Point Attractor Research Articles

Long-time behavior for a class of weighted equations with degeneracy

In this paper we study the existence and some properties of the global attractors for a class of weighted equations when the weighted Sobolev space \begin{document}$ H_0^{1,a}(\Omega) $\end{document} (see Definition 1.1) cannot be bounded embedded into \begin{document}$ L^2(\Omega) $\end{document} . We claim that the dimension of the global attractor is infinite by estimating its lower bound of \begin{document}$ Z_2 $\end{document} -index. Moreover, we prove that there is an infinite sequence of stationary points in the global attractor which goes to 0 and the corresponding critical value sequence of the Lyapunov functional also goes to 0.

Black Holes and Higher Depth Mock Modular Forms

By enforcing invariance under S-duality in type IIB string theory compactified on a Calabi-Yau threefold, we derive modular properties of the generating function of BPS degeneracies of D4-D2-D0 black holes in type IIA string theory compactified on the same space. Mathematically, these BPS degeneracies are the generalized Donaldson-Thomas invariants counting coherent sheaves with support on a divisor $\cal D$, at the large volume attractor point. For $\cal D$ irreducible, this function is closely related to the elliptic genus of the superconformal field theory obtained by wrapping M5-brane on $\cal D$ and is therefore known to be modular. Instead, when $\cal D$ is the sum of $n$ irreducible divisors ${\cal D}_i$, we show that the generating function acquires a modular anomaly. We characterize this anomaly for arbitrary $n$ by providing an explicit expression for a non-holomorphic modular completion in terms of generalized error functions. As a result, the generating function turns out to be a (mixed) mock modular form of depth $n-1$.

Limit cycle dynamics can guide the evolution of gene regulatory networks towards point attractors

Developmental dynamics in Boolean models of gene networks self-organize, either into point attractors (stable repeating patterns of gene expression) or limit cycles (stable repeating sequences of patterns), depending on the network interactions specified by a genome of evolvable bits. Genome specifications for dynamics that can map specific gene expression patterns in early development onto specific point attractor patterns in later development are essentially impossible to discover by chance mutation alone, even for small networks. We show that selection for approximate mappings, dynamically maintained in the states comprising limit cycles, can accelerate evolution by at least an order of magnitude. These results suggest that self-organizing dynamics that occur within lifetimes can, in principle, guide natural selection across lifetimes.

Control of spiral drift by using feedback signals from a circular measuring domain in oscillatory media

In this paper, a non-local feedback is used to force spiral waves in the oscillatory system modeled by the complex Ginzburg-Landau equation, where the feedback signal is derived from a circular measuring domain. It is found that the spiral tip could be driven to follow an attractor after a period of drift. The existence of multiple types of attractors, which include a point attractor at the measuring center and a set of circular attractors concentric with the measuring domain, is illustrated. Which to be chosen depends on the direction of the spiral initial drift and the distance between the initial tip position and the measuring center. Here the diameter of the circular attractor has a periodic change with the size of the measuring domain, but is not affected by the movement of the measuring domain. Correlation between the tip drift and the feedback signal is analyzed, and the modulus function of the feedback signal will has a constant value after the spiral tip arrives at the attractors. If the feedback signal is generated from the measuring domain far away from the position of the initial spiral tip, the spiral breakup can be caused by the feedback control, and it fist occurs near the boundary and then invades inward until the area of the remaining spiral is small enough. The effect of the feedback gain, the delay time and the scaling parameter is also considered.

Bridging Dynamical Systems and Optimal Trajectory Approaches to Speech Motor Control With Dynamic Movement Primitives.

Current models of speech motor control rely on either trajectory-based control (DIVA, GEPPETO, ACT) or a dynamical systems approach based on feedback control (Task Dynamics, FACTS). While both approaches have provided insights into the speech motor system, it is difficult to connect these findings across models given the distinct theoretical and computational bases of the two approaches. We propose a new extension of the most widely used dynamical systems approach, Task Dynamics, that incorporates many of the strengths of trajectory-based approaches, providing a way to bridge the theoretical divide between what have been two separate approaches to understanding speech motor control. The Task Dynamics (TD) model posits that speech gestures are governed by point attractor dynamics consistent with a critically damped harmonic oscillator. Kinematic trajectories associated with such gestures should therefore be consistent with a second-order dynamical system, possibly modified by blending with temporally overlapping gestures or altering oscillator parameters. This account of observed kinematics is powerful and theoretically appealing, but may be insufficient to account for deviations from predicted kinematics—i.e., changes produced in response to some external perturbations to the jaw, changes in control during acquisition and development, or effects of word/syllable frequency. Optimization, such as would be needed to minimize articulatory effort, is also incompatible with the current TD model, though the idea that the speech production systems economizes effort has a long history and, importantly, also plays a critical role in current theories of domain-general human motor control. To address these issues, we use Dynamic Movement Primitives (DMPs) to expand a dynamical systems framework for speech motor control to allow modification of kinematic trajectories by incorporating a simple, learnable forcing term into existing point attractor dynamics. We show that integration of DMPs with task-based point-attractor dynamics enhances the potential explanatory power of TD in a number of critical ways, including the ability to account for external forces in planning and optimizing both kinematic and dynamic movement costs. At the same time, this approach preserves the successes of Task Dynamics in handling multi-gesture planning and coordination.

