institutional change towards and beyond ‘institutional’ tipping points. Different stages of institutional change, where social groups switch from one dominant attractor regime to another, are discussed and illustrated. Empirical research conducted in two organizations in the local government and primary health care sector in Australia are used as examples for how institutional adaptation occurs and how institutional tipping points can be identified. Reflecting on these examples, the paper reviews the conceptual value-add of the institutional tipping points concept, while also discussing its epistemological and methodological limitations.

Disformal invariance of curvature perturbation

We show that under a general disformal transformation the linear comoving curvature perturbation is not identically invariant, but is invariant on superhorizon scales for any theory that is disformally related to Horndeski's theory. The difference between disformally related curvature perturbations is found to be given in terms of the comoving density perturbation associated with a single canonical scalar field. In General Relativity it is well-known that this quantity vanishes on superhorizon scales through the Poisson equation that is obtained on combining the Hamiltonian and momentum constraints, and we confirm that a similar result holds for any theory that is disformally related to Horndeski's scalar-tensor theory so long as the invertibility condition for the disformal transformation is satisfied. We also consider the curvature perturbation at full nonlinear order in the unitary gauge, and find that it is invariant under a general disformal transformation if we assume that an attractor regime has been reached. Finally, we also discuss the counting of degrees of freedom in theories disformally related to Horndeski's.

Superconformal inflationary α-attractors

Recently a broad class of superconformal inflationary models was found leading to a universal observational prediction $n_s=1-2/N$ and $r=12/N^2$. Here we generalize this class of models by introducing a parameter $\alpha$ inversely proportional to the curvature of the inflaton Kahler manifold. In the small curvature (large $\alpha$) limit, the observational predictions of this class of models coincide with the predictions of generic chaotic inflation models. However, for sufficiently large curvature (small $\alpha$), the predictions converge to the universal attractor regime with $n_s=1-2/N$ and $r=12\alpha/N^2$, which corresponds to the part of the $n_s-r$ plane favored by the Planck data.

A Common Neurodynamical Mechanism Could Mediate Externally Induced and Intrinsically Generated Transitions in Visual Awareness

The neural correlates of conscious visual perception are commonly studied in paradigms of perceptual multistability that allow multiple perceptual interpretations during unchanged sensory stimulation. What is the source of this multistability in the content of perception? From a theoretical perspective, a fine balance between deterministic and stochastic forces has been suggested to underlie the spontaneous, intrinsically driven perceptual transitions observed during multistable perception. Deterministic forces are represented by adaptation of feature-selective neuronal populations encoding the competing percepts while stochastic forces are modeled as noise-driven processes. Here, we used a unified neuronal competition model to study the dynamics of adaptation and noise processes in binocular flash suppression (BFS), a form of externally induced perceptual suppression, and compare it with the dynamics of intrinsically driven alternations in binocular rivalry (BR). For the first time, we use electrophysiological, biologically relevant data to constrain a model of perceptual rivalry. Specifically, we show that the mean population discharge pattern of a perceptually modulated neuronal population detected in electrophysiological recordings in the lateral prefrontal cortex (LPFC) during BFS, constrains the dynamical range of externally induced perceptual transitions to a region around the bifurcation separating a noise-driven attractor regime from an adaptation-driven oscillatory regime. Most interestingly, the dynamical range of intrinsically driven perceptual transitions during BR is located in the noise-driven attractor regime, where it overlaps with BFS. Our results suggest that the neurodynamical mechanisms of externally induced and spontaneously generated perceptual alternations overlap in a narrow, noise-driven region just before a bifurcation where the system becomes adaptation-driven.

New local duals in eternal inflation

Global-local duality is the equivalence of seemingly different regulators in eternal inflation. For example, the light-cone time cutoff (a global measure, which regulates time) makes the same predictions as the causal patch (a local measure that cuts off space). We show that global-local duality is far more general. It rests on a redundancy inherent in any global cutoff: at late times, an attractor regime is reached, characterized by the unlimited exponential self-reproduction of a certain fundamental region of spacetime. An equivalent local cutoff can be obtained by restricting to this fundamental region. We derive local duals to several global cutoffs of interest. The New Scale Factor Cutoff is dual to the Short Fat Geodesic, a geodesic of fixed infinitesimal proper width. Vilenkin's CAH Cutoff is equivalent to the Hubbletube, whose width is proportional to the local Hubble volume. The famous youngness problem of the Proper Time Cutoff can be readily understood by considering its local dual, the Incredible Shrinking Geodesic.

