Chaos, Information Processing and Paradoxical Games

The following sections are included:

• Preface
• Contents

The scattering of ingoing particles by a target is classically given by Rutherford's law. The orbits of particles with small impact parameter are deflected to larger angles than the orbits with larger impact parameter. On the other hand in Bohmian quantum mechanics the opposite is true. Orbits with smaller impact parameter undergo smaller deflections. We study Bohmian orbits in two cases (1) the case of a plane ingoing wave with a gaussian distribution with dispersion D perpendicular to the initial direction of propagation z, and (2) the case of a wave packet that is also gaussian beyond the z-direction with dispersion l, and we consider two subcases l > D and l < D. The ingoing wave is deflected radially outwards from the target along a line called the “separator”. Close to the separator there are many “nodal points” and unstable “Xpoints”. The ingoing flow is initially close to the stable manifolds of the X-points and then it goes around the nodal points and escapes close to the unstable manifolds of the X-points…

In the framework of Nambu Mechanics, we have recently argued that Non-Hamiltonian Chaotic Flows in R³, are dissipation induced deformations of integrable volume preserving flows, specified by pairs of Intersecting Surfaces in R³. In the present work we focus our attention on the Lorenz system with a linear dissipative sector in its phase space dynamics. In this case the intersecting surfaces are quadratic. We parametrize its dissipation strength through a continuous control parameter ∈, acting homogeneously over the whole 3-dim. phase space. In the extended ∈-Lorenz system we find a scaling relation between the dissipation strength ∈ and Reynolds number parameter r, resulting from the scale covariance that we impose on the Lorenz equations under arbitrary rescalings of all its dynamical coordinates. Its integrable limit, (∈ = 0, fixed r), which is described in terms of intersecting Quadratic Nambu “Hamiltonians” Surfaces, gets mapped on the infinite value limit of the Reynolds number parameter (r → , ∈ = 1). In effect weak dissipation, through small ∈ values, generates and controls the well explored Route to Chaos in the large r-value regime. The non-dissipative ∈ = 0 integrable limit is therefore the gateway to Chaos for the Lorenz system.

The spatiotemporal complexity induced by perturbed initial excitations through the development of modulational instability in nonlinear lattices with or without disorder, may lead to the formation of very high amplitude, localized transient structures that can be named as extreme events. We analyze the statistics of the appearance of these collective events in two different universal lattice models; a one-dimensional nonlinear model that interpolates between the intergable Ablowitz-Ladik (AL) equation and the nonintegrable discrete nonlinear Schrödinger (DNLS) equation, and a two-dimensional disordered DNLS equation. In both cases, extreme events arise in the form of discrete rogue waves as a result of nonlinear interaction and rapid coalescence between mobile discrete breathers. In the former model, we find power-law dependence of the wave amplitude distribution and significant probability for the appearance of extreme events close to the integrable limit. In the latter model, more importantly, we find a transition in the return time probability of extreme events from exponential to power-law regime. Weak nonlinearity and moderate levels of disorder, corresponding to weak chaos regime, favor the appearance of extreme events in that case.

A systematic method of performing dynamical coarse graining for a variety of low dimensional chaotic systems is presented. The approach, also known as generalized Markov coarse graining, allows dynamical relaxation times to be extracted as spectra of the associated transition matrices. For many systems, including a number of conservative “microscopically reversible systems”, the probabilistic predictions as a function of time are statistically exact, as are the expectation values of large classes of observables

Using bi-parametric sweeping based on symbolic representation we reveal self-similar fractal structures induced by hetero- and homoclinic bifurcations of saddle singularities in the parameter space of two systems with deterministic chaos. We start with the system displaying a few homoclinic bifurcations of higher codimension: resonant saddle, orbitflip and inclination switch that all can give rise to the onset of the Lorenz-type attractor. It discloses a universal unfolding pattern in the case of systems of autonomous ordinary differential equations possessing two-fold symmetry or “ℤ₂-systems” with the characteristic separatrix butterfly. The second system is the classic Lorenz model of 1963, originated in fluid mechanics.