Non-ideal memristor synapse-coupled bi-neuron Hopfield neural network: Numerical simulations and breadboard experiments

This paper presents a third-order non-ideal memristor synapse-coupled bi-neuron Hopfield neural network (HNN), in which a non-ideal memristor synapse is employed to generate an electromagnetic induction current caused by the potential difference between two neurons. The memristive HNN model possesses an odd number of equilibrium points including one (or three, dependent on the memristor coupling strength) unstable saddle point, two unstable index-2 saddle-foci, and two (or four) stable node-foci. Bifurcation plots, local attraction basins, and phase plane plots show that the memristive HNN model behaves bi-stability of coexisting chaotic and stable point attractors, which is perfectly validated by breadboard experiments based on discrete components.

Asymptotics for a class of iterated random cubic operators

We consider a class of cubic stochastic operators that are motivated by models for evolution of frequencies of genetic types in populations. We take populations with three mutually exclusive genetic types.The long term dynamics of single maps, starting with a generic initial condition, is asymptotic to equilibria where either only one genetic type survives, or where all three genetic types occur.We consider a family of independent and identically distributed maps from this class and study its long term dynamics, in particular its random point attractors. The long term dynamics of the random composition of maps is asymptotic, almost surely, to equilibria. In contrast to the deterministic system, for generic initial conditions these can be equilibria with one or two or three types present (depending only on the distribution).

Digital memcomputing: From logic to dynamics to topology

Digital memcomputing machines (DMMs) are a class of computational machines designed to solve combinatorial optimization problems. A practical realization of DMMs can be accomplished via electrical circuits of highly non-linear, point-dissipative dynamical systems engineered so that periodic orbits and chaos can be avoided. A given logic problem is first mapped into this type of dynamical system whose point attractors represent the solutions of the original problem. A DMM then finds the solution via a succession of elementary instantons whose role is to eliminate solitonic configurations of logical inconsistency (“logical defects”) from the circuit. By employing a supersymmetric theory of dynamics, a DMM can be described by a cohomological field theory that allows for computation of certain topological matrix elements on instantons that have the mathematical meaning of intersection numbers on instantons. We discuss the “dynamical” meaning of these matrix elements, and argue that the number of elementary instantons needed to reach the solution cannot exceed the number of state variables of DMMs, which in turn can only grow at most polynomially with the size of the problem. These results shed further light on the relation between logic, dynamics and topology in digital memcomputing.

Critical branching processes in digital memcomputing machines

Memcomputing is a novel computing paradigm that employs time non-locality (memory) to solve combinatorial optimization problems. It can be realized in practice by means of non-linear dynamical systems whose point attractors represent the solutions of the original problem. It has been previously shown that during the solution search digital memcomputing machines go through a transient phase of avalanches (instantons) that promote dynamical long-range order. By employing mean-field arguments we predict that the distribution of the avalanche sizes follows a Borel distribution typical of critical branching processes with exponent . We corroborate this analysis by solving various random 3-SAT instances of the Boolean satisfiability problem. The numerical results indicate a power-law distribution with exponent , in very good agreement with the mean-field analysis. This indicates that memcomputing machines self-tune to a critical state in which avalanches are characterized by a branching process, and that this state persists across the majority of their evolution.

