Dynamical Systems Point Research Articles

Estimation of Regions of Attraction for Time-Varying Uncertain Nonlinear Systems Modeled by a Particular Class of Linear Differential Inclusion

This paper proposes the use of a particular class of linear differential inclusion (LDI) for the estimation of regions of attraction of asymptotically stable equilibrium points of nonlinear dynamical systems with time-varying parametric uncertainties bounded by a polytope. In this paper, we provide a systematic procedure based on the mean value theorem in conjunction to a linear matrix inequality based approximation technique to obtain the description of the uncertain nonlinear system via the proposed LDI form, which can be written equivalently as a particular class of a time-varying linear system. Through this suitable system representation in the form of this time-varying linear system, a numerical algorithm based on convex optimization is proposed to determine these regions of stability. Two numerical examples are given to demonstrate the effectiveness of the proposed approach.

The Compound Poisson Limit Ruling Periodic Extreme Behaviour of Non-Uniformly Hyperbolic Dynamics

We prove that the distributional limit of the normalised number of returns to small neighbourhoods of periodic points of non-uniformly hyperbolic dynamical systems is compound Poisson. The returns to small balls around a fixed point in the phase space correspond to the occurrence of rare events, or exceedances of high thresholds, so that there is a connection between the laws of Return Times Statistics and Extreme Value Laws. The fact that the fixed point in the phase space is a repelling periodic point implies that there is a tendency for the exceedances to appear in clusters whose average sizes is given by the Extremal Index, which depends on the expansion of the system at the periodic point. We recall that for generic points, the exceedances, in the limit, are singular and occur at Poisson times. However, around periodic points, the picture is different: the respective point processes of exceedances converge to a compound Poisson process, so instead of single exceedances, we have entire clusters of exceedances occurring at Poisson times with a geometric distribution ruling its multiplicity. The systems to which our results apply include: general piecewise expanding maps of the interval (Rychlik maps), maps with indifferent fixed points (Manneville-Pomeau maps) and Benedicks-Carleson quadratic maps.

Erratum to: 2-D Duffing Oscillator: Elliptic Functions from a Dynamical Systems Point of View

Erratum to: 2-D Duffing Oscillator: Elliptic Functions from a Dynamical Systems Point of View

Rigorous A Posteriori Computation of (Un)Stable Manifolds and Connecting Orbits for Analytic Maps

This work is concerned with high order polynomial approximation of stable and unstable manifolds for analytic discrete time dynamical systems. We develop a posteriori theorems for these polynomial approximations which allow us to obtain rigorous bounds on the truncation errors via a computer assisted argument. Moreover, we represent the truncation error as an analytic function, so that the derivatives of the truncation error can be bounded using classical estimates of complex analysis. As an application of these ideas we combine the approximate manifolds and rigorous bounds with a standard Newton-Kantorovich argument in order to obtain a kind of result for connecting orbits between fixed points of discrete time dynamical systems. A feature of this method is that we obtain the transversality of the connecting orbit automatically. Examples of the manifold computation are given for invariant manifolds which have dimension between two and ten. Examples of the a posteriori error bounds and the analytic-shadowing argument for connecting orbits are given for dynamical systems in dimension three and six.

On spin dependence of relativistic acoustic geometry

This work makes the first ever attempt to understand the influence of the black hole background spacetime in determining the fundamental properties of the embedded relativistic acoustic geometry. To accomplish such task, we investigate the role of the spin angular momentum of the astrophysical black hole (the Kerr parameter a—a representative feature of the background black hole metric) in estimating the value of the acoustic surface gravity (the representative feature of the corresponding analogue spacetime). Since almost all astrophysical black holes are supposed to posses some degree of intrinsic rotation, the influence of the Kerr parameter on classical analogue models is very important to understand. We study the general relativistic, axially symmetric, non-self-gravitating inflow of the hydrodynamic fluid onto a rotating astrophysical black hole from the dynamical systems point of view. In this work the location of the acoustic horizon inside such fluid flow is identified and the associated acoustic surface gravity is estimated. We study the dependence of such surface gravity as a function of the Kerr parameter as well as with other dynamical and thermodynamic variables governing the fluid flow under strong gravity, and demonstrate that for retrograde flow, the surface gravity (and hence the associated analogue Hawking temperature) correlates with the black hole spin in general, whereas for the prograde flow, the surface gravity as well as the analogue temperature correlates with the black hole spin for slow to moderately rotating holes, but anti-correlates with the spin for fast to extremely rotating holes. We found that for certain values of the initial boundary conditions, more than one acoustic horizons, namely two black hole types and one white hole type, may form, and the surface gravity may become formally infinite at the acoustic white hole. We discuss the possible connection between the corresponding analogue Hawking temperature and astrophysically relevant observables associated with the spectral signature of the black hole candidates. Our result indicates that the modified dispersion relation evaluated at the close proximity of the acoustic horizon (and hence the nonuniversal feature of Hawking-like effects) is a sensitive function of the spin angular momentum of the astrophysical black hole. We propose that the black hole spin dependence of such dispersion relation may be used to distinguish a corotating flow from a counter rotating flow for axisymmetric accretion onto a Kerr black hole.

