Narrow Results

We study the asymptotic relaxation of Kuramoto oscillators in an analytic potential field. The Kuramoto model describes the synchronization of weakly coupled limit-cycle oscillators and it casts as a gradient flow on the Euclidean space with an analytic potential. The Kuramoto model corresponds to our proposed model with a constant force field. As a nontrivial local potential field, we consider two types of local potential fields corresponding to quadratic and linear-plus-oscillatory potentials. For the asymptotic relaxation, we provide sufficient frameworks leading to the uniform boundedness of state for each local potential field. The proposed framework is given in terms of system parameters and initial data. Then, we use the gradient flow formulation and uniform boundedness of state to find the existence of equilibrium and convergence toward them, which implies the existence of asymptotic relaxation.

We extend the Microscopic Representation approach to the quantitative study of religious and folk stories: A story encrypting symbolically the creation is deconstructed into its simplest conceptual elements and their relationships. We single out a particular kind of relationship which we call "diagonal (or transitive) link": given two relations between the couples of elements AB and respectively BC, the "diagonal link" is the (composite) relation AC. We find that the diagonal links are strongly and systematically correlated with the events in the story that are considered crucial by the experts. We further compare the number of diagonal links in the symbolic creation story with a folk tale, which ostensibly narrates the same overt succession of events (but without pretensions of encrypting additional meanings). We find that the density of diagonal links per word in the folk story is lower by a factor of two. We speculate that, as in other fields, the simple transitive operations acting on elementary objects are at the core of the emergence and recognition of macroscopic meaning and novelty in complex systems.

We present a cellular-automaton model of a reaction-diffusion excitable system with concentration dependent inhibition of the activator, and study the dynamics of mobile localizations (gliders) and their generators. We analyze a three-state totalistic cellular automaton on a two-dimensional lattice with hexagonal tiling, where each cell connects with 6 others. We show that a set of specific rules support spiral glider-guns (rotating activator-inhibitor spirals emitting mobile localizations) and stationary localizations which destroy or modify gliders, along with a rich diversity of emergent structures with computational properties. We describe how structures are created and annihilated by glider collisions, and begin to explore the necessary processes that generate this kind of complex dynamics.

Conventional lattice-gas automata consist of particles moving discretely on a fixed lattice. While such models have been quite successful for a variety of fluid flow problems, there are other systems, e.g., flow in a flexible membrane or chemical self-assembly, in which the geometry is dynamical and coupled to the particle flow. Systems of this type seem to call for lattice gas models with dynamical geometry. We construct such a model on one-dimensional (periodic) lattices and describe some simulations illustrating its nonequilibrium dynamics.

The accelerating expansion of the Universe points to a small positive value for the cosmological constant or vacuum energy density. We discuss recent ideas that the cosmological constant plus Large Hadron Collider (LHC) results might hint at critical phenomena near the Planck scale.

A possible resolution of the incompatibility of quantum mechanics and general relativity is that the relativity principle is emergent. I show that the central paradox of black holes also occurs at a liquid-vapor critical surface of a bose condensate but is resolved there by the phenomenon of quantum criticality. I propose that real black holes are actually phase boundaries of the vacuum analogous to this, and that the Einstein field equations simply fail at the event horizon the way quantum hydrodynamics fails at a critical surface. This can occur without violating classical general relativity anywhere experimentally accessible to external observers. Since the low-energy effects that occur at critical points are universal, it is possible to make concrete experimental predictions about such surfaces without knowing much, if anything about the true underlying equations. Many of these predictions are different from accepted views about black holes — in particular the absence of Hawking radiation and the possible transparency of cosmological black hole surfaces.

In the present interdisciplinary review, we focus on the applications of the symmetry principles to quantum and statistical physics in connection with some other branches of science. The profound and innovative idea of quasiaverages formulated by N. N. Bogoliubov, gives the so-called macro-objectivation of the degeneracy in the domain of quantum statistical mechanics, quantum field theory and quantum physics in general. We discuss the complementary unifying ideas of modern physics, namely: spontaneous symmetry breaking, quantum protectorate and emergence. The interrelation of the concepts of symmetry breaking, quasiaverages and quantum protectorate was analyzed in the context of quantum theory and statistical physics. The chief purposes of this paper were to demonstrate the connection and interrelation of these conceptual advances of the many-body physics and to try to show explicitly that those concepts, though different in details, have certain common features. Several problems in the field of statistical physics of complex materials and systems (e.g., the chirality of molecules) and the foundations of the microscopic theory of magnetism and superconductivity were discussed in relation to these ideas.