After a reminder of Jaynes' maximum entropy principle and of my quantum theoretical extension, I consider two coupled quantum systems A,B and formulate a quantum version of Bayes' theorem. The application of Feynman's disentangling theorem allows me to calculate the conditional density matrix , if system A is an oscillator (or a set of them), linearly coupled to an arbitrary quantum system B. Expectation values can simply be calculated by means of the normalization factor of that is derived.

In order to perceive that a physical system evolves in time, two requirements must be met: (a) it must be possible to define a “clock” and (b) it must be possible to make a copy of the state of the system, that can be reliably retrieved to make a comparison.We investigate what constraints quantum mechanics poses on these issues, in light of recent experiments with entangled photons.

A dynamical system giving rise to multiple steady states and subjected to noise and a periodic forcing is analyzed from the standpoint of information theory. It is shown that stochastic resonance has a clearcut signature on information entropy, information transfer and other related quantities characterizing information transduction within the system.

Following the spirit of the late John Nicolis, the purpose of this paper is to develop an evolutionary approach for the basic problem of generation, storage and dissipation of information in physical and biological systems via dynamical processes. After analysing the relation of entropy and information we develop our view that information is in general a nonphysical, emergent quantity, in spite of the fact that information transfer is always connected with flows of physical energy and entropy. We argue that information can have two basic forms: free information (like that of disks, tapes, books), that is what is transferred between sender and receiver, and bound information, that is a physical non-equilibrium structure which retains potential information reflecting the history of its formation (like fossils, geological strata or galaxies). As a basic concept we consider a kinetic phase transition of the second kind, termed the ritualization transition, which leads to the self-organized emergence of symbols, the key elements of free information. Ritualization occurs only in the context of life. Hence, the simplest physical example for a ritualization process is a system that starts as a physical and ends as a biological one, in other words, the origin of life. Our interest in this transition is focussed on the self-organization of information, on the way how a physical system can be enabled to create symbols and the related symbol-processing machinery out of ordinary pre-biological roots.

A basic question in evolution is dealing with the nature of an evolutionary memory. At thermodynamic equilibrium, at stable stationary states or other stable attractors the memory on the path leading to the long-time solution is erased, at least in part. Similar arguments hold for unique optima. Optimality in biology is discussed on the basis of microbial metabolism. Biology, on the other hand, is characterized by historical contingency, which has recently become accessible to experimental test in bacterial populations evolving under controlled conditions. Computer simulations give additional insight into the nature of the evolutionary memory, which is ultimately caused by the enormous space of possibilities that is so large that it escapes all attempts of visualization. In essence, this contribution is dealing with two questions of current evolutionary theory: (i) Are organisms operating at optimal performance? and (ii) How is the evolutionary memory built up in populations?

The late Professor J.S. Nicolis always emphasized, both in his writings and in presentations and discussions with students and friends, the relevance of a dynamical systems approach to biology. In particular, viewing the genome as a “biological text” captures the dynamical character of both the evolution and function of the organisms in the form of correlations indicating the presence of a long-range order. This genomic structure can be expressed in forms reminiscent of natural languages and several temporal and spatial traces left by the functioning of dynamical systems: Zipf laws, self-similarity and fractality. Here we review several works of our group and recent unpublished results, focusing on the chromosomal distribution of biologically active genomic components: Genes and protein-coding segments, CpG islands, transposable elements belonging to all major classes and several types of conserved non-coding genomic elements. We report the systematic appearance of power-laws in the size distribution of the distances between elements belonging to each of these types of functional genomic elements. Moreover, fractality is also found in several cases, using box-counting and entropic scaling.We present here, for the first time in a unified way, an aggregative model of the genomic dynamics which can explain the observed patterns on the grounds of known phenomena accompanying genome evolution. Our results comply with recent findings about a “fractal globule” geometry of chromatin in the eukaryotic nucleus.

