Narrow Results

In this paper the existence of quantum wave-packets in neural structures, such as automata and associative memories, is studied. The synaptic weights are considered to be stochastic variables, with probability density functions given by the solution of Schrödinger's equation. It is shown that the weights' update can be performed with the use of unitary operators, as indicated by the postulates of quantum mechanics. Moreover, it is proved that the number of attractors of quantum associative memories increases exponentially with respect to conventional associative memories. Finally, simulation tests are used to demonstrate the improved pattern storage capabilities of quantum associative memories.

Random Boolean networks, originally introduced as simplified models for the genetic regulatory networks, are abstract models widely applied for the study of the dynamical behaviors of self-organizing complex systems. In these networks, connectivity and the bias of the Boolean functions are the most important factors that can determine the behavioral regime of the systems. On the other hand, it has been found that topology and some structural elements of the networks such as the reciprocity, self-loops and source nodes, can have relevant effects on the dynamical properties of critical Boolean networks. In this paper, we study the impact of source and sink nodes on the dynamics of homogeneous and heterogeneous Boolean networks. Our research shows that an increase of the source nodes causes an exponentially growing of the different behaviors that the system can exhibit regardless of the network topology, while the amount of order seems to undergo modifications depending on the topology of the system. Indeed, with the increase of the source nodes the orderliness of the heterogeneous networks also increases, whereas it diminishes in the homogeneous ones. On the other hand, although the sink nodes seem not to have effects on the dynamic of the homogeneous networks, for the heterogeneous ones we have found that an increase of the sinks gives rise to an increasing of the order, although the different potential behaviors of the system remains approximately the same.

We study the critical points of the black hole scalar potential V_BH in N = 2, d = 4 supergravity coupled to n_V vector multiplets, in an asymptotically flat extremal black hole background described by a 2(n_V+1)-dimensional dyonic charge vector and (complex) scalar fields which are coordinates of a special Kähler manifold.

For the case of homogeneous symmetric spaces, we find three general classes of regular attractor solutions with nonvanishing Bekenstein–Hawking entropy. They correspond to three (inequivalent) classes of orbits of the charge vector, which is in a 2(n_V+1)-dimensional representation R_V of the U-duality group. Such orbits are nondegenerate, namely they have nonvanishing quartic invariant (for rank-3 spaces). Other than the ½-BPS one, there are two other distinct non-BPS classes of charge orbits, one of which has vanishing central charge.

The three species of solutions to the N = 2 extremal black hole attractor equations give rise to different mass spectra of the scalar fluctuations, whose pattern can be inferred by using invariance properties of the critical points of V_BH and some group theoretical considerations on homogeneous symmetric special Kähler geometry.

We review recent developments in understanding quantum/string corrections to BPS black holes and strings in five-dimensional supergravity. These objects are solutions to the effective action obtained from M-theory compactified on a Calabi–Yau threefold, including the one-loop corrections determined by anomaly cancellation and supersymmetry. We introduce the off-shell formulation of this theory obtained through the conformal supergravity method and review the methods for investigating supersymmetric solutions. This leads to quantum/string corrected attractor geometries, as well as asymptotically flat black strings and spinning black holes. With these solutions in hand, we compare our results with analogous studies in four-dimensional string-corrected supergravity, emphasizing the distinctions between the four- and five-dimensional theories.

While stable polyrhythmic multifrequency $n:l$ dynamics has traditionally been an important element in music performance, recently, this type of dynamics has been discovered in the human brain in terms of elementary temporal neural activity patterns. In this context, the canonical-dissipative systems framework is a promising modeling approach due to its two key features to bridge the gap between classical mechanics and life sciences, on the one hand, and to provide analytical or semi-analytical solutions, on the other hand. Within this framework, a family of testbed models is constructed that exhibit $n:l$ multifrequency limit cycle attractors describing two components oscillating with frequencies at $n:l$ ratios and stable polyrhythmic phase relationships. The attractors are super-integrable due to the existence of third invariants of motion for all $n:l$ ratios. Strikingly, all $n:l$ attractors models satisfy the same generic bifurcation diagram. The study generalizes earlier work on super-integrable systems, on the one hand, and canonical-dissipative limit cycle oscillators, on the other hand. Explicit worked-out models for 1:4 and 2:3 frequency ratios are presented.