Many scientists have struggled to uncover the elusive origin of "complexity", and its many equivalent jargons, such as emergence, self-organization, synergetics, collective behaviors, nonequilibrium phenomena, etc. They have provided some qualitative, but not quantitative, characterizations of numerous fascinating examples from many disciplines. For example, Schrödinger had identified "the exchange of energy" from open systems as a necessary condition for complexity. Prigogine has argued for the need to introduce a new principle of nature which he dubbed "the instability of the homogeneous". Turing had proposed "symmetry breaking" as an origin of morphogenesis. Smale had asked what "axiomatic" properties must a reaction–diffusion system possess to make the Turing interacting system oscillate.

The purpose of this paper is to show that all the jargons and issues cited above are mere manifestations of a new fundamental principle called local activity, which is mathematically precise and testable. The local activity theorem provides the quantitative characterization of Prigogine's "instability of the homogeneous" and Smale's quest for an axiomatic principle on Turing instability.

Among other things, a mathematical proof is given which shows none of the complexity-related jargons cited above is possible without local activity. Explicit mathematical criteria are given to identify a relatively small subset of the locally-active parameter region, called the edge of chaos, where most complex phenomena emerge.

The space-time phases of a complex dynamic system are the probability distributions for state as a function of space and time which arise by evolving initial probability distributions from the distant past. Toom proved important results about space-time phases for a class of majority voter probabilistic cellular automata (PCA). Here, variants of the majority voter PCA are presented which are proved to exhibit a variety of types of space-time phase. These examples are expected to serve as useful steps on the way to a general theory of space-time phases.

One of the main characteristics of complexity is the emergence of properties due to dynamical processes. This special issue has put together a unique collection of articles written by leading researchers and experts around the globe on recent advances in complex systems and applications, in various fields of science and engineering. It focuses not only on equation-based modeling of eco- or bio-systems analysis but also on the study of eco- or bio-complexity and global emergent properties and self-organization, resulting from various interactions.

In systems theory and science, emergence is the way complex systems and patterns arise out of a multiplicity of relatively simple interactions. Emergence is central to the theories of integrative levels and of complex systems [Aziz-Alaoui & Bertelle, 2009]. In this paper, we use the emergent property of the ultra weak multidimensional coupling of p 1-dimensional dynamical chaotic systems which leads from chaos to randomness.

Generation of random or pseudorandom numbers, nowadays, is a key feature of industrial mathematics. Pseudorandom or chaotic numbers are used in many areas of contemporary technology such as modern communication systems and engineering applications. More and more European or US patents using discrete mappings for this purpose are obtained by researchers of discrete dynamical systems [Petersen & Sorensen, 2007; Ruggiero et al., 2006]. Efficient Chaotic Pseudo Random Number Generators (CPRNG) have been recently introduced. They use the ultra weak multidimensional coupling of p 1-dimensional dynamical systems which preserve the chaotic properties of the continuous models in numerical experiments. Together with chaotic sampling and mixing processes, ultra weak coupling leads to families of (CPRNG) which are noteworthy [Hénaff et al., 2009a, 2009b, 2009c, 2010].

In this paper we improve again these families using a double threshold chaotic sampling instead of a single one.

We analyze numerically the properties of these new families and underline their very high qualities and usefulness as CPRNG when very long series are computed. Moreover, a determining property of such improved CPRNG is the high number of parameters used and the high sensitivity to the parameters value which allows choosing it as cipher-keys. It is why we call these families multiparameter chaotic pseudo-random number generators (M-p CPRNG).

This research draws on theories of emergence to inform the creation of an artistic and direct visualization. This is an interactive artwork and drawing tool for creative participant experiences. Emergence is characteristically creative and many different models of emergence exist. It is therefore possible to effect creativity through the application of emergence mechanisms from these different disciplines. A review of theories of emergence and examples of visualization in the arts, is provided. An art project led by the author is then discussed in this context. This project, Iterative Intersections, is a collaboration with community artists from Cerebral Palsy League. It has resulted in a number of creative outcomes including the interactive art application, Of me with me. Analytical discussion of this work shows how its construction draws on aspects of experience design, fractal and emergent theory to effect perceptual emergence and creative experience as well as to facilitate self-efficacy.

The Self-Organized Hydrodynamics model of collective behavior is studied on an annular domain. A modal analysis of the linearized model around a perfectly polarized steady-state is conducted. It shows that the model has only pure imaginary modes in countable number and is hence stable. Numerical computations of the low-order modes are provided. The fully nonlinear model is numerically solved and nonlinear mode-coupling is then analyzed. Finally, the efficiency of the modal decomposition to analyze the complex features of the nonlinear model is demonstrated.

