Different Basins Of Attraction Research Articles

Inducing Niching Behavior in Differential Evolution Through Local Information Sharing

In practical situations, it is very often desirable to detect multiple optimally sustainable solutions of an optimization problem. The population-based evolutionary multimodal optimization algorithms can be very helpful in such cases. They detect and maintain multiple optimal solutions during the run by incorporating specialized niching operations to aid the parallel localized convergence of population members around different basins of attraction. This paper presents an improved information-sharing mechanism among the individuals of an evolutionary algorithm for inducing efficient niching behavior. The mechanism can be integrated with stochastic real-parameter optimizers relying on differential perturbation of the individuals (candidate solutions) based on the population distribution. Various real-coded genetic algorithms (GAs), particle swarm optimization (PSO), and differential evolution (DE) fit the example of such algorithms. The main problem arising from differential perturbation is the unequal attraction toward the different basins of attraction that is detrimental to the objective of parallel convergence to multiple basins of attraction. We present our study through DE algorithm owing to its highly random nature of mutation and show how population diversity is preserved by modifying the basic perturbation (mutation) scheme through the use of random individuals selected probabilistically. By integrating the proposed technique with DE framework, we present three improved versions of well-known DE-based niching methods. Through an extensive experimental analysis, a statistically significant improvement in the overall performance has been observed upon integrating of our technique with the DE-based niching methods.

Deflation Techniques for Finding Distinct Solutions of Nonlinear Partial Differential Equations

Nonlinear systems of partial differential equations (PDEs) may permit several distinct solutions. The typical current approach to finding distinct solutions is to start Newton's method with many different initial guesses, hoping to find starting points that lie in different basins of attraction. In this paper, we present an infinite-dimensional deflation algorithm for systematically modifying the residual of a nonlinear PDE problem to eliminate known solutions from consideration. This enables the Newton--Kantorovitch iteration to converge to several different solutions, even starting from the same initial guess. The deflated Jacobian is dense, but an efficient preconditioning strategy is devised, and the number of Krylov iterations is observed not to grow as solutions are deflated. The power of the approach is demonstrated on several problems from special functions, phase separation, differential geometry, and fluid mechanics that permit distinct solutions.

Constructing the Lyapunov Function by Quasi-interpolation

The Lyapunov function plays a vital role in dynamical systems. One method of constructing the Lyapunov function is the meshless collocation by radial basis functions. However, the meshless collocation method gives a non-local Lyapunov function (with unwanted nonnegative orbital derivative in some neighborhood of an equilibrium). To overcome the difficulty of the method, the paper proposes a scheme for constructing the Lyapunov function by multiquadric trigonometric B-spline quasi-interpolation. The scheme is efficient, simple and easy to compute. More importantly, it provides two additional shape parameters for the constructed Lyapunov function. This implies that, by adjusting these two shape parameters suitably, one can get different shapes of the Lyapunov function and thus obtain different basins of attraction for a dynamical system. Moreover, by taking the union of these different basins, one can even obtain a larger basin of attraction that provides better approximation to the exact one.

Gaussian Classifier-Based Evolutionary Strategy for Multimodal Optimization

This paper presents a Gaussian classifier-based evolutionary strategy (GCES) to solve multimodal optimization problems. An evolutionary technique for them must answer two crucial questions to guarantee its success: how to distinguish among the different basins of attraction and how to safeguard the already discovered good-quality solutions including both global and local optima. In GCES, multimodal optimization problems are regarded as classification ones, and Gaussian mixture models are employed to save the locations and basins of already and presently identified local or global optima. A sequential estimation technique for the covariance of a Gaussian model is introduced into GCES. To best adjust the global step size, a strategy named top-ranked sample selection is introduced, and a classification method instead of a common but problematic radius-triggered manner is proposed. Experiments are performed on a series of benchmark test functions to compare GCES with the state-of-the-art multimodal optimization approaches. The results show that GCES is not only simple to program and understand, but also provides better and consistent performance.

