Quantum Synchronization and Entanglement of Dissipative Qubits Coupled to a Resonator.

In a dissipative regime, we study the properties of several qubits coupled to a driven resonator in the framework of a Jaynes-Cummings model. The time evolution and the steady state of the system are numerically analyzed within the Lindblad master equation, with up to several million components. Two semi-analytical approaches, at weak and strong (semiclassical) dissipations, are developed to describe the steady state of this system and determine its validity by comparing it with the Lindblad equation results. We show that the synchronization of several qubits with the driving phase can be obtained due to their coupling to the resonator. We establish the existence of two different qubit synchronization regimes: In the first one, the semiclassical approach describes well the dynamics of qubits and, thus, their quantum features and entanglement are suppressed by dissipation and the synchronization is essentially classical. In the second one, the entangled steady state of a pair of qubits remains synchronized in the presence of dissipation and decoherence, corresponding to the regime non-existent in classical synchronization.

Opinion Formation in the World Trade Network.

We extend the opinion formation approach to probe the world influence of economical organizations. Our opinion formation model mimics a battle between currencies within the international trade network. Based on the United Nations Comtrade database, we construct the world trade network for the years of the last decade from 2010 to 2020. We consider different core groups constituted by countries preferring to trade in a specific currency. We will consider principally two core groups, namely, five Anglo-Saxon countries that prefer to trade in US dollar and the 11 BRICS+ that prefer to trade in a hypothetical currency, hereafter called BRI, pegged to their economies. We determine the trade currency preference of the other countries via a Monte Carlo process depending on the direct transactions between the countries. The results obtained in the frame of this mathematical model show that starting from the year 2014, the majority of the world countries would have preferred to trade in BRI than USD. The Monte Carlo process reaches a steady state with three distinct groups: two groups of countries preferring to trade in whatever is the initial distribution of the trade currency preferences, one in BRI and the other in USD, and a third group of countries swinging as a whole between USD and BRI depending on the initial distribution of the trade currency preferences. We also analyze the battle between three currencies: on one hand, we consider USD, BRI and EUR, the latter currency being pegged by the core group of nine EU countries. We show that the countries preferring EUR are mainly the swing countries obtained in the frame of the two currencies model. On the other hand, we consider USD, CNY (Chinese yuan), OPE, the latter currency being pegged to the major OPEC+ economies for which we try to probe the effective economical influence within international trade. Finally, we present the reduced Google matrix description of the trade relations between the Anglo-Saxon countries and the BRICS+.

Nonlinear Perturbation of Random Matrix Theory.

We consider a system of linear oscillators, or quantum states, described by random matrix theory and analyze how its time evolution is affected by a nonlinear perturbation. Our numerical results show that above a certain chaos border a weak or moderate nonlinearity leads to a dynamical thermalization of a finite number of degrees of freedom with energy equipartition over linear eigenmodes as expected from the laws of classical statistical mechanics. The system temperature is shown to change in a broad range from positive to negative values, and the dependence of system characteristics on the initial injected energy is determined. Below the chaos border the dynamics is described by the Kolmogorov-Arnold-Moser integrability. Owing to universal features of random matrix theory we argue that the obtained results describe the generic properties of its nonlinear perturbation.

Dollar-Yuan Battle in the World Trade Network.

From the Bretton Woods agreement in 1944 till the present day, the US dollar has been the dominant currency in world trade. However, the rise of the Chinese economy has recently led to the emergence of trade transactions in Chinese yuan. Here, we mathematically analyze how the structure of international trade flows would favor a country to trade whether in US dollar or in Chinese yuan. The trade currency preference of a country is modeled as a binary variable with the properties of a spin in an Ising model. The computation of this trade currency preference is based on the world trade network built from the 2010-2020 UN Comtrade data and is determined by two multiplicative factors: the relative weight of trade volume exchanged by the country with its direct trade partners and the relative weight of its trade partners in global international trade. The performed analysis, based on the convergence of the Ising spin interactions, shows that from 2010 to present a transition took place, and the majority of the world countries would now have a preference to trade in Chinese yuan if one only considers the world trade network structure.

COVID-19's Impact on International Trade.

