Complicated Attractors Research Articles

Excitable media store and transfer complicated information via topological defect motion

Excitable media are prevalent models for describing interesting effects in physical, chemical, and biological systems such as pattern formation, chaos, and wave propagation. In this manuscript, we propose a spatially extended variant of the FitzHugh–Nagumo model that exhibits new effects. In this excitable medium, waves of new kinds propagate. We show that the time evolution of the medium state at the wavefronts is determined by complicated attractors which can be chaotic. The dimension of these attractors can be large and we can control the attractor structure by initial data and a few parameters. These waves are capable transfer complicated information given by a Turing machine or associative memory. We show that these waves are capable to perform cell differentiation creating complicated patterns.

Models of Genetic Networks with Given Properties

A multi-parameter system of ordinary differential equations, modelling genetic networks, is considered. Attractors of this system correspond to future states of a network. Sufficient conditions for the non-existence of stable critical points are given. Due to the special structure of the system, attractors must exist. Therefore the existence of more complicated attractors was expected. Several examples are considered, confirming this conclusion.

Phase tipping: how cyclic ecosystems respond to contemporary climate.

We identify the phase of a cycle as a new critical factor for tipping points (critical transitions) in cyclic systems subject to time-varying external conditions. As an example, we consider how contemporary climate variability induces tipping from a predator–prey cycle to extinction in two paradigmatic predator–prey models with an Allee effect. Our analysis of these examples uncovers a counterintuitive behaviour, which we call phase tipping or P-tipping, where tipping to extinction occurs only from certain phases of the cycle. To explain this behaviour, we combine global dynamics with set theory and introduce the concept of partial basin instability for attracting limit cycles. This concept provides a general framework to analyse and identify easily testable criteria for the occurrence of phase tipping in externally forced systems, and can be extended to more complicated attractors.

Dynamics of a multiplex neural network with delayed couplings.

Multiplex networks have drawn much attention since they have been observed in many systems, e.g., brain, transport, and social relationships. In this paper, the nonlinear dynamics of a multiplex network with three neural groups and delayed interactions is studied. The stability and bifurcation of the network equilibrium are discussed, and interesting neural activities of the network are explored. Based on the neuron circuit, transfer function circuit, and time delay circuit, a circuit platform of the network is constructed. It is shown that delayed couplings play crucial roles in the network dynamics, e.g., the enhancement and suppression of the stability, the patterns of the synchronization between networks, and the generation of complicated attractors and multi-stability coexistence.

Knots and Solenoids that Cannot Be Attractors of Self-homeomorphisms of ℝ3

AbstractAs a 1st step to understand how complicated attractors for dynamical systems can be, one may consider the following realizability problem: given a continuum $K \subseteq \mathbb{R}^3$, decide when $K$ can be realized as an attractor for a homeomorphism of $\mathbb{R}^3$. In this paper we introduce toroidal sets as those continua $K \subseteq \mathbb{R}^3$ that have a neighbourhood basis comprised of solid tori and, generalizing the classical notion of genus of a knot, give a natural definition of the genus of toroidal sets and study some of its properties. Using these tools we exhibit knots and solenoids for which the answer to the realizability problem stated above is negative.

Instability Influenced by CO2 and Equivalence Ratio in Oxyhydrogen Flames on Flat Burner

ABSTRACTOxyhydrogen (H2/O2) combustion with carbon dioxide (CO2) was used widely because the replacement of nitrogen (N2) with CO2 was applied to reduce emissions of nitrogen oxide (NOx). However, the replacement of N2 with CO2 brings about cellular premixed flames owing to intrinsic instability. This study focuses on the effects of CO2 ratio on the characteristics of cellular premixed flames on flat burner at atmospheric pressure. Oxyhydrogen flames were experimentally investigated with variation of CO2 ratio, equivalence ratio, and total gas flow rate. Then the cell size, power spectral density, and reconstructed attractor were obtained. When the CO2 ratio increased, greater cell size, lower sharp peak frequency of power spectral density, and the more complicated doughnut ring of reconstructed attractor were found. It was caused by the replacement of N2 with CO2, which affected diffusive-thermal instability. As the equivalence ratio increased, the smaller cell size, higher sharp peak frequency, and smaller doughnut ring were obtained owing to the decrease of instability intensity. Moreover, we obtained smaller cell size, higher sharp peak frequency, and smaller doughnut ring when the total gas flow rate increased. The results showed that the increase of CO2 ratio and the decrease of equivalence ratio and total gas flow rate induced greater cell size, lower sharp peak frequency, and a more complicated attractor as higher instability intensity.