Numerical analysis of parallel replica dynamics

Parallel replica dynamics is a method for accelerating the computation of processes charac- terized by a sequence of infrequent events. In this work, the processes are governed by the overdamped Langevin equation. Such processes spend much of their time about the minima of the underlying poten- tial, occasionally transitioning into different basins of attraction. The essential idea of parallel replica dynamics is that the exit distribution from a given well for a single process can be approximated by the distribution of the first exit of N independent identical processes, each run for only 1/N -th the amount of time. While promising, this leads to a series of numerical analysis questions about the accuracy of the exit distributions. Building upon the recent work in (C. Le Bris, T. Lelievre, M. Luskin and D. Perez, Monte Carlo Methods Appl. 18 (2012) 119-146), we prove a unified error estimate on the exit distributions of the algorithm against an unaccelerated process. Furthermore, we study a dephasing mechanism, and prove that it will successfully complete.

One‐Dimensional Birth‐Death Process and Delbrück‐Gillespie Theory of Mesoscopic Nonlinear Chemical Reactions

As a mathematical theory for the stochastic, nonlinear dynamics of individuals within a population, Delbrück‐Gillespie process (DGP) is a birth–death system with state‐dependent rates which contain the system size V as a natural parameter. For large V, it is intimately related to an autonomous, nonlinear ODE as well as a diffusion process. For nonlinear dynamical systems with multiple attractors, the quasi‐stationary and stationary behavior of such a birth–death process can be understood in terms of a separation of time scales by a T*∼eαV (α > 0): a relatively fast, intra‐basin diffusion for t≪T* and a much slower inter‐basin Markov jump process for t≫T*. In this paper for one‐dimensional systems, we study both stationary behavior (t=∞) in terms of invariant distribution , and finite time dynamics in terms of the mean first passsage time (MFPT) . We obtain an asymptotic expression of MFPT in terms of the “stochastic potential”. We show in general no continuous diffusion process can provide asymptotically accurate representations for both the MFPT and the for a DGP. When n1 and n2 belong to two different basins of attraction, the MFPT yields the T*(V) in terms of Φ (x, V) ≈φ0(x) + (1/V)φ1(x). For systems with saddle‐node bifurcations and catastrophe, discontinuous “phase transition” emerges, which can be characterized by Φ (x, V) in the limit of . In terms of timescale separation, the relation between deterministic local nonlinear bifurcations, and stochastic global phase transition is discussed. The one‐dimensional theory is a pedagogic first step toward a general theory of DGP.

Characterization of the Ligand Receptor Encounter Complex and Its Potential for in Silico Kinetics-Based Drug Development.

The study of drug-receptor interactions has largely been framed in terms of the equilibrium thermodynamic binding affinity, an in vitro measure of the stability of the drug-receptor complex that is commonly used as a proxy measure of in vivo biological activity. In response to the growing realization of the importance of binding kinetics to in vivo drug activity we present a computational methodology for the kinetic characterization of drug-receptor interactions in terms of the encounter complex. Using trajectory data from multiple Brownian dynamics simulations of ligand diffusion, we derive the spatial density of the ligand around the receptor and show how it can be quantitatively partitioned into different basins of attraction. Numerical integration of the ligand densities within the basins can be used to estimate the residence time of the ligand within these diffusive binding sites. Simulations of two structurally similar inhibitors of Hsp90 exhibit diffusive binding sites with similar spatial structure but with different ligand residence times. In contrast, a pair of structurally dissimilar inhibitors of MDM2, a peptide and a small molecule, exhibit spatially distinct basins of attraction around the receptor, which in turn reveal differences in ligand orientational order. Thus, our kinetic approach provides microscopic details of drug-receptor dynamics that provide novel insight into the observed differences in the thermodynamic binding affinities for the two inhibitors, such as the differences in the entropic contributions to binding. The characterization of the encounter complex, in terms of the structure, topology, and dynamics of diffusive binding sites, offers a new perspective on ligand-receptor interactions and the potential for greater insight into drug action. The method, which requires no prior knowledge of the bound state, is a first step toward the incorporation of ligand kinetics into in silico drug development protocols.