On Dynamical Systems and Phase Transitions for q + 1-state p-adic Potts Model on the Cayley Tree

In the present paper, we study a new kind of p-adic measures for q + 1-state Potts model, called p-adic quasi Gibbs measure. For such a model, we derive a recursive relations with respect to boundary conditions. Note that we consider two mode of interactions: ferromagnetic and antiferromagnetic. In both cases, we investigate a phase transition phenomena from the associated dynamical system point of view. Namely, using the derived recursive relations we define a fractional p-adic dynamical system. In ferromagnetic case, we establish that if q is divisible by p, then such a dynamical system has two repelling and one attractive fixed points. We find basin of attraction of the fixed point. This allows us to describe all solutions of the nonlinear recursive equations. Moreover, in that case there exists the strong phase transition. If q is not divisible by p, then the fixed points are neutral, and this yields that the existence of the quasi phase transition. In antiferromagnetic case, there are two attractive fixed points, and we find basins of attraction of both fixed points, and describe solutions of the nonlinear recursive equation. In this case, we prove the existence of a quasi phase transition.

Modeling and forecasting of stock markets under a system adaptation framework

This paper adopts the concept of dynamic feedback systems to model the behavior of financial markets, or more specifically, the stock market from a dynamic system point of view. Based on a feedback adaptation scheme, the authors model the movement of a stock market index within a framework that is composed of an internal dynamic model and an adaptive filter. The output-error model is adopted as the internal model whereas the adaptive filter is a time-varying state space model with instrumental variables. Its input-output behavior, and internal as well as external forces are then identified. Special attention has also been paid to the recent financial crisis by examining the movement of Dow Jones Industrial Average (DJIA) as an example to illustrate the advantage of the proposed framework. Supported by time-varying causality tests, five influential factors from economic and sentiment aspects are introduced as the input of this framework. Testing results show that the proposed framework has a much better prediction performance than the existing methods, especially in complicated economic situations. An application of this framework is also presented with focuses on forecasting the turning periods of the market trend. Realizing that a market trend is about to change when the external force begins to exhibit clear patterns in its frequency responses, the authors develop a set of rules to recognize this kind of clear patterns. These rules work well for stock indexes from US, China and Singapore.

Monomial Dynamical Systems in # P-complete

In this paper, we study boolean monomial dynamical systems. Colón-Reyes, Jarrah, Laubenbacher, and Sturmfels(2006) studied fixed point structure of boolean monomial dynamical systems of f by associating the dynamical systems of f with its dependency graph χf and Jarrah, Laubenbacher, and Veliz-Cuba(2010) extended it and presented lower and upper bound for the number of cycles of a given length for general boolean monomial dynamics. But, it is even difficult to determine the exact number of fixed points of boolean monomial dynamics. We show that the problem of counting fixed points of a boolean monomial dynamical systems is #P-complete, for which no efficient algorithm is known. This is proved by a 1-1 correspondence between fixed points of f sand antichains of the poset of strongly connected components of χf..

2-D Duffing Oscillator: Elliptic Functions from a Dynamical Systems Point of View

K. Meyer has advocated for the study of elliptic functions and integrals from a dynamical systems point of view. Here, we follow his advice and we propose the bidimensional Hamiltonian Duffing oscillator as a model; it allows us to deal with the elliptic integral of third kind directly. Focusing on bounded trajectories we do a detailed analysis of the solutions in the three regions defined by the parameters. In our opinion, for the study of elliptic functions, the model presented here represents an alternative to the pendulum or the free rigid body systems.