Coexistence of attractors in integer- and fractional-order three-dimensional autonomous systems with hyperbolic sine nonlinearity: Analysis, circuit design and combination synchronisation

This paper reports the results of the analytical, numerical and analogical analyses of integer- and fractional-order chaotic systems with hyperbolic sine nonlinearity (HSN). By varying a parameter, the integer order of the system displays transcritical bifurcation and new complex shapes of bistable double-scroll chaotic attractors and four-scroll chaotic attractors. The coexistence among four-scroll chaotic attractors, a pair of double-scroll chaotic attractors and a pair of point attractors is also reported for specific parameter values. Numerical results indicate that commensurate and incommensurate fractional orders of the systems display bistable double-scroll chaotic attractors, four-scroll chaotic attractors and coexisting attractors between a pair of double-scroll chaotic attractors and a pair of point attractors. Moreover, the physical existence of chaotic attractors and coexisting attractors found in the integer-order and commensurate fractional-order chaotic systems with HSN is verified using PSIM software. Numerical simulations and PSIM results have a good qualitative agreement. The results obtained in this work have not been reported previously in three-dimensional autonomous system with HSN and thus represent an enriching contribution to the understanding of the dynamics of this class of systems. Finally, combination synchronisation of such three-coupled identical commensurate fractional-order chaotic systems is analysed using the active backstepping method.

Stochastic Euler–Bernoulli beam driven by additive white noise: Global random attractors and global dynamics

This paper concerns with the long time dynamical behavior of a stochastic Euler–Bernoulli beam driven by additive white noise. By verifying the existence of absorbing set and obtaining the stabilization estimation of the dynamical system induced by the beam, the existence of global random attractors that attracts all bounded sets in phase space is proved. Furthermore, the finite Hausdorff dimension for the global random attractors is attained. In light of the relationship between global random attractor and random invariant probability measure, the global dynamics of the beam are analyzed according to numerical simulation on global random basic attractors and global random point attractors.

Degenerating the butterfly attractor in a plasma perturbation model using nonlinear controllers

In this work, the dynamical behaviors of a low-dimensional model, which governs the interplay between a driver associated with pressure gradient and relaxation of instability due to magnetic field perturbations, are investigated. Besides that, two nonlinear controllers are constructed precisely to shift the equilibria of the plasma model apart from each other. Simulation results show that shifting the equilibria can change the spacing of chaotic attractors, and subsequently break the butterfly wings into one or two symmetric pair of coexisting chaotic attractors. Furthermore, stretching the equilibria of the system apart enough from each other gives rise to degenerate the butterfly wings into several periodic orbits. In addition, with appropriate initial conditions, the complex multistability behaviors including the coexistence of butterfly chaotic attractor with two point attractors, the coexistence of transient transition chaos with completely quasi-periodic behavior, and the coexistence of symmetric Hopf bifurcations are also observed.

Higgs inflation in the Palatini formulation with kinetic terms for the metric

We consider scalar field inflation in the Palatini formulation of general relativity. The covariant derivative of the metric is then non-zero. From the effective theory point of view it should couple to other fields. We write down the most general couplings between it and a scalar field that are quadratic in derivatives. We consider both the case when the torsion is determined by the field equations and the case when it is assumed to be zero a priori. We find the metric derivative terms can significantly modify inflationary predictions. We specialise to Higgs inflation and terms of only up to dimension 4. Transforming to the Einstein frame, we show that by tuning the coefficients of the new terms, we can generate various effective inflationary potentials, including quadratic, hilltop-type, $\alpha$-attractor and inflection point. Some of these can give inflation in agreement with observations, including with a large tensor-to-scalar ratio, even if the non-minimal coupling is zero.

Existence of multiple equilibrium points in global attractor for damped wave equation

This paper is a continuation of Meng and Zhong in (Discrete Contin. Dyn. Syst., Ser. B 19:217–230, 2014). We go on studying the property of the global attractor for some damped wave equation with critical exponent. The difference between this paper and Meng and Zhong in (Discrete Contin. Dyn. Syst., Ser. B 19:217–230, 2014) is that the origin is not a local minimum point but rather a saddle point of the Lyapunov function F for the symmetric dynamical systems. Using the abstract result established in Zhang et al. in (Nonlinear Anal., Real World Appl. 36:44–55, 2017), we prove the existence of multiple equilibrium points in the global attractor for some wave equations under some suitable assumptions in the case that the origin is an unstable equilibrium point.

Role of Self-Loop in Cell-Cycle Network of Budding Yeast

Study of network dynamics is very active area in biological and social sciences. However, the relationship between the network structure and the attractors of the dynamics has not been fully understood yet. In this study, we numerically investigated the role of degenerate self-loops on the attractors and its basin size using the budding yeast cell-cycle network model. In the network, all self-loops negatively surpress the node (self-inhibition loops) and the attractors are only fixed points, i.e. point attractors. It is found that there is a simple division rule of the state space by removing the self-loops when the attractors consist only of point attractors. The point attractor with largest basin size is robust against the change of the self-inhibition loop. Furthermore, some limit cycles of period 2 appear as new attractor when a self-activation loop is added to the original network. It is also shown that even in that case, the point attractor with largest basin size is robust.