We analyze how the COVID-19 pandemic affected the trade of products between countries. With this aim, using the United Nations Comtrade database, we perform a Google matrix analysis of the multiproduct World Trade Network (WTN) for the years 2018-2020, comprising the emergence of the COVID-19 as a global pandemic. The applied algorithms-PageRank, CheiRank and the reduced Google matrix-take into account the multiplicity of the WTN links, providing new insights into international trade compared to the usual import-export analysis. These complex networks analysis algorithms establish new rankings and trade balances of countries and products considering all countries on equal grounds, independent of their wealth, and every product on the basis of its relative exchanged volumes. In comparison with the pre-COVID-19 period, significant changes in these metrics occurred for the year 2020, highlighting a major rewiring of the international trade flows induced by the COVID-19 pandemic crisis. We define a new PageRank-CheiRank product trade balance, either export or import-oriented, which is significantly perturbed by the pandemic.

Collective intelligence defines biological functions in Wikipedia as communities in the hidden protein connection network.

English Wikipedia, containing more than five millions articles, has approximately eleven thousands web pages devoted to proteins or genes most of which were generated by the Gene Wiki project. These pages contain information about interactions between proteins and their functional relationships. At the same time, they are interconnected with other Wikipedia pages describing biological functions, diseases, drugs and other topics curated by independent, not coordinated collective efforts. Therefore, Wikipedia contains a directed network of protein functional relations or physical interactions embedded into the global network of the encyclopedia terms, which defines hidden (indirect) functional proximity between proteins. We applied the recently developed reduced Google Matrix (REGOMAX) algorithm in order to extract the network of hidden functional connections between proteins in Wikipedia. In this network we discovered tight communities which reflect areas of interest in molecular biology or medicine and can be considered as definitions of biological functions shaped by collective intelligence. Moreover, by comparing two snapshots of Wikipedia graph (from years 2013 and 2017), we studied the evolution of the network of direct and hidden protein connections. We concluded that the hidden connections are more dynamic compared to the direct ones and that the size of the hidden interaction communities grows with time. We recapitulate the results of Wikipedia protein community analysis and annotation in the form of an interactive online map, which can serve as a portal to the Gene Wiki project.

Deconfinement of classical Yang-Mills color fields in a disorder potential.

We study numerically and analytically the behavior of classical Yang-Mills color fields in a random one-dimensional potential described by the Anderson model with disorder. Above a certain threshold, the nonlinear interactions of Yang-Mills fields lead to chaos and deconfinement of color wavepackets with their subdiffusive spreading in space. The algebraic exponent of the second moment growth in time is found to be in the range of 0.3-0.4. Below the threshold, color wavepackets remain confined even if a very slow spreading at very long times is not excluded due to subtle nonlinear effects and the Arnold diffusion for the case when initially color packets are located in close vicinity. In the case of large initial separation of color wavepackets, they remain well confined and localized in space. We also present the comparison with the behavior of the one-component field model of discrete Anderson nonlinear Schrödinger equation with disorder.

Some elements for a history of the dynamical systems theory.

Writing a history of a scientific theory is always difficult because it requires to focus on some key contributors and to "reconstruct" some supposed influences. In the 1970s, a new way of performing science under the name "chaos" emerged, combining the mathematics from the nonlinear dynamical systems theory and numerical simulations. To provide a direct testimony of how contributors can be influenced by other scientists or works, we here collected some writings about the early times of a few contributors to chaos theory. The purpose is to exhibit the diversity in the paths and to bring some elements-which were never published-illustrating the atmosphere of this period. Some peculiarities of chaos theory are also discussed.

Fibrosis Protein-Protein Interactions from Google Matrix Analysis of MetaCore Network.

Protein-protein interactions is a longstanding challenge in cardiac remodeling processes and heart failure. Here, we use the MetaCore network and the Google matrix algorithms for prediction of protein-protein interactions dictating cardiac fibrosis, a primary cause of end-stage heart failure. The developed algorithms allow identification of interactions between key proteins and predict new actors orchestrating fibroblast activation linked to fibrosis in mouse and human tissues. These data hold great promise for uncovering new therapeutic targets to limit myocardial fibrosis.

Interdependence of Sectors of Economic Activities for World Countries from the Reduced Google Matrix Analysis of WTO Data.