About the oscillatory possibilities of the dynamical systems

This paper may be ultimately described as an attempt to make feasible the evolutionary emergence of novelty in a supposedly deterministic world whose behavior is associated with that of the mathematical dynamical systems. It means philosophical implications that the paper needs to address, subsidiarily at least. The work was motivated by the observation of complex oscillatory behaviors in a family of physical devices and related mathematical models, for which there is no known explanation in the mainstream of nonlinear dynamics. The paper begins by describing a nonlinear mechanism of oscillatory mode mixing explaining such behaviors and, through its generalization to richer nonlinear vector fields, establishes a generic dynamical scenario with extraordinary oscillatory possibilities, including expansive growing scalability toward high dimensionalities and through nonlinear multiplicities. The scenario is then used to tentatively explain complex oscillatory behaviors observed in nature like those of turbulent fluids and living brains. Finally, by considering the scenario as a dynamic substrate underlying generic aspects of both the functioning and the genesis of complex behaviors in a supposedly deterministic world, a theoretical framework covering the evolutionary development of structural transformations in the time evolution of that world is built up. The analysis includes attempts to clarify the roles of items often invoked apropos of pathways to complexity like chaos, pattern formation, externally-driven bifurcations, hysteresis, irreversibility, and order through random fluctuations. Thermodynamics, as the exclusive field of physics in providing generic evolutionary criteria, is briefly and synthetically considered from the dynamical systems point of view by trying to elucidate its explanatory possibilities concerning the emergence of complexity. Quantum mechanics gets involved in two different ways: the lack of a dynamical systems perspective in the currently accepted interpretations of that fundamental theory and the indeterminacy issues, and both questions are discussed to point out their consequences. The reported evolutionary framework is far from a complete theory but includes both the elements and the skeleton for its tentative building within feasible philosophical grounds. In the lack of alternatives, one should imagine how it could be one of such theories and how it could be built, in order to evaluate our approach. In particular, notice that our approach is to a theory of nothing of the physical world but of the underlying reasons for its ordered and creative functioning, which we interpret to be independent of that world, i.e., a theory of what the Catalan expression ”l’entrellat del món” describes so well.

F-limit points in dynamical systems defined on the interval

AbstractGiven a free ultrafilter p on ℕ we say that x ∈ [0, 1] is the p-limit point of a sequence (x n)n∈ℕ ⊂ [0, 1] (in symbols, x = p -limn∈ℕ x n) if for every neighbourhood V of x, {n ∈ ℕ: x n ∈ V} ∈ p. For a function f: [0, 1] → [0, 1] the function f p: [0, 1] → [0, 1] is defined by f p(x) = p -limn∈ℕ f n(x) for each x ∈ [0, 1]. This map is rarely continuous. In this note we study properties which are equivalent to the continuity of f p. For a filter F we also define the ω F-limit set of f at x. We consider a question about continuity of the multivalued map x → ω fF(x). We point out some connections between the Baire class of f p and tame dynamical systems, and give some open problems.

On the existence of J 2 invariant relative orbits from the dynamical system point of view

The present paper addresses the existence of J2 invariant relative orbits with arbitrary relative magnitude over the infinite time using the Routh reduction and Poincare techniques in the J2 Hamiltonian problem. The current research also proposes a novel numerical searching approach for J2 invariant relative orbits from the dynamical system point of view. A new type of Poincare mapping is defined from different central manifolds of the pseudo-circular orbits (parameterized by the Jacobi energy E, the polar component of momentum Hz and the measure of distance Δr between the fixed point and its central manifolds) to the nodal periods Td and the drifts of longitude of the ascending node during one period (ΔΩ), which differs from Koon et al.’s (AIAA 2001) definition on central manifolds parameterized by the same fixed point. The Poincare mapping is surjective because it compresses the three-dimensional variables into two-dimensional images, and the mapping degenerates into a bijective mapping in consideration of the fixed points. An iteration algorithm to the degenerated bijective mapping is proposed from the continuation procedure to perform the ergodic representation of E- and Hz-contour maps on the space of Td–ΔΩ. For the surjective mapping with Δr ≠ 0, different pseudo-circular or elliptical orbits may share the same images. Hence, the inverse surjective mapping may achieve non-unique variables from a single image, which makes the generation of J2 invariant relative orbits possible. The pseudo-circular or elliptical orbits generated from the surjective mapping will be defined in different meridian planes. Hence, the critical contribution of the present paper is the assignment of J2 invariant relative orbits to different invariant parameters E and Hz depending on the E- and Hz-contour map, which will hold J2 invariant relative orbits for extended durations. To investigate the high-order nonlinearity neglected by previous studies, a formation configuration with a large magnitude of 500 km is successfully generated from the theory developed in the present work, which is beyond the scope of the linear conditions of J2 invariant relative orbits. Therefore, the existence of J2 invariant relative orbit with an arbitrary relative magnitude over the infinite time is achieved from the dynamical system point of view.