Self-excited and hidden attractors in a novel chaotic system with complicated multistability

In this paper, a new 3D chaotic system with trigonometric function term as a nonlinear controller is proposed. Depending on the nonlinear controller and the value of the parameters, the proposed system exhibits self-excited attractor with an unstable equilibrium, and hidden attractor with no equilibrium or a stable equilibrium. In addition, the unusual and striking dynamic behavior of the coexistence of self-excited chaotic with periodic attractors, two self-excited chaotic attractors with periodic attractor, three periodic attractors, hidden chaotic with point attractors, two hidden chaotic attractors, and four hidden chaotic attractors are explored by selecting appropriate initial values. Consequently, the proposed system has high sensitivity to its initial values and parameters, hence it can be applied in chaos-based cryptographic applications. Thus, the non-periodicity of coexisting attractors of the system is investigated through Lyapunov exponents and Sample Entropy. To demonstrate the performance of the system in real applications, we construct a pseudo-random number generator (PRNG) based on the hidden attractor case. The randomness test results show that the generated PRNG pass all the statistical tests.

Trivariate non‐Gaussian copulas to analyze the spatial behavior of rat sightings

The brown rat lives with man in a wide variety of environmental contexts and adversely affects public health by transmission of diseases, bites, and allergies. Understanding behavioral and spatial correlation aspects of pest species can contribute to their effective management and control. Rat sightings can be described by spatial coordinates in a particular region of interest defining a spatial point pattern. In this paper, we investigate the spatial structure of rat sightings in the Latina district of Madrid (Spain) and its relation to a number of distance‐based covariates that relate to the proliferation of rats. Given a number of locations, biologically considered as attractor points, the spatial dependence is modeled by distance‐based covariates and angular orientations through copula functions. We build a particular spatial trivariate distribution using univariate margins coming from the covariate information and provide predictive distributions for such distances and angular orientations.

Motion Switching With Sensory and Instruction Signals by Designing Dynamical Systems Using Deep Neural Network

To ensure that a robot is able to accomplish an extensive range of tasks, it is necessary to achieve a flexible combination of multiple behaviors. This is because the design of task motions suited to each situation would become increasingly difficult as the number of situations and the types of tasks performed by them increase. To handle the switching and combination of multiple behaviors, we propose a method to design dynamical systems based on point attractors that accept (i) “instruction signals” for instruction-driven switching. We incorporate the (ii) “instruction phase” to form a point attractor and divide the target task into multiple subtasks. By forming an instruction phase that consists of point attractors, the model embeds a subtask in the form of trajectory dynamics that can be manipulated using sensory and instruction signals. Our model comprises two deep neural networks: A convolutional autoencoder and a multiple time-scale recurrent neural network. In this study, we apply the proposed method to manipulate soft materials. To evaluate our model, we design a cloth-folding task that consists of four subtasks and three patterns of instruction signals, which indicate the direction of motion. The results depict that the robot can perform the required task by combining subtasks based on sensory and instruction signals. And, our model determined the relations among these signals using its internal dynamics.

Experimental design of a system to investigate chaotic dripping

We report here the construction of a system designed to investigate chaotic dripping behaviour. The benchtop system has been designed for the purpose of measuring chaotic dripping in a microgravity test facility, although measurements under normal gravity only are reported here. The results confirm the main predictions of a simple 1-D mass-spring-damper theoretical model of the system, including the formation of point attractors and associated limit cycles, although there are significant departures from this model within specific flow rate regions resulting in the formation of what are termed here ‘mid’ drops (drops of a smaller size than normal). It is hypothesized that the origin of these mid drops arises from the development of a ‘wetting’ mass, namely a mass of liquid that is evacuated from the delivery tube from the previous drop excision.

Dynamic analysis, circuit realization, control design and image encryption application of an extended Lü system with coexisting attractors

This paper introduces an extended Lü system with coexisting attractors. The number and stability of equilibria are determined. The coexisting attractors of the system are displayed by the bifurcation diagrams, Lyapunov exponent spectrum, phase portraits. It is shown that the system has a pair of strange attractors, a pair of limit cycles, a pair of point attractors for different initial conditions. The circuit implementation of the chaotic attractor and coexisting attractors of the system are presented. The control problem of the system is studied as well. A controller is designed to stabilize the system to the origin and realize the switching between two chaotic attractors based on the passive control method. Moreover, a chaotic image encryption algorithm is proposed according to the system. The performance of the algorithm is numerically analyzed.