In this paper we survey some recent results on the behavior of solutions of parabolic equations subjected to nonlinear boundary conditions.

The results range from local existence and regularity of solutions, to global existence, dissipativeness and existence of attractors, and to blow-up in finite time.

Some applications are given to some singular perturbation problems and to pattern formation from boundary flux.

In practice it is very difficult to distinguish between stochastic and chaotic nonlinearity. It is also difficult to identify nonlinear structures in time series when the hidden dynamics are complex. In this paper, we provide evidence that in the presence of specific high-dimensional underlying structures, White [1989] and Theiler et al. [1992] tests fail to detect nonlinearity.

This paper proves, via an analytical approach, that 170 (out of 256) Boolean CA rules in a one-dimensional cellular automata (CA) are time-reversible in a generalized sense. The dynamics on each attractor of a time-reversible rule N is exactly mirrored, in both space and time, by its bilateral twin ruleN^†. In particular, all 69 period-1 rules, 17 (out of 25) period-2 rules, and 84 (out of 112) Bernoulli rules are time-reversible.

The remaining 86 CA rules are time-irreversible in the sense that N and N^† mirror their dynamics only in space, but not in time. In this case, each attractor of N defines a unique arrow of time.

A simple "time-reversal test" is given for testing whether an attractor of a CA rule is time-reversible or time-irreversible. For a time-reversible attractor of a CA rule N the past can be uniquely recovered from the future of N^†, and vice versa. This remarkable property provides 170 concrete examples of CA time machines where time travel can be routinely achieved by merely hopping from one attractor to its bilateral twin attractor, and vice versa. Moreover, the time-reversal property of some local rules can be programmed to mimic the matter–antimatter "annihilation" or "pair-production" phenomenon from high-energy physics, as well as to mimic the "contraction" or "expansion" scenarios associated with the Big Bang from cosmology.

Unlike the conventional laws of physics, which are based on a unique universe, most CA rules have multiple universes (i.e. attractors), each blessed with its own laws. Moreover, some CA rules are endowed with both time-reversible attractors and time-irreversible attractors.

Using an analytical approach, the time-τ return map of each Bernoulli στ-shift attractor of all 112 Bernoulli rules are shown to obey an ultra-compact formula in closed form, namely,.

or its inverse map.

These maps completely characterize the time-asymptotic (steady state) behavior of the nonlinear dynamics on the attractors. In-depth analysis of all but 18 global equivalence classes of CA rules have been derived, along with their basins of attraction, which characterize their transient regimes.

Above all, this paper provides a rigorous nonlinear dynamics foundation for a paradigm shift from an empirical-based approach à la Wolfram to an attractor-based analytical theory of cellular automata.

Normally, conservative systems do not have attractors. However, in a system with escapes, the infinity acts as an attractor. Furthermore, attractors may appear as singularities at a finite distance. We consider the basins of escape in a particular Hamiltonian system with escapes and the rates of escape for various values of the parameters. Then we consider the basins of attraction of a system of two fixed black holes, with particular emphasis on the asymptotic curves of its unstable periodic orbits.

The parameter dependence of the various attractive solutions of the three variable nonlinear Lorenz equations is studied as a function of r, the normalized Rayleigh number, and of σ, the Prandtl number. Previous work, either for fixed σ and all r or along σ ∝ r and , is extended to the entire (r, σ) parameter plane. An onion-like periodic pattern is found which is due to the alternating stability of symmetric and nonsymmetric periodic orbits. This periodic pattern is explained by considering non-trivial limits of large r and σ and thus interpolating between the above mentioned cases. The mathematical analysis uses Airy functions as introduced in previous work, but instead of concentrating on the Lorenz map we analyze the trajectories in full phase space. The periodicity of the Airy function allows to calculate analytically the periodic onion structure in the (r, σ)-plane. Previous observations about sequences of bifurcations are confirmed, and more details regarding their symmetry are reported.

This paper continues our quest to develop a rigorous analytical theory of 1-D cellular automata via a nonlinear dynamics perspective. The 18 yet uncharacterized local rules are henceforth partitioned into ten complex Bernoulliσ_τ-shift rules and eight hyper Bernoulliσ_τ-shift rules, the latter including such famous rules and . All exhibit a bizarre composite wave dynamics with arbitrarily large Bernoulli velocity σ and Bernoulli return time τ as the length L → ∞.