We present emergent dynamics of continuous and discrete thermomechanical Cucker–Smale (TCS) models equipped with temperature as an extra observable on general digraph. In previous literature, the emergent behaviors of the TCS models were mainly studied on a complete graph, or symmetric connected graphs. Under this symmetric setting, the total momentum is a conserved quantity. This determines the asymptotic velocity and temperature a priori using the initial data only. Moreover, this conservation law plays a crucial role in the flocking analysis based on the elementary ${\ell }_2$ energy estimates. In this paper, we consider a more general connection topology which is registered by a general digraph, and the weights between particles are given to be inversely proportional to the metric distance between them. Due to this possible symmetry breaking in communication, the total momentum is not a conserved quantity, and this lack of conservation law makes the asymptotic velocity and temperature depend on the whole history of solutions. To circumvent this lack of conservation laws, we instead employ some tools from matrix theory on the scrambling matrices and some detailed analysis on the state-transition matrices. We present two sufficient frameworks for the emergence of mono-cluster flockings on a digraph for the continuous and discrete models. Our sufficient frameworks are given in terms of system parameters and initial data.

We present a stochastic Justh–Krishnaprasad flocking model describing interactions among individuals in a planar domain with their positions and heading angles. The deterministic counterpart of the proposed model describes the formation of nematic alignment in an ensemble of planar particles moving with a unit speed. When the noise is turned off, we show that the nematic alignment state, in which all heading angles are either same or the opposite, is nonlinearly stable using a Lyapunov functional approach. We employed a diameter-like functional via the rearrangement of heading angles in the $2\pi$-interval. In contrast, under the additive noise, a continuous angle configuration will be deviated asymptotically from the nematic state. Nevertheless, in any finite-time interval, we will see that some part of angle configuration will stay close to the nematic state with a positive probability, where we call this phenomenon as stochastic persistency. We provide a quantitative estimate on the probability for stochastic persistency and compare several numerical examples with analytical results.

The Lorentzian metric structure allows one to implement the relativistic notion of causality in any field theory and to define a notion of time dimension. We propose that at the microscopic level the metric is Riemannian and that the Lorentzian structure, usually thought as fundamental, is in fact an effective property, that emerges in some regions of a 4-dimensional space with a positive definite metric. We argue that a decent classical field theory for scalars, vectors and spinors in flat spacetime can be constructed, and that gravity can be included under the form of a covariant Galileon theory instead of general relativity.

Traditional derivations of general relativity (GR) from the graviton degrees of freedom assume spacetime Lorentz covariance as an axiom. In this paper, we survey recent evidence that GR is the unique spatially-covariant effective field theory of the transverse, traceless graviton degrees of freedom. The Lorentz covariance of GR, having not been assumed in our analysis, is thus plausibly interpreted as an accidental or emergent symmetry of the gravitational sector. From this point of view, Lorentz covariance is a necessary feature of low-energy graviton dynamics, not a property of spacetime. This result has revolutionary implications for fundamental physics.

The aim of this work is to show that at the population level, emerging properties may occur as a result of the coupling between the fast micro-dynamics and the slow macrodynamics. We studied a prey-predator system with different time scales in a heterogeneous environment. A fast time scale is associated to the migration process on spatial patches and a slow time scale is associated to the growth and the interactions between the species. Preys go on the spatial patches on which some resources are located and can be caught by the predators on them. The efficiency of the predators to catch preys is patch-dependent. Preys can be more easily caught on some spatial patches than others. Perturbation theory is used in order to aggregate the initial system of ordinary differential equations for the patch sub-populations into a macro-system of two differential equations governing the total populations. Firstly, we study the case of a linear process of migration for which the aggregated system is formally identical to the slow part of the full system. Then, we study an example of a nonlinear process of migration. We show that under these conditions emerging properties appear at the population level.

We consider a predator-prey model in a multi-patch environment. We assume the existence of two time scales: the migration process takes place on the behavioural level and is thus much faster than the population dynamics. Each population is subdivided into subpopulations which correspond to the spatial distribution. The model is thus a large system of ordinary differential equations. We assume that the migration rates are fastly oscillating: it is the case for some aquatic populations for example. Indeed, these populations undergo regular vertical movements in the water column every day. In order to study our model, we use a reduction method which allows us to simplify the initial model. It is then possible to bring to light that some properties emerge from the coupling between the fast migration process and the slow population dynamics. We give an explicit example of the emerging property.

Group behavior emergent from the systems composed of two types of agents are investigated. The agents are defined on a two-dimensional grid system and move under the influence of the attractive and/or repulsive interactions. Depending on the intensity and the sense of the interactions, a wide variety of spatiotemporal patterns emerge on the system. Those patterns are discussed in terms of the well-known phenomena in real systems such as the residential segregation in cities, cell sorting in multicellular system, self-running droplet, group behavior of a fish school under the attack of a predator and the fission in a cell division process.