The development of normal patterns along the primary and secondary vertebrate axes depends on the regularity of the early HOX gene expressions. During the initial developmental stages these expressions form a sequential pattern of partially overlapping domains along the anterior-posterior axis of the embryo in coincidence with the 3′ to 5′ order of the genes in the chromosome (spatial collinearity). In addition, the HOX genes are activated one after the other in the same 3′ to 5′ order (temporal collinearity). Genetic engineering experiments were performed in order explore the mechanism responsible for these remarkable collinearity phenomena. Several biomolecular models were proposed explaining some of the experimental findings. A biophysical model has been also proposed which is based on the hypothesis that physical forces are created which act on the Hox cluster. This cluster is initially inactive, located inside the chromosome territory. The physical forces translocate sequentially the Hox genes one after the other from inside the chromosome territory towards the interchromosome domain where they are activated in the area of the transcription factories. The above biophysical model mechanism has been strongly supported by recent experimental evidence and some evolutionary considerations. In this model realization, pulling forces are created between the ‘negatively’ charged Hox cluster and its ‘positively’ charged chromatin environment.

The term criticality has multiple meanings in different contexts. For interpretation of recordings of electroencephalographic (EEG) and electrocorticographic (ECoG) potentials from sensory cortices of humans and animals we adopt the convention established in thermodynamics for the critical point that terminates the boundary between gaseous and liquid states, and the region beyond the point. Our recordings reveal intermittent bursts of beta and gamma oscillations. In doing so we define state variables to replace the conventional variables of pressure, volume and temperature with measures of ECoG power, entropy, and feedback gain. Each burst of oscillation has a narrow distribution of frequencies that carries a spatial pattern of amplitude modulation (AM), but only when the subjects hold themselves in a stance of expectancy, waiting for one of multiple conditioned stimuli (CS) that will direct them into one of several actions (CR), and then only when they receive an expected CS. Local 1/f fluctuations have the form of phase modulation (PM) patterns that resemble fog in vapor. Large-scale, spatially coherent AM patterns emerge from and dissolve into this random background activity but only on receiving a CS. They do so by spontaneous symmetry breaking in a phase transition that resembles the condensation of a raindrop, in that it requires a large distribution of components, a source of transition energy, a singularity in the dynamics, and a connectivity that can sustain interaction over relatively immense correlation distances with respect to particle size. We conclude that the background activity at the pseudoequilibrium state conforms to fractal distributions of phase patterns corresponding to a phase transition from a gas-like, disorganized, lowdensity phase to a liquid-like high-density, more organized phase, that the activation of a Hebbian assembly is required for the phase transition, that a singularity is required to initiate and terminate a phase transition, and that the high-density phase can help to explain the richness of perceptual experience on recall and recognition of a stimulus.

The multifractal spectrum D_q (Rényi dimensions) is used for the analysis and comparison between the Neuron Axons Network (NAN) of healthy and pathological human brains because it conveys information about the statistics in many scales, from the very rare to the most frequent network configurations. Comparison of the Fractional Anisotropy Magnetic Resonance Images between healthy and pathological brains is performed with and without noise reduction. Modelling the complex structure of the NAN in the human brain is undertaken using the dynamics of the Lorenz model in the chaotic regime. The Lorenz multifractal spectra capture well the human brain characteristics in the large negative q's which represent the rare network configurations. In order to achieve a closer approximation in the positive part of the spectrum (q > 0) two independent modifications are considered: a) redistribution of the dense parts of the Lorenz model's phase space into their neighbouring areas and b) inclusion of additive uniform noise in the Lorenz model. Both modifications, independently, drive the Lorenz spectrum closer to the human NAN one in the positive q region without destroying the already good correspondence of the negative spectra. The modelling process shows that the unmodified Lorenz model in its full chaotic regime has a phase space distribution with high fluctuations in its dense parts, while the fluctuations in the human brain NAN are smoother. The induced modifications (phase space redistribution or additive noise) moderate the fluctuations only in the positive part of the Lorenz spectrum leading to a faithful representation of the human brain axons network in all scales.