On the calculation of absolute free energies from molecular-dynamics or Monte Carlo data

We propose a method for calculating absolute free energies from Monte Carlo or molecular-dynamics data. The method is based on the identity that expresses the partition function Q as a Boltzmann average: 1Q=w(p,x)exp[betaH(p,x)], where w(p,x) is an arbitrary weight function such that its integral over the phase space is equal to 1. In practice, to minimize statistical errors the weight function is chosen such that the regions of the phase space where sampling statistics are poor are excluded from the average. The "ideal" weight function would be the equilibrium phase-space density exp[-betaH(p,x)]Q itself. We consider two methods for constructing the weight function based on different estimates of the equilibrium phase-space density from simulation data. In the first method, it is chosen to be a Gaussian function, whose parameters are obtained from the covariance matrix of the atomic coordinates. In the second, a clustering algorithm is used to attempt partitioning the data into clusters corresponding to different basins of attraction visited by the system. The weight function is then constructed as a superposition of Gaussians calculated for each cluster separately. We show that these strategies can be used to improve upon previous methods of estimating absolute entropies from covariance matrices.

Soil nutrient cycles as a nonlinear dynamical system

Abstract. An analytical model for the soil carbon and nitrogen cycles is studied from the dynamical system point of view. Its main nonlinearities and feedbacks are analyzed by considering the steady state solution under deterministic hydro-climatic conditions. It is shown that, changing hydro-climatic conditions, the system undergoes dynamical bifurcations, shifting from a stable focus to a stable node and back to a stable focus when going from dry, to well-watered, and then to saturated conditions, respectively. An alternative degenerate solution is also found in cases when the system can not sustain decomposition under steady external conditions. Different basins of attraction for "normal" and "degenerate" solutions are investigated as a function of the system initial conditions. Although preliminary and limited to the specific form of the model, the present analysis points out the importance of nonlinear dynamics in the soil nutrient cycles and their possible complex response to hydro-climatic forcing.

Stability and interaction in flatline games

Starting from a given one-shot game played by a finite population of agents living in flatline, a circular or constrained grid structured by the classical definitions of neighborhood, we define transformation rules for cellular automata, which are determined by the best-reply behavior in standard two-person symmetric matrix games. A meaningful concept of solution for the underlying population games will necessarily include robustness against any possible unilateral deviation undertaken by a single player. By excluding the invisible hand of mutation we obtain a purely deterministic population model. The resulting process of cellular transformation is then analyzed for chicken and stag-hunt type cellular games and finally compared with the outcomes of more prominent evolutionary models. Special emphasis is given to an exhaustive combinatorial description of the different basins of attraction corresponding to stable stationary states.

Mechanistic view of the relaxation dynamics of a simple glass-former. A bridge between the topographic and the dynamic approaches

We provide a link between the two main approaches to the relaxation dynamics of glassy systems: The `real dynamics' scheme and the inherent dynamics or topographic formalism. The first approach is based on molecular dynamics (MD) simulations, whilst the second one reflects the underlying influence of the energy `landscape' (within a timescale separation and activated dynamics scenario) and constitutes a widespread picture within the realm of complex systems ranging from glasses to biopolymers. For a model glass-former (a binary Lennard-Jones system), MD studies which characterized in detail the movements of the different particles led to the discovery of dynamical heterogeneities. On the other hand, the topographic approach identified activated events on the potential energy surface of this system corresponding to transitions between different basins of attraction or inherent structures. In this work we demonstrate that at low temperature the relevant events identified by both methods conform to a basic mechanistic phenomenology with elementary steps involving ballistic string-like particle movements. We also show that as temperature increases and the timescales characterizing events of different range become comparable, these elemental steps loose their nature of rare activated events. Concurrently, the system looses diversity and complexity, signatures of glassy behavior. This fact enables us to furnish for the first time the microscopic structural and dynamical basis and conditions for the prevalence of the `landscape paradigm' for this class of systems.

Critical collapse of a cylindrically symmetric scalar field in four-dimensional Einstein’s theory of gravity

Four-dimensional cylindrically symmetric spacetimes with homothetic self-similarity are studied in the context of Einstein's Theory of Gravity, and a class of exact solutions to the Einstein-massless scalar field equations is found. Their local and global properties are investigated and found that they represent gravitational collapse of a massless scalar field. In some cases the collapse forms black holes with cylindrical symmetry, while in the other cases it does not. The linear perturbations of these solutions are also studied and given in closed form. From the spectra of the unstable eigen-modes, it is found that there exists one solution that has precisely one unstable mode, which may represent a critical solution, sitting on a boundary that separates two different basins of attraction in the phase space.

