True Attractor 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.

Три столицы «Русьского» мира

The author studies the probabilistic attractor of postbifurcation movement of Russia, Ukraine and Belorussia. The essence of the attractor is defined as presence in the unified "Rus'skoy" civilization. The purpose of the work is to give argumentation of the need of choosing the given civilization. The research is methodologically completed using dialectic, hermeneutic and historical and logical approaches. The results of the study represent actual conclusions from the dialectic of "egoistic" and "collectivist" in recent history of the three countries; from the long-standing contradiction of leaders and peoples of these states; from actual confrontation "Nazi-Ukraine" and "Russo-Ukraine". The author demonstrates the essence of this confrontation – geopolitical struggle of Western and Russian civilization at the area of Belorussia, Ukraine and Russia. The novelty of the study is implied in understanding the antinomic truth of Rus'skoy civilization – catch-up, as well as advanced; in formulating new definitions of "Nazi-Ukraine" and "Russo-Ukraine" and in explaining their actual contradictions; in clarifying the specific of the opposition between the West and Russia. The author shows the "orthodoxy" of the West's antislavism, the reasons of the Eastern-Slavic space destruction, axiological degradation of the West and potentiality of "empire decline" (USA). The article reveals the orthodoxy of territorial and property claims to Slavic peoples; the Germans' desire to settle new territories and use these areas as colonies. The researcher proves that the attitude towards Ukraine is that of a partner-vassal. The conclusions of the research are grounded on the argumentation of the true attractor of the three countries movement – towards the unified "Rus'skaya" civilization. The essential causes of this process are immanent contradictions of the Western civilization, especially between the American atlanticism and Germanocentric continentalism. Moreover, there is the necessity for common cultural and historical fate connected with the faith to invaluable traditions being trampled and disappearing in the Western civilization. At the same time, the author demonstrates the possibility of maintaining democratic values and specific "bourgeois – non-bourgeois" social and economic projects.

A spiral attractor network drives rhythmic locomotion.

The joint activity of neural populations is high dimensional and complex. One strategy for reaching a tractable understanding of circuit function is to seek the simplest dynamical system that can account for the population activity. By imaging Aplysia's pedal ganglion during fictive locomotion, here we show that its population-wide activity arises from a low-dimensional spiral attractor. Evoking locomotion moved the population into a low-dimensional, periodic, decaying orbit - a spiral - in which it behaved as a true attractor, converging to the same orbit when evoked, and returning to that orbit after transient perturbation. We found the same attractor in every preparation, and could predict motor output directly from its orbit, yet individual neurons' participation changed across consecutive locomotion bouts. From these results, we propose that only the low-dimensional dynamics for movement control, and not the high-dimensional population activity, are consistent within and between nervous systems.

The error of representation: basic understanding

Representation error arises from the inability of the forecast model to accurately simulate the climatology of the truth. We present a rigorous framework for understanding this kind of error of representation. This framework shows that the lack of an inverse in the relationship between the true climatology (true attractor) and the forecast climatology (forecast attractor) leads to the error of representation. A new gain matrix for the data assimilation problem is derived that illustrates the proper approaches one may take to perform Bayesian data assimilation when the observations are of states on one attractor but the forecast model resides on another. This new data assimilation algorithm is the optimal scheme for the situation where the distributions on the true attractor and the forecast attractors are separately Gaussian, and there exists a linear map between them. The results of this theory are illustrated in a simple Gaussian multivariate model.

Attractor solutions in scalar-field cosmology

Models of cosmological scalar fields often feature "attractor solutions" to which the system evolves for a wide range of initial conditions. There is some tension between this well-known fact and another well-known fact: Liouville's theorem forbids true attractor behavior in a Hamiltonian system. In universes with vanishing spatial curvature, the field variables (\phi,\dot\phi) specify the system completely, defining an effective phase space. We investigate whether one can define a unique conserved measure on this effective phase space, showing that it exists for m^2\phi^2 potentials and deriving conditions for its existence in more general theories. We show that apparent attractors are places where this conserved measure diverges in the (\phi,\dot\phi) variables and suggest a physical understanding of attractor behavior that is compatible with Liouville's theorem.

Instability of Optical Solitons in the Boundary Value Problem for a Medium of Finite Extension

We consider an integrable nonlinear wave system (anisotropic chiral field model) which exhibits a soliton solution when the Cauchy problem for an infinitely long medium is posed. Whenever the boundary value problem is formulated for the same system but for a medium of finite extension, we reveal that the soliton becomes unstable and the true attractor is a different structure which is called polarization attractor. In contrast to the localized nature of solitons, the polarization attractor occupies the entire length of the medium. By demonstrating the qualitative difference between nonlinear wave propagation in an infinite medium and in a medium of finite extension (with simultaneous change of the initial value problem to the boundary value problem), we would like to point out that solitons may loose their property of being stable attractors. Additionally, our findings show the interest of developing methods of integration for boundary value problems.