Basin tree diagrams of all ten complex Bernoulli σ_τ-shift rules are exhibited for lengths L = 3, 4, …, 8. Superficial as it may seem, these basin tree diagrams suggest general qualitative properties which have since been proved to be true in general. Two such properties form the main results of this paper; namely,

Explicit global state transition formulas are given for local rules , and . Such formulas led to the rigorous proof of several surprising periodicity constraints for rule , and to the discovery of a new global, quasi-equivalence class, defined via an alternating transformation. In particular, local rules and are globally quasi-equivalent where corresponding space-time patterns can be derived from each other by simply complementing every other row.

Another important result of this paper is the discovery of a scale-free phenomenon exhibited by the local rules , and . In particular, the period "T" of all attractors of rules , and , as well as of all isles of Eden of rules and , increases linearly with unit slope, in logarithmic scale, with the length L.

This paper presents the basin tree diagrams of all hyper Bernoulli σ_τ-shift rules for string lengths L = 3, 4, …, 8. These diagrams have revealed many global and time-asymptotic properties that we have subsequently proved to be true for all L < ∞. In particular, we have proved that local rule has no Isles of Eden for all L, and that local rules and are inhabited by a dense set (continuum) of Isles of Eden if, and only if, L is an odd integer. A novel and powerful graph-theoretic tool, called Isles-of-Eden digraph, has been developed and can be used to test the existence of dense Isles of Eden of any local rule which satisfies certain constraints, such as rules , , , , as well as all invariant local rules, such as rules , , and , subject to no constraints.

Our scientific odyssey through the theory of 1-D cellular automata is enriched by the definition of quasi-ergodicity, a new empirical property discovered by analyzing the time-1 return maps of local rules. Quasi-ergodicity plays a key role in the classification of rules into six groups: in fact, it is an exclusive characteristic of complex and hyper Bernoulli-shift rules. Besides introducing quasi-ergodicity, this paper answers several questions posed in the previous chapters of our quest. To start with, we offer a rigorous explanation of the fractal behavior of the time-1 characteristic functions, finding the equations that describe this phenomenon. Then, we propose a classification of rules according to the presence of Isles of Eden, and prove that only 28 local rules out of 256 do not have any of them; this result sheds light on the importance of Isles of Eden. A section of this paper is devoted to the characterization of Bernoulli basin-tree diagrams through modular arithmetic; the formulas obtained allow us to shorten drastically the number of cases to take into consideration during numerical simulations. Last but not least, we present some theorems about additive rules, including an analytical explanation of their scale-free property.

This stage of our journey through the universe of one-dimensional binary Cellular Automata is devoted to period-1 rules, constituting the first of the six groups in which we systematized the 88 globally-independent CA rules.

The first part of this article is mainly dedicated to reviewing the terminology and the empirical results found in the previous papers of our quest. We also introduce the concept of the ω-limit orbit with the purpose of linking our work to the classical theory of nonlinear dynamical systems. Moreover, we present the basin tree diagrams of all period-1 rules — except for rule , which is trivial — along with their Boolean cubes and time-1 characteristic functions.

In the second part, we prove a theorem demonstrating that all rules belonging to group 1 have robust period-1 rules for any finite, and infinite, bit-string length L. This is the first time we give analytical results on the behavior of CA local rules for large values of L and, consequently, for bi-infinite bit strings.

The theoretical treatment is complemented by two remarkable practical results: an explicit formula for generating isomorphic basin trees, and an algorithm for creating new periodic orbits by concatenation. We also provide several examples of both of them, showing how they help to avoid tedious simulations.

The 11th part of our tour through one-dimensional binary Cellular Automata concerns period-2 rules, which form the second group in our classification of the 88 globally-independent CA rules according to the properties of their periodic orbits. In this article, we display the basin tree diagrams of all period-2 rules along with their time-2 characteristic functions, and then we prove that all rules belonging to group 2 have robust period-2 ω-limit orbits for any finite, and infinite, bit string length. This rigorous result, which pairs with the one about period-1 rules given in the tenth installment of our chronicle, confirms what we stated about period-2 rules on the basis of empirical evidence. In the second part of this tutorial, we introduce the notion of quasi global-equivalence and prove that there are only 82 quasi globally-independent CA rules. For the first time, we show that the space-time patterns of globally-independent local rules can depend on each other, and we present an example of quasi-global transformation. We also define the super string 𝄞, and its unique decimal representation x_𝄞, dubbed the super decimal, which provides a completely transparent yet rigorous proof that rule is chaotic when L → ∞. Moreover, we present the basin tree generation formulas, which uncover the analytical relationships between basin trees of globally-equivalent rules. Last but not least, for pedagogical and epistemological reasons, we conclude this paper with the selection of rule, instead of rule , as the prototypic universal Turing machine for our future discourse.