On equilibrium properties of evolutionary multi-player games with random payoff matrices

The analysis of equilibrium points in biological dynamical systems has been of great interest in a variety of mathematical approaches to biology, such as population genetics, theoretical ecology or evolutionary game theory. The maximal number of equilibria and their classification based on stability have been the primary subjects of these studies, for example in the context of two-player games with multiple strategies. Herein, we address a different question using evolutionary game theory as a tool. If the payoff matrices are drawn randomly from an arbitrary distribution, what are the probabilities of observing a certain number of (stable) equilibria? We extend the domain of previous results for the two-player framework, which corresponds to a single diploid locus in population genetics, by addressing the full complexity of multi-player games with multiple strategies. In closing, we discuss an application and illustrate how previous results on the number of equilibria, such as the famous Feldman–Karlin conjecture on the maximal number of isolated fixed points in a viability selection model, can be obtained as special cases of our results based on multi-player evolutionary games. We also show how the probability of realizing a certain number of equilibria changes as we increase the number of players and number of strategies.

COLORFUL SYMMETRIC IMAGES IN THREE-DIMENSIONAL SPACE FROM DYNAMICAL SYSTEMS

Functions that are invariant with respect to the tetrahedral and cubic symmetries are determined. These invariant mappings are applied to serve as the density functions for automatic generation of the colorful images with such symmetries in three dimensional space from a dynamical system's point of view.

Optimal region of latching activity in an adaptive Potts model for networks ofneurons

In statistical mechanics, the Potts model is a model for interacting spins with more thantwo discrete states. Neural networks which exhibit features of learning and associativememory can also be modeled by a system of Potts spins. A spontaneous behavior ofhopping from one discrete attractor state to another (referred to as latching) has beenproposed to be associated with higher cognitive functions. Here we propose a model inwhich both the stochastic dynamics of Potts models and an adaptive potentialfunction are present. A latching dynamics is observed in a limited region of thenoise(temperature)–adaptation parameter space. We hence suggest noise as a fundamentalfactor in such alternations alongside adaptation. From a dynamical systems point of view,the noise–adaptation alternations may be the underlying mechanism for multi-stabilityin attractor-based models. An optimality criterion for realistic models is finallyinferred.

Preperiodic points and unlikely intersections

In this article, we combine complex-analytic and arithmetic tools to study the preperiodic points of one-dimensional complex dynamical systems. We show that for any fixed complex numbers a and b, and any integer d at least 2, the set of complex numbers c for which both a and b are preperiodic for z^d+c is infinite if and only if a^d = b^d. This provides an affirmative answer to a question of Zannier, which itself arose from questions of Masser concerning simultaneous torsion sections on families of elliptic curves. Using similar techniques, we prove that if two complex rational functions f and g have infinitely many preperiodic points in common, then they must have the same Julia set. This generalizes a theorem of Mimar, who established the same result assuming that f and g are defined over an algebraic extension of the rationals. The main arithmetic ingredient in the proofs is an adelic equidistribution theorem for preperiodic points over number fields and function fields, with non-archimedean Berkovich spaces playing an essential role.

The gentlest ascent dynamics

Dynamical systems that describe the escape from the basins of attraction of stable invariant sets are presented and analysed. It is shown that the stable fixed points of such dynamical systems are the index-1 saddle points. Generalizations to high index saddle points are discussed. Both gradient and non-gradient systems are considered. Preliminary results on the nature of the dynamical behaviour are presented.

Two-dimensional attractors in the border-collision normal form

New techniques are developed to show that the two-dimensional normal form for codimension one border-collision bifurcations of fixed points of discrete time piecewise smooth dynamical systems has attractors which are themselves two-dimensional regions with polygonal boundaries. This makes it possible to prove the existence of these attractors for a countable set of parameter values which cannot be treated using the essentially one-dimensional methods in the literature.