Speed versus accuracy in spiking attractor networks

Event Abstract Back to Event Speed versus accuracy in spiking attractor networks Networks of recurrently coupled neurons have been suggested to be the biological substrates for many of the sophisticated computations the nervous system can perform. In particular, the computations carried out by such networks are often considered to take the form of dynamically evolving patterns of activity that tend towards attractor states. The rate of convergence to these attractors is of crucial importance because it implies a fundamental trade-off between the time within which the state of a network comes close enough to the desired attractor (speed), and how close it comes to that attractor (accuracy). Here, we present an analysis of this trade-off and show how it depends on the time constants governing the dynamics of attractor networks. Importantly, the time-accuracy trade-off cannot be analysed meaningfully in the most common formulation of attractor networks that characterises neural activities solely by continuous firing rates. Under that formulation, the time constants determining the speed of convergence can be set to arbitrarily small values in theory, and biophysical limits on time constants can only be treated as constraints. Thus, we consider a more realistic scenario in which neurons interact through spikes that are stochastically generated based on their underlying `firing rates', or membrane potentials. We find that under such conditions the speed-accuracy trade-off is optimally balanced by finite, rather than infinitely small, time constants. This is because each neuron effectively needs time to estimate the firing rate of its presynaptic partners, and longer time constants will lead to more accurate estimates but slower convergence. We derive analytical relationships between the time constants governing the dynamics of spiking networks and the bias and variance of network states relative to the true attractor state. We also show how these relations change with other parameters of interest, such as the size of the network, and the average firing rates of neurons. In particular, we show that the bigger the network is, the faster it will come close enough to the attractor. We also report the results of numerical simulations that show good agreement with the theory. Computing the bias and variance also allows us to compute the error the network is making in reaching its attractor state, which in turn is used to determine the best time constants to minimize the error. Our results on the dependence of optimal time constants on average firing rates also suggest interesting normative roles for adaptation of time constants by slow components of membrane dynamics and by short-term synaptic plasticity. This work was supported by the Wellcome trust. Conference: Computational and systems neuroscience 2009, Salt Lake City, UT, United States, 26 Feb - 3 Mar, 2009. Presentation Type: Poster Presentation Topic: Poster Presentations Citation: (2009). Speed versus accuracy in spiking attractor networks. Front. Syst. Neurosci. Conference Abstract: Computational and systems neuroscience 2009. doi: 10.3389/conf.neuro.06.2009.03.131 Copyright: The abstracts in this collection have not been subject to any Frontiers peer review or checks, and are not endorsed by Frontiers. They are made available through the Frontiers publishing platform as a service to conference organizers and presenters. The copyright in the individual abstracts is owned by the author of each abstract or his/her employer unless otherwise stated. Each abstract, as well as the collection of abstracts, are published under a Creative Commons CC-BY 4.0 (attribution) licence (https://creativecommons.org/licenses/by/4.0/) and may thus be reproduced, translated, adapted and be the subject of derivative works provided the authors and Frontiers are attributed. For Frontiers’ terms and conditions please see https://www.frontiersin.org/legal/terms-and-conditions. Received: 02 Feb 2009; Published Online: 02 Feb 2009. Login Required This action requires you to be registered with Frontiers and logged in. To register or login click here. Abstract Info Abstract The Authors in Frontiers Google Google Scholar PubMed Related Article in Frontiers Google Scholar PubMed Abstract Close Back to top Javascript is disabled. Please enable Javascript in your browser settings in order to see all the content on this page.

Out of equilibrium dynamics of supersymmetry at high energy density

We investigate the out of equilibrium dynamics of global chiral supersymmetry at finite energy density. We concentrate on two specific models. The first is the massive Wess–Zumino model which we study in a self-consistent one-loop approximation. We find that for energy densities above a certain threshold, the fields are driven dynamically to a point in field space at which the fermionic component of the superfield is massless. The state, however, is found to be unstable, indicating a breakdown of the one-loop approximation. To investigate further, we consider an O( N) massive chiral model which is solved exactly in the large N limit. For sufficiently high energy densities, we find that for late times the fields reach a nonperturbative minimum of the effective potential degenerate with the perturbative minimum. This minimum is a true attractor for O( N) invariant states at high energy densities, and this provides a mechanism for determining which of the otherwise degenerate vacua is chosen by the dynamics. The final state for large energy density is a cloud of massless particles (both bosons and fermions) around this new nonperturbative supersymmetric minimum. By introducing boson masses which softly break the supersymmetry, we demonstrate a see-saw mechanism for generating small fermion masses. We discuss some of the cosmological implications of our results.

Spatial discretization of mappings

State space discretization occurs in the discrete finite arithmetic of a computer. When a dynamical system is simulated by numerical computations, it consequently evolves in a discretized space of this kind. Where attractors are seen in the simulation, what is their relation to the theoretical structures? If a theoretical attractor occurs, should we expect always to see a computational attractor? We address these questions by giving sufficient conditions for a discretized attractor to be present, and show that it converges to the true attractor in the sense of convergence of compact sets in the Hausdorff metric.

Computer dynamics and shadowing of chaotic orbits

We report results of digital integration of a piecewise linear system operating in a chaotic, parameter sensitive region, using different precisions of floating number representation in a digital computer. We compare these results to the “exact” closed form solution. Resulting discrepancies are explained by treating the computer as a deterministic system superimposing its dynamics on the dynamics of the investigated system. Good approximations of the true attractor are obtained by injecting controlled amounts of uniformly distributed random noise during digital integration of the system trajectories.