This 12th part of our Nonlinear Dynamics Perspective of Cellular Automata concludes a series of three articles devoted to CA local rules having robust periodic ω-limit orbits. Here, we consider only the two rules, and , constituting the third of the six groups in which we classified the 1D binary Cellular Automata. Among the numerous theoretical results contained in this article, we emphasize the complete characterization of the ω-limit orbits, both robust and nonrobust, of these two rules and the proof that period-3 and period-6 ω-limit orbits are dense for and , respectively. Furthermore, we will also introduce the fundamental concepts of perfect period-T orbitsets and riddled basins, and see how they emerge in rule .

As stated in the title, we also focus on permutive rules, which have been introduced in a previous installment of our series but never thoroughly studied. Indeed, we will review some of the well-known properties of such rules, like the surjectivity, examining their implications for finite and bi-infinite Cellular Automata.

Finally, we propose a new list of the 88 globally-independent local rules, which is slightly different from the one we have used so far but has the great advantage of being selected via a rigorous methodology and not an arbitrary choice. For the sake of completeness, we display in the appendix the basin tree diagrams and the portraits of the ω-limit orbits of the rules from this refined table which have not yet been reported in our previous articles.

More than one third of the 88 globally-independent Cellular Automata rules exhibit robust simple Bernoulli-shift dynamics. Among them we find rule , which we proved to be chaotic in the previous episodes of our chronicle, and rule , the famous global majority rule. Therefore, we cannot overstate the importance of the Bernoulli σ_τ-shift rules which we will present in two parts of our continuing odyssey on the Nonlinear Dynamics Perspective of Cellular Automata. This paper covers the first 15 of the 30 Bernoulli σ_τ-shift rules. In this paper, after recalling the main concepts of Bernoulli rules — such as the role of the three Bernoulli parameters σ, τ and β — we will display the basin tree diagrams of these rules together with a convenient summary of the results extracted from them. Then, we will show that the superstring is an excellent testing signal to find the robust behavior of a given rule. Finally, we will conclude this paper with a discussion about the difference between robust and nonrobust ω-limit orbits of the Bernoulli σ_τ-shift rules.

Over the past eight years, we have studied one of the simplest, yet extremely interesting, dynamical systems; namely, the one-dimensional binary Cellular Automata. The most remarkable results have been presented in a series of papers which is concluded by the present article. The final stop of our odyssey is devoted to the analysis of the second half of the 30 Bernoulli σ_τ-shift rules, which constitute the largest among the six groups in which we classified the 256 local rules. For all these 15 rules, we present the basin-tree diagrams obtained by using each bit string with L ≤ 8 as initial state, a summary of the characteristics of their ω-limit orbits, and the space-time patterns generated from the superstring. Also, in the last section we summarize the main results we obtained by means of our "nonlinear dynamics perspective".

In this paper we consider a lattice dynamical system generated by a parabolic equation modeling suspension flows. We prove the existence of a global compact connected attractor for this system and the upper semicontinuity of this attractor with respect to finite-dimensional approximations. Also, we obtain a sequence of approximating discrete dynamical systems by the implementation of the implicit Euler method, proving the existence and the upper semicontinuous convergence of their global attractors.

One of the prime paradigms for complex temporal dynamics, the motion of an inelastic ball bouncing on a sinusoidally oscillating table, is revisited. Using extensive numerical simulations, we address the not yet conclusively settled problem of the occurrence of chaos in the partially elastic case. We systematically investigate the spectrum of long-time solutions as function of the initial conditions and system parameters. Subsequently, we generalize the bouncing ball system by taking the velocity dependence of the coefficient of restitution into account and exemplarily demonstrate the drastic impact of such a generalization on the overall dynamics.