We critically discuss the two moments of human cognition, namely, apprehension (A), whereby a coherent perception emerges from the recruitment of neuronal groups, and judgment (B), that entails the comparison of two apprehensions acquired at different times, coded in a suitable language and recalled by memory. (B) requires selfconsciousness, in so far as the agent who expresses the judgment must be aware that the two apprehensions are submitted to his/her own scrutiny and that it is his/her duty to extract a mutual relation. Since (B) lasts around 3 seconds, the semantic value of the pieces under comparison must be decided within this time. This implies a fast search of the memory contents. As a fact, exploring human subjects with sequences of simple words, we find evidence of a limited time window, corresponding to the memory retrieval of a linguistic item in order to match it with the next one in a text flow (be it literary, or musical,or figurative). Classifying the information content of spike trains, an uncertainty relation emerges between the bit size of a word and its duration. This uncertainty is ruled by a constant that can be given a numerical value and that has nothing to do with Planck's constant. A “quantum conjecture” in the above sense might explain the onset and decay of the memory window connecting successive pieces of a linguistic text. The conjecture here formulated is applicable to other reported evidences of quantum effects in human cognitive processes, so far lacking a plausible framework since no efforts to assign a quantum constant have been associated.

Biological dynamical systems can autonomously change their rule governing the dynamics. To deal with the change in their rule, possible approaches to extend dynamical-systems theory are discussed: They include chaotic itinerancy in high-dimensional dynamical systems, discreteness-induced switches of states, and interference between slow and fast modes. Applications of these concepts to cell differentiation, adaptation, and memory are briefly reviewed, while biological evolution is discussed as selection of dynamical systems by dynamical systems. Finally, necessity of mathematical framework to deal with self-referential dynamics for the rule formation is stressed.

We propose a dynamical model that represents a process of deductive inference. We discuss the stability of logic dynamics and a neural basis for the dynamics. We propose a new concept of descriptive stability, thereby enabling a structure of stable descriptions of mathematical models concerning dynamic phenomena to be clarified. The present theory is based on the wider and deeper thoughts of John S. Nicolis. In particular, it is based on our joint paper on the chaos theory of human short-term memories with a magic number of seven plus or minus two.

An outstanding problem in complex systems and mathematical biology is to explore and understand the fundamental mechanisms of species coexistence. Existing approaches are based on niche partitioning, dispersal, chaotic evolutionary dynamics, and more recently, evolutionary games. Here we briefly review a number of fundamental issues associated with the coexistence of mobile species under cyclic competitions in the framework of evolutionary games.

The aim of the present paper is to elucidate the transition from collective to random behavior exhibited by various mathematical models of bird flocking. In particular, we compare Vicsek's model [Vicsek et al., Phys. Rev. Lett. 75, 1226–1229 (1995)] with one based on topological considerations. The latter model is found to exhibit a first order phase transition from flocking to decoherence, as the “noise parameter” of the problem is increased, whereas Vicsek's model gives a second order transition. Refining the topological model in such a way that birds are influenced mostly by the birds in front of them, less by the ones at their sides and not at all by those behind them (because they do not see them), we find a behavior that lies in between the two models. Finally, we propose a novel mechanism for preserving the flock's cohesion, without imposing artificial boundary conditions or attractive forces.

We suggest that the main features of animal construction can be understood as the sum of locally independent actions of non-interacting individuals subjected to the global constraints imposed by the nascent structure. We first formulate an analytically tractable macroscopic description of construction which predicts a 1/3 power law for how the length of the structure grows with time. We further show how the power law is modified when biases in random walk performed by the constructors as well as halting times between consecutive construction steps are included.

Systems at a critical point in phase transitions can be regarded as being relevant to biological complex behaviour. Such a perspective can only result, in a mathematical consistent manner, from a recursive structure. We implement a recursive structure based on updating by asynchronously tuned elementary cellular automata (AT ECA), and show that a large class of elementary cellular automata (ECA) can reveal critical behavior due to the asynchronous updating and tuning.We show that the obtained criticality coincides with the criticality in phase transitions of asynchronous ECA with respect to density decay, and that multiple distributed ECAs, synchronously updated, can emulate critical behavior in AT ECA. Our approach draws on concepts and tools from category and set theory, in particular on “adjunction dualities” of pairs of adjoint functors.

The reader is taken into the heart of a fictitious dialogue between two friends who never talked long enough with each other during the lifetime of both. It is the fearlessness of the mind of John that prompted the hopefully not too erratic thoughts that are going to be offered. The central figure is Heraclitus, the Great.

The following sections are included:

• Selected References from John Nicolis' Bibliography
• Index