Koopman Invariant Subspaces and Finite Linear Representations of Nonlinear Dynamical Systems for Control.

In this work, we explore finite-dimensional linear representations of nonlinear dynamical systems by restricting the Koopman operator to an invariant subspace spanned by specially chosen observable functions. The Koopman operator is an infinite-dimensional linear operator that evolves functions of the state of a dynamical system. Dominant terms in the Koopman expansion are typically computed using dynamic mode decomposition (DMD). DMD uses linear measurements of the state variables, and it has recently been shown that this may be too restrictive for nonlinear systems. Choosing the right nonlinear observable functions to form an invariant subspace where it is possible to obtain linear reduced-order models, especially those that are useful for control, is an open challenge. Here, we investigate the choice of observable functions for Koopman analysis that enable the use of optimal linear control techniques on nonlinear problems. First, to include a cost on the state of the system, as in linear quadratic regulator (LQR) control, it is helpful to include these states in the observable subspace, as in DMD. However, we find that this is only possible when there is a single isolated fixed point, as systems with multiple fixed points or more complicated attractors are not globally topologically conjugate to a finite-dimensional linear system, and cannot be represented by a finite-dimensional linear Koopman subspace that includes the state. We then present a data-driven strategy to identify relevant observable functions for Koopman analysis by leveraging a new algorithm to determine relevant terms in a dynamical system by ℓ1-regularized regression of the data in a nonlinear function space; we also show how this algorithm is related to DMD. Finally, we demonstrate the usefulness of nonlinear observable subspaces in the design of Koopman operator optimal control laws for fully nonlinear systems using techniques from linear optimal control.

Stochastic equilibria of an asset pricing model with heterogeneous beliefs and random dividends

We investigate dynamical properties of a heterogeneous agent model with random dividends and further study the relationship between dynamical properties of the random model and those of the corresponding deterministic skeleton, which is obtained by setting the random dividends as their constant mean value. Based on our recent mathematical results, we prove the existence and stability of random fixed points as the perturbation intensity of random dividends is sufficiently small. Furthermore, we prove that the random fixed points converge almost surely to the corresponding fixed points of the deterministic skeleton as the perturbation intensity tends to zero. Moreover, simulations suggest similar behaviors in the case of more complicated attractors. Therefore, the corresponding deterministic skeleton is a good approximation of the random model with sufficiently small random perturbations of dividends. Given that dividends in real markets are generally very low, it is reasonable and significant to some extent to study the effects of heterogeneous agents’ behaviors on price fluctuations by the corresponding deterministic skeleton of the random model.

Dynamic compartment approach for modelling regimes of carbon cycle functioning in bog ecosystems

Svirezhev's method of dynamic model design by a given “storage-flow” diagram [Svirezhev Y.M., 1997. On some general properties of trophic networks. Ecol. Model. 99, 7–17] is developed and used for investigating dynamic regimes of carbon cycle functioning in a typical boreal transitional bog ecosystem. Ecosystems are often represented by static “storage-flow” diagrams reflecting their structure and matter or energy transfer between components at fixed time moments. Using the data of such diagrams aggregated in ecological field studies one can construct a dynamic model of the ecosystem to predict its future behaviour and to estimate a response to external perturbations—natural and human. Stability of both current equilibrium and possible alternative steady states and more complicated attractors are studied under two types of parameter perturbation: CO 2 atmospheric concentration increase initiated by greenhouse effect, and change in the rate of carbon output from dead organic matter and litter which depends on the water table level and possible peat excavation. Calculation of bifurcation curves gives areas in the parameter space where stable functioning of carbon cycle is provided. Steady states can be interpreted as raised bog, meadow, forest and fen. CO 2 concentration increase leads the current state of transitional bog to loose stability with appearance of oscillatory dynamics and further evolution to the chaotic attractor. The model is rich by chaotic solutions serving as transition regimes between regular steady and periodic attractors. Another chaotic regime is formed from forest equilibrium and exists in the same area of phase space where current equilibrium is stable.