We apply the recently developed reduced Google matrix algorithm for the analysis of the OECD-WTO World Network of Economic Activities. This approach allows to determine interdependencies and interactions of economy sectors of several countries, including China, Russia and the USA, properly taking into account the influence of all the other world countries and their economic activities. Within this analysis, we also obtain the sensitivity of EU countries' economies to the petroleum activity sector. We show that this approach takes into account the multiplicity of economical interactions between countries and activity sectors, thus providing a richer analysis compared to the usual export-import analysis.

Kolmogorov Turbulence Defeated by Anderson Localization for a Bose-Einstein Condensate in a Sinai-Oscillator Trap.

We study the dynamics of a Bose-Einstein condensate in a Sinai-oscillator trap under a monochromatic driving force. Such a trap is formed by a harmonic potential and a repulsive disk located in the center vicinity corresponding to the first experiments of condensate formation by Ketterle and co-workers in 1995. We allow that the external driving allows us to model the regime of weak wave turbulence with the Kolmogorov energy flow from low to high energies. We show that in a certain regime of weak driving and weak nonlinearity such a turbulent energy flow is defeated by the Anderson localization that leads to localization of energy on low energy modes. This is in a drastic contrast to the random phase approximation leading to energy flow to high modes. A critical threshold is determined above which the turbulent flow to high energies becomes possible. We argue that this phenomenon can be studied with ultracold atoms in magneto-optical traps.

Random matrix model of Kolmogorov-Zakharov turbulence.

We introduce and study a random matrix model of Kolmogorov-Zakharov turbulence in a nonlinear purely dynamical finite-size system with many degrees of freedom. For the case of a direct cascade, the energy and norm pumping takes place at low energy scales with absorption at high energies. For a pumping strength above a certain chaos border, a global chaotic attractor appears with a stationary energy flow through a Hamiltonian inertial energy interval. In this regime, the steady-state norm distribution is described by an algebraic decay with an exponent in agreement with the Kolmogorov-Zakharov theory. Below the chaos border, the system is located in the quasi-integrable regime similar to the Kolmogorov-Arnold-Moser theory and the turbulence is suppressed. For the inverse cascade, the system rapidly enters a strongly nonlinear regime where the weak turbulence description is invalid. We argue that such a dynamical turbulence is generic, showing that it is present in other lattice models with disorder and Anderson localization. We point out that such dynamical models can be realized in multimode optical fibers.

Goal quest for an intelligent surfer moving in a chaotic flow.

We consider a model of an intelligent surfer moving on the Ulam network generated by a chaotic dynamics in the Chirikov standard map. This directed network is obtained by the Ulam method with a division of the phase space in cells of fixed size forming the nodes of a Markov chain. The goal quest for this surfer is to determine the network path from an initial node A to a final node B with minimal resistance given by the sum of inverse transition probabilities. We develop an algorithm for the intelligent surfer that allows us to perform the quest in a small number of transitions which grows only logarithmically with the network size. The optimal path search is done on a fractal intersection set formed by nodes with small Erdös numbers of the forward and inverted networks. The intelligent surfer exponentially outperforms a naive surfer who tries to minimize its phase space distance to target B. We argue that such an algorithm provides unique hints for motion control in chaotic flows.

Delocalization induced by nonlinearity in systems with disorder.

We study numerically the effects of nonlinearity on the Anderson localization in lattices with disorder in one and two dimensions. The obtained results show that at moderate strength of nonlinearity a spreading over the lattice in time takes place with an algebraic growth of number of populated sites Deltan proportional to tnu. This spreading continues up to a maximal dimensionless time scale t=10(9) reached in the numerical simulations. The numerical values of nu are found to be approximately 0.15-0.2 and 0.25 for the dimension d=1 and 2, respectively, being in a satisfactory agreement with the theoretical value d/(3d+2). During the computational times t

Delocalization transition for the Google matrix.

We study the localization properties of eigenvectors of the Google matrix, generated both from the world wide web and from the Albert-Barabási model of networks. We establish the emergence of a delocalization phase for the PageRank vector when network parameters are changed. For networks with localized PageRank, eigenvalues of the matrix in the complex plane with a modulus above a certain threshold correspond to localized eigenfunctions while eigenvalues below this threshold are associated with delocalized relaxation modes. We argue that, for networks with delocalized PageRank, the efficiency of information retrieval by Google-type search is strongly affected since the PageRank values have no clear hierarchical structure in this case.