Existence of travelling wavefronts of the KdV–Burgers equation

In this letter, we study the existence of travelling wavefronts of the KdV–Burgers equation from a monotone dynamical systems point of view. For any constant wave velocity, we obtain a sufficient condition for the existence. The conditions under which some exact travelling wavefronts have been found are special cases of ours.

A neurocomputational model of nicotine addiction based on reinforcement learning

Event Abstract Back to Event A neurocomputational model of nicotine addiction based on reinforcement learning Selin Metin1* and Neslihan Serap Sengor1 1 Istanbul Technical University, Türkiye In modeling nicotine addiction, we consider that addiction develops due to malfunctioning of reinforcement learning processes driven by metabolical modifications in the brain. We base our approach on the fact that addiction develops as a result of damaged reward mechanism. Excessive fondness for a substance causes compulsive seeking of that substance and opponent process theory is used in explaining the development of addiction. Nicotine blood level triggers the secretion of dopamine (DA) from the ventral tegmental area (VTA) due to a race with the naturally secreted acetylcholine (Ach) neurotransmitter. In return, glutamate secretion in learning processes is affected and it modifies behavioral choices. Chronic nicotine exposure stamps in the behavioral patterns modified by DA secretion and causes addiction development. The proposed computational model is developed using MATLAB in-house codes and realizes DA secretion from the VTA to the cortico-striato-thalamic (CST) loop using reinforcement learning. DA secretion, action selection, action evaluation, and value assignment subsystems are modeled as nonlinear dynamical systems. The dynamic behavior of these systems is investigated by bifurcation analysis using XPP in order to give an explanation of processes going on from dynamical system point of view. The action selection system uses competitive learning which is modified with VTA DA signaling affected by nicotine. Arithmetically increasing reward value affects the system in favor of selecting the smoking action. Error in expectation symbolizes the modifying effects of the neurotransmitters and changes the output of dorsal striatum, amplifies the emotional input, and updates the stimulus value. Past actions contribute to the evaluation process as the input from the previous action so the cumulative effects of the previous actions trigger the present action selection. During model execution, addictive, non-addictive, and indecisive behaviors are observed in simulation results. Addiction developed in 20/50 trials with an average of 346/1000 steps and standard deviation of 265.76. To improve the proposed model, reward should be computed as a dynamically changing function of model parameters. Also, the approach should be expanded to include the molecular basis of DA secretion mechanism from the VTA to the CST loop involving the relevant neurotransmitters to be able to give a more realistic model. Funding Information: This work is partially supported by BAP-ITU References [1] Gutkin, B.S., Dehaene, S., Changeux, J.P., “A Neurocomputational Hypothesis for Nicotine Addiction”, PNAS, vol.103, no.4, 1106-1111, Jan 24, 2006. [2] Metin, S., Sengor, N.S., A Neurocomputational Model of Nicotine Addiction Based on Reinforcement Learning, ICANN 2010, 20th International Conference on Artificial Neural Networks, (2010), pp: 15-18. Keywords: Dopamine, Neurons, networks and dynamical systems, reinforcement learning Conference: BC11 : Computational Neuroscience & Neurotechnology Bernstein Conference & Neurex Annual Meeting 2011, Freiburg, Germany, 4 Oct - 6 Oct, 2011. Presentation Type: Poster Topic: neurons, networks and dynamical systems (please use "neurons, networks and dynamical systems" as keywords) Citation: Metin S and Sengor N (2011). A neurocomputational model of nicotine addiction based on reinforcement learning. Front. Comput. Neurosci. Conference Abstract: BC11 : Computational Neuroscience & Neurotechnology Bernstein Conference & Neurex Annual Meeting 2011. doi: 10.3389/conf.fncom.2011.53.00058 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: 23 Aug 2011; Published Online: 04 Oct 2011. * Correspondence: Dr. Selin Metin, Istanbul Technical University, Istanbul, Türkiye, selinmetin@gmail.com 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 Selin Metin Neslihan Serap Sengor Google Selin Metin Neslihan Serap Sengor Google Scholar Selin Metin Neslihan Serap Sengor PubMed Selin Metin Neslihan Serap Sengor 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.