Abstract: Controllability of networked systems with non-linear dynamics remains an interesting challenge with widespread applications to problems ranging from engineering to biology. As a step in this direction, this paper explores global controllability criteria for discrete-time non-linear networks. We identify two classes of non-linear networks: those with non-linear edge dynamics and those with non-linear node dynamics. For each of these classes, we formulate the global controllability matrix and discuss corresponding controllability conditions. In the first case, we obtain a time-dependent controllability matrix, whereas, in the second, we obtain a non-linear operator. We point to a network interpretation of controllability associated to linear independence of sets of paths from driver nodes to every node of the network and comment on possible applications of our formalism.

Abstract: Synchronization phenomena has attracted the interest of the scientific communities of different fields since old times. It appears of a decisive role especially as a self-organizing mechanism which manifests in biology e.g. mating of fireflies or in physics e.g. Josephson array [1]. In fact, the resonance effect shown for such behaviour results in many cases of vital importance in Nature [1]. However, not always the synchronization is the desiderable expectation in a physical process. This is the case for example of the Millenium Bridge of London [2]. Due to strong coupling of the bridge mechanical parts it started swaying after a given number of pedestrians tempted to cross it. In this paper we propose, a completely unconventional and novative control method to the synchronization problem. The idea is to prevent the set of weakly coupled nonlinear oscillators from phase-synchronizing. Based in a recent work [3] where a Hamiltonian formulation of the seminal Kuramoto model [4] was presented, we were able to construct a control technique making use of Hamiltonian control methods. Adding a control term of magnitude O(ε2) (where ε is the size of the coupling strength) in the Hamiltonian of the Kuramoto equation the system not only does not synchronize but it is also robust to resonance phenomena which never occur. The results we obtained using a simple paradigmatic model of synchronization, show that it is possible to design complex systems e.g. mechanical structures, immune to any resonance effect by simply making a small modification to the original system. References [1] S.H. Strogatz, Sync : The Emerging Science of Spontaneous Order, Hyperion (2003). [2] Strogatz, Steven et al., Nature 438, 43–44 (2005). [3] D. Witthaut, M. Timme Phys. Rev. E 90, 032917 (2014). [4] Y. Kuramoto, Chemical Oscillations, Waves, and Turbulence, New York, Springer-Verlag (1984).

Abstract: Co-evolutionary adaptive mechanisms are not only ubiquitous in nature, but also beneficial for the functioning of a variety of systems. We here consider an adaptive network of oscillators with a stochastic, fitness-based, rule of connectivity, and show that it self-organizes from fragmented and incoherent states to connected and synchronized ones. The synchronization and percolation are associated to abrupt transitions, and they are concurrently (and significantly) enhanced as compared to the non-adaptive case. Finally we provide evidence that only partial adaptation is sufficient to determine these enhancements. Our study, therefore, indicates that inclusion of simple adaptive mechanisms can efficiently describe some emergent features of networked systems' collective behaviors, and suggests also self-organized ways to control synchronization and percolation in natural and social systems.

Abstract: The air-water interface is of huge importance to a wide range of environmental, biological and industrial chemistry. It shows complex behaviour and continues to surprise both experimental and theoretical communities. For many years the biological physical chemistry community has highlighted the different behaviour of water in and close to hydrophobic surfaces, such as proteins or lipid membranes. Recent work on ellipsometry at the air-water interface has suggested that the refractive index of the surface region may be significantly higher than that of the bulk water. This higher refractive index, would not only infer a significant change of interactions in water at a hydrophobic region, but also impact on the interpretation of many of the non-linear spectroscopic studies as they rely on the linear optical properties being understood. We attempt to investigate this behaviour using the Amber 12 molecular mechanics software. However, classical molecular mechanics simulations are generally parameterised to accurately recreate bulk mechanical, electronic and thermodynamic properties. The interfacial and surface regions of atomistic and molecular systems tend to be neglected. In order to ensure accurate surface behaviour we have implemented ways to deal with long range Lennard-Jones corrections in systems containing interfaces based on the methodology of Janecek. We present how these corrections are important for replicating surface behaviour in water, and a novel way to thermodynamically estimate surface energetic and entropic terms.

Abstract: The Collatz conjecture is named after Lothar Collatz, who first proposed it in 1937. The conjecture is also known as the (3x+1) conjecture, the Ulam conjecture, Kakutani's problem, the Thwaites conjecture, Hasse's algorithm or the Syracuse problem. This can be formulated as an innocent problem of arithmetics. Take any positive integer n. If n is even, divide it by 2 to get n/2. If n is odd, multiply it by 3 and add 1 to obtain 3n+1.Repeating the process iteratively, the map is believed to converge to a period 3 orbit formed by the triad {1,2,4}. Equivalently, the conjecture states that the Collatz map will always reach 1, no matter what integer number one starts with. Numerical experiments have confirmed the validity of the conjecture for extraordinarily large values of the starting integer n. The beauty of the conjecture emanates indeed from its apparent, tantalising, simplicity, which however hides formidable challenges, when one tries to cast it on solid roots. In this paper, we provide a novel argument to support the validity of the Collatz conjecture, which, to the best of our knowledge, configures as the first proof of the claim. The proof exploits the formalism of stochastic maps defined on directed graphs. More specifically, the proof articulates along the following lines: (i) define the (forward) third iterate of the Collatz map and consider the equivalence classes of integer numbers modulo 8; (ii) employ a stochastic approach based on a Markov process to prove the contracting property of such map on generic orbits; (iii) demonstrate that diverging orbits are not allowed because they will not be compatible with the stationary equilibrium distribution of the Markov process. The proof will be illustrated with emphasis to the methological aspects that require resorting to the concept of directed graph.

Abstract: A typical feature of many biological and ecological complex systems is their capability to be highly sensitive and responsive to small changes of the values of specific key variables, while being at the same time extremely resilient to a large class of disturbances. The possibility to build artificial systems with these characteristics is of extreme importance for the development of nanomachines and biological circuits with potential medical and environmental applications. The main theoretical difficulty toward the realisation of these devices lies in the lack of a mathematical methodology to design the blueprint of a self-controlled system composed of a large number of microscopic interacting constituents that should operate in a prescribed fashion. Here a general methodology is proposed to engineer a system of interacting components (particles) which is able to self-regulate their concentrations in order to produce any prescribed output in response to a particular input. The methodology is based on the mathematical equivalence between artificial neurons in neural networks and species in autocatalytic reactions, and it specifies the relationship between the artificial neural network’s parameters and the rate coefficients of the reactions between particle species. Such systems are characterised by a high degree of robustness as they are able to reach the desired output despite disturbances and perturbations of the concentrations of the various species. Relating concepts from artificial intelligence to dynamical systems, the results presented here demonstrate the possibility to employ approaches and techniques developed in one field to the other, bringing potential advancements in both disciplines and related applications. Preprint: https://arxiv.org/abs/1602.09070