On the structure of attractors for discrete, periodically forced systems with applications to population models

This work discusses the effects of periodic forcing on attracting cycles and more complicated attractors for autonomous systems of nonlinear difference equations. Results indicate that an attractor for a periodically forced dynamical system may inherit structure from an attractor of the autonomous (unforced) system and also from the periodicity of the forcing. In particular, a method is presented which shows that if the amplitude of the k-periodic forcing is small enough then the attractor for the forced system is the union of k homeomorphic subsets. Examples from population biology and genetics indicate that each subset is also homeomorphic to the attractor of the original autonomous dynamical system.

Complicated Endogenous Business Cycles under Gross Substitutability

In this paper we investigate the global behaviour of the output equilibrium time paths in a two-dimensional overlapping generations model with productive investment and capital accumulation, where agents behave rationally and markets are competitive. We show that equilibrium time paths can converge to complicated (chaotic) attractors as well as periodic attractors when the substitution effect dominates the income effect everywhere. The occurrence of complicated equilibrium paths is guaranteed by homoclinic intersections of the stable and unstable manifolds of the autarkic steady state.Journal of Economic LiteratureClassificiation Number: E32.

ON SELF-ORGANIZED CRITICALITY AND SYNCHRONIZATION IN LATTICE MODELS OF COUPLED DYNAMICAL SYSTEMS

Lattice models of coupled dynamical systems lead to a variety of complex behaviors. Between the individual motion of independent units and the collective behavior of members of a population evolving synchronously, there exist more complicated attractors. In some cases, these states are identified with self-organized critical phenomena. In other situations, they are identified with clusterization or phase-locking. The conditions leading to such different behaviors in models of integrate-and-fire oscillators and stick-slip processes are reviewed.

Chaotic and strange attractors of a two-dimensional map

A family of continuous, invertible standard maps of the torus, cylinder and plane is considered in this paper. Sequences of bifurcations are studied which correspond to the transformation of an invariant curve to chaotic and strange attractors. The characteristic variations of complicated attractors are considered. Hyperbolicity conditions are obtained for the case of piecewise-smooth maps. The maps generating the Henon, Lozi, Belykh attractors belong to our class of standard maps.

Experiment on Chaotic Vibrations of a Cantilevered Beam Deformed by a Stretched Cable.

This paper presents the experimental results on chaotic vibrations of a cantilevered beam deformed by a stretched cable. The beam is deformed to a postbuckled configuration by a cable that is extended from the top to bottom of the beam. We call the deformed beam a 'tendon beam'. As a lateral load is applied to the beam statically, the beam changes to another equilibrium position accompanied with a snap through deformation. When the beam is subjected to periodic excitation, a large-ampltiude response is generated by resonance. Chaotic motion appears near the fundamental resonance region as well as the region in the higher mode of vibration. Both chaotic motions are confirmed as the chaos due to maximum Lyapunov exponent with a positive number. By recording the Poincare section of the chaos behavior, fractal attractors are clearly focused in the fundamental resonance region. The more complicated attractor is obtained in the resonance region of the second mode of vibration. The correlation dimension of the chaos in the second mode is higher than that of the fundamental mode. It is generated by the two-to-thirtees internal resonance between the fundamental mode and second mode.