Chaotic neural network applied to two-dimensional motion control

Chaotic dynamics generated in a chaotic neural network model are applied to 2-dimensional (2-D) motion control. The change of position of a moving object in each control time step is determined by a motion function which is calculated from the firing activity of the chaotic neural network. Prototype attractors which correspond to simple motions of the object toward four directions in 2-D space are embedded in the neural network model by designing synaptic connection strengths. Chaotic dynamics introduced by changing system parameters sample intermediate points in the high-dimensional state space between the embedded attractors, resulting in motion in various directions. By means of adaptive switching of the system parameters between a chaotic regime and an attractor regime, the object is able to reach a target in a 2-D maze. In computer experiments, the success rate of this method over many trials not only shows better performance than that of stochastic random pattern generators but also shows that chaotic dynamics can be useful for realizing robust, adaptive and complex control function with simple rules.

Complementarity in the multiverse

In the multiverse, as in AdS space, light cones relate bulk points to boundary scales. This holographic UV-IR connection defines a preferred global time cutoff that regulates the divergences of eternal inflation. An entirely different cutoff, the causal patch, arises in the holographic description of black holes. Remarkably, I find evidence that these two regulators define the same probability measure in the multiverse. Initial conditions for the causal patch are controlled by the late-time attractor regime of the global description.

Consequences of dark matter-dark energy interaction on cosmological parameters derived from type Ia supernova data

Models where the dark matter component of the Universe interacts with the dark energy field have been proposed as a solution to the cosmic coincidence problem, since in the attractor regime both dark energy and dark matter scale in the same way. In these models the mass of the cold dark matter particles is a function of the dark energy field responsible for the present acceleration of the Universe, and different scenarios can be parametrized by how the mass of the cold dark matter particles evolves with time. In this article we study the impact of a constant coupling $\ensuremath{\delta}$ between dark energy and dark matter on the determination of a redshift dependent dark energy equation of state ${w}_{\mathrm{DE}}(z)$ and on the dark matter density today from SNIa data. We derive an analytical expression for the luminosity distance in this case. In particular, we show that the presence of such a coupling increases the tension between the cosmic microwave background data from the analysis of the shift parameter in models with constant ${w}_{\mathrm{DE}}$ and SNIa data for realistic values of the present dark matter density fraction. Thus, an independent measurement of the present dark matter density can place constraints on models with interacting dark energy.

Self-reproduction ink-inflation

We study cosmological self-reproduction in models of inflation driven by a scalar field $\ensuremath{\phi}$ with a noncanonical kinetic term ($k$-inflation). We develop a general criterion for the existence of attractors and establish conditions selecting a class of $k$-inflation models that admit a unique attractor solution. We then consider quantum fluctuations on the attractor background. We show that the correlation length of the fluctuations is of order ${c}_{s}{H}^{\ensuremath{-}1}$, where ${c}_{s}$ is the speed of sound. By computing the magnitude of field fluctuations, we determine the coefficients of Fokker-Planck equations describing the probability distribution of the spatially averaged field $\ensuremath{\phi}$. The field fluctuations are generally large in the inflationary attractor regime; hence, eternal self-reproduction is a generic feature of $k$-inflation. This is established more formally by demonstrating the existence of stationary solutions of the relevant Fokker-Planck equations. We also show that there exists a (model-dependent) range ${\ensuremath{\phi}}_{R}<\ensuremath{\phi}<{\ensuremath{\phi}}_{\mathrm{max}}$ within which large fluctuations are likely to drive the field towards the upper boundary $\ensuremath{\phi}={\ensuremath{\phi}}_{\mathrm{max}}$, where the semiclassical consideration breaks down. An exit from inflation into reheating without reaching ${\ensuremath{\phi}}_{\mathrm{max}}$ will occur almost surely (with probability 1) only if the initial value of $\ensuremath{\phi}$ is below ${\ensuremath{\phi}}_{R}$. In this way, strong self-reproduction effects constrain models of $k$-inflation.

Age constraints and fine tuning in variable-mass particle models

VAMP (variable-mass particles) scenarios, in which the mass of the cold dark matter particles is a function of the scalar field responsible for the present acceleration of the Universe, have been proposed as a solution to the cosmic coincidence problem, since in the attractor regime both dark energy and dark matter scale in the same way. We find that only a narrow region in parameter space leads to models with viable values for the Hubble constant and dark energy density today. In the allowed region, the dark energy density starts to dominate around the present epoch and consequently such models cannot solve the coincidence problem. We show that the age of the Universe in this scenario is considerably higher than the age for noncoupled dark energy models, and conclude that more precise independent measurements of the age of the Universe would be useful in distinguishing between coupled and noncoupled dark energy models.

Dealing with final state sensitivity for synchronous communication

We show di2erences ofthe synchronization basins oftwo Matsumoto-Chua’s circuit (operating in either the R6 ossler-type attractor regime or the Double-Scroll-type attractor regime), coupled unidirectionally or bidirectionally through a negative feed-back controller. The knowledge of the structure ofthe synchronization basins allows one to design a communication system robust to imperfections on the setting of the initial condition. c � 2002 Elsevier Science B.V. All rights reserved.