Diffusion and localization for the Chirikov typical map.

We consider the classical and quantum properties of the "Chirikov typical map," proposed by Boris Chirikov in 1969. This map is obtained from the well-known Chirikov standard map by introducing a finite-number T of random phase-shift angles. These angles induce a random behavior for small time-scales (tT) . We identify the classical chaos border k(c) approximately T (-3/2)1 for the kick parameter k and two regimes with diffusive behavior on short and long time scales. The quantum dynamics is characterized by the effect of Chirikov localization (or dynamical localization). We find that the localization length depends in a subtle way on the two classical diffusion constants in the two time-scale regime.

Incommensurate standard map.

We introduce and study the extension of the Chirikov standard map when the kick potential has two and three incommensurate spatial harmonics. This system is called the incommensurate standard map. At small kick amplitudes, the dynamics is bounded by the isolating Kolmogorov-Arnold-Moser surfaces, whereas above a certain kick strength, it becomes unbounded and diffusive. The quantum evolution at small quantum kick amplitudes is somewhat similar to the case of the Aubru-André model studied in mathematics and experiments with cold atoms in a static incommensurate potential. We show that for the quantum map there is also a metal-insulator transition in space whereas in momentum we have localization similar to the case of two-dimensional Anderson localization. In the case of three incommensurate frequencies of the space potential, the quantum evolution is characterized by the Anderson transition similar to the three-dimensional case of the disordered potential. We discuss possible physical systems with such a map description including dynamics of comets and dark matter in planetary systems.

Wikipedia network analysis of cancer interactions and world influence.

We apply the Google matrix algorithms for analysis of interactions and influence of 37 cancer types, 203 cancer drugs and 195 world countries using the network of 5 416 537 English Wikipedia articles with all their directed hyperlinks. The PageRank algorithm provides a ranking of cancers which has 60% and 70% overlaps with the top 10 deadliest cancers extracted from World Health Organization GLOBOCAN 2018 and Global Burden of Diseases Study 2017, respectively. The recently developed reduced Google matrix algorithm gives networks of interactions between cancers, drugs and countries taking into account all direct and indirect links between these selected 435 entities. These reduced networks allow to obtain sensitivity of countries to specific cancers and drugs. The strongest links between cancers and drugs are in good agreement with the approved medical prescriptions of specific drugs to specific cancers. We argue that this analysis of knowledge accumulated in Wikipedia provides useful complementary global information about interdependencies between cancers, drugs and world countries.

Interactions of pharmaceutical companies with world countries, cancers and rare diseases from Wikipedia network analysis.

Using the English Wikipedia network of more than 5 million articles we analyze interactions and interlinks between the 34 largest pharmaceutical companies, 195 world countries, 47 rare renal diseases and 37 types of cancer. The recently developed algorithm using a reduced Google matrix (REGOMAX) allows us to take account both of direct Markov transitions between these articles and also of indirect transitions generated by the pathways between them via the global Wikipedia network. This approach therefore provides a compact description of interactions between these articles that allows us to determine the friendship networks between them, as well as the PageRank sensitivity of countries to pharmaceutical companies and rare renal diseases. We also show that the top pharmaceutical companies in terms of their Wikipedia PageRank are not those with the highest market capitalization.

Inferring hidden causal relations between pathway members using reduced Google matrix of directed biological networks.

Signaling pathways represent parts of the global biological molecular network which connects them into a seamless whole through complex direct and indirect (hidden) crosstalk whose structure can change during development or in pathological conditions. We suggest a novel methodology, called Googlomics, for the structural analysis of directed biological networks using spectral analysis of their Google matrices, using parallels with quantum scattering theory, developed for nuclear and mesoscopic physics and quantum chaos. We introduce analytical "reduced Google matrix" method for the analysis of biological network structure. The method allows inferring hidden causal relations between the members of a signaling pathway or a functionally related group of genes. We investigate how the structure of hidden causal relations can be reprogrammed as a result of changes in the transcriptional network layer during cancerogenesis. The suggested Googlomics approach rigorously characterizes complex systemic changes in the wiring of large causal biological networks in a computationally efficient way.