The Parameter Space Structure of the Kicked Logistic Map and Its Stability

In this paper, we analyzed the three-dimension parameter space of the Kicked Logistic Map (KLM), which is the Logistic Map perturbed by periodic kicks with constant amplitude. In this space parameter, diagrams are numerically determined, identifying the regions with finite attractors and their topology. For the identified periodic regions, isoperiodic diagrams are also computed. Examples of these diagrams are presented for fixed kick periods. Dynamical properties of the KLM are characterized by the forms observed in these diagrams. Furthermore, the considered map has different basins of attraction. Thus, for the kick period t = 2, an analytical analysis shows the coexistence of two basins of attraction. In addition, for this kick period, a stability diagram is presented for the period-two orbits, without iterating the KLM, reproducing the corresponding regions in the isoperiodic diagrams. Lastly, the coexistence of two basins, one for a periodic and another for a chaotic attractor, causes, in critical regions of the parameter space, the appearance of a type of crisis named transfer crisis.

A method for training recurrent neural networks for classification by building basins of attraction

The main contribution of this report is the suggestion, development, and trial of a new way of using a Hopfield network for classification. Rather than storing the members of the training sets (sets of exemplars) as stable points, a connection matrix is determined, using perceptron-type learning, such that the exemplars are put into basins of attraction. The elements from different training sets are placed in different basins of attraction, with usually more than one basin per training class. The attractors for the basins of attraction are determined at the same time that the exemplars are placed in these basins. An arbitrary element is then classified by the basin of attraction it falls into. The method is successfully applied to artificially generated classes and to the classification of cervical cells for cancer detection.

Basins of attraction for stationary points on a potential-energy surface

Methods to locate stationary points on model potential-energy surfaces, especially minima and transition states, are of great importance. Here we visualise the basins of attraction of such points, that is, the regions of the surface for which searches converge to a given structure. The results demonstrate the superiority of the eigenvector-following (EF) method over simple Newton–Raphson (NR) iteration. Furthermore, the boundaries between different basins of attraction may apparently be multifractal in character and highly complex. The results also provide some general rules for the efficient characterisation of potential-energy surfaces which are of practical value.

Small random perturbations of dynamical systems: Exponential loss of memory of the initial condition

We consider Markov processes arising from small random perturbations of non-chaotic dynamical systems. Under rather general conditions we prove that, with large probability, the distance between two arbitrary paths starting close to a same attractor of the unperturbed system decreases exponentially fast in time. The case of paths starting in different basins of attraction is also considered as well as some applications to the analysis of the invariant measure and to elliptic problems with small parameter in front to the second derivatives. The proof is based on a multiscale analysis of the typical trajectories of the Markov process; this analysis is done using techniques involved in the proof of Anderson localization for disordered quantum systems.

Chaos in quantum optics hyperchaos and long memory times

The onset of deterministic chaos in lasers is studied by referring to the invariant properties of low dimensional attractors, in order to isolate the characteristics of chaos from the random fluctuations due to the coupling with a thermal reservoir. For this purpose, attention is focused on single model homogeneous line lasers, whose dynamics is ruled by a low number of coupled variables. In the examined cases, experiments and theoretical model are in close agreement. In particular, when many attractors co-exist for the same parameter values (generalized multistability) the presence of random noise induces long lived transients with 1/f like low frequency spectra. These are due to jumps over different basins of attraction (so called "hyperchaos").

Observation of order and chaos in a nuclear spin–flip laser

Experimental observations and computer simulations of the nonlinear response are discussed for a single-mode, solid-state, nuclear spin–flip, ruby nuclear-magnetic-resonance (NMR) laser. A theoretical model is derived that is based on the classical Bloch equations and that demonstrates one-to-one correspondence to a homogeneously broadened, single-mode ring laser. Experimental evidence is presented for limit cycle behavior, sequences of sub-harmonic bifurcations, transitions to chaos, noisy bands, windows of regular behavior, intermittency, abrupt transitions between different basins of attraction, and hysteresis when a physical system parameter of the NMR laser is modulated at a low frequency. First experimental results are shown for a NMR laser with an external, detuned high-frequency signal below the injection-locking threshold. In this region, the output exhibits transitions from regular to chaotic oscillations and phase-locked spiking somewhat of the nature of what has been theoretically proposed for optical systems.