Chaotic solitons and the foundations of physics: a potential revolution

This paper will present a skeleton outline of new mathematical tools, based on concepts from quantum field theory (QFT), which could open up a new approach to understanding chaotic solitons and other chaotic modes for classical PDE in four dimensions. It will argue that a further development of these tools could lead to a radical reformulation of physics, with large potential implications both for technology and for biology. The central ideas include: the use of statistical moments as a way of mapping a complicated attractor to a point in Fock space; the use of reification or dressing operators to transform the statistical dynamics into a linear Schrodinger-like equation with a Hermitian “Hamiltonian”; the use of localized eigenvectors in Fock space to characterize chaotic solitons; in applications to QFT, the assumption of time- symmetry (rather than forwards time) in microscopic causality, which makes it possible to derive the measurement formalism of QFT; the use of field operators to calculate expectation values, and a truly four-dimensional approach that is equivalent but simpler than the usual approach of QFT. The potential implications, while speculative, could be extremely large; however, they cannot be realized or defined more precisely without additional mathematical research. There are many new research opportunities here in many different directions, all of them with large potential impact.

Generation of n-double scrolls (n=1, 2, 3, 4,...)

Chua's double scroll is probably the best known and most extensively studied example of chaotic behaviour generated by electrical circuits. It is illustrated in this paper that by modifying the characteristic of the nonlinear resistor with additional break points even more complicated attractors can be obtained, called n-double scrolls (n=1, 2, 3, 4,...). The new circuit can be seen as a generalization of Chua's circuit such that the 1-double scroll corresponds to the classical double scroll. The construction of the attractors was partially based on a combination of linearization around equilibrium points and an alternative method for studying nonlinear systems that we called a quasilinear approach. This method is heuristic and qualitative but may give additional global insight into the state space behaviour and may open new views towards the construction of attractors. >

Generalized horseshoe maps and inverse limits

The now-classical example due to Smale, the horseshoe map, displays interesting dynamics as well as a topologically complicated attractor. In 1986 Marcy Barge showed that the full attracting sets of horseshoe maps are homeomorphic to inverse limits of the unit interval with a single bonding map. Here we extend Barge's results to a more general class of maps. 1. Introduction. In [Ba], Barge describes the attracting sets of horseshoe maps as inverse limits of the unit interval with a single bonding map. Topologically these spaces are chainable continua known as Knaster continua. In this paper we consider a more general class of maps which we will refer to as generalized horseshoe maps. We will show that the attractors of these maps are homeomorphic to inverse limits of the unit interval with a single bonding map. Both the generalized horseshoe map and the bonding map which defines the inverse limit space described above follow a pattern in a sense we will define in the next section. In §3 we will prove two theorems about inverse limit spaces which will be needed in the proof of the main result given in §4. In the final section of the paper we will give some examples, and show that the horseshoe maps which Barge studied in [Ba] are special cases of the generalized horseshoes we consider here. For basic information on attractors and inverse limits see [S].

The exponential leveling in elliptic singular perturbation problems with complicated attractors

The exponential leveling in elliptic singular perturbation problems with complicated attractors

Stability of cosmological models

A new notion of stability recently introduced by Zeeman is applied in the cosmological setting by considering a number of simple dissipative cosmological models which are expressible as gradient fields. Despite the simplicity and the unrealisticness of some these models, this study we hope will give some idea as to the potential usefulness of the basic idea for models with more complicated attractors especially where the underlying dynamics may be chaotic and structurally unstable. Italso highlights the role of dissipation in connection with stability and its potential relevance in dealing with the question of measure on the initial states of the universe.

Bifurcations and chaos in differential systems representing the electrical activity of nerve cell membranes

Excitation equations are systems of nonlinear o.d.es that represent the biophysical properties of nerve and muscle membrane. They can have equilibrium, periodic, quasi-periodic and chaotic solutions. Numerical evaluation or eigenvalues show that periodic solutions develop from equilibria by Hopf bifurcation. Further bifurcations can lead to quasi-periodic or chaotic attractors. Methods of identifying complicated attractors are illustrated.