# Abstracts

**Peter Ashwin** (University of Exeter, Exeter, UK)

*Tipping points for parameter shifts*

Bifurcation theory for autonomous systems (and to a less extent, for systems with stationary forcing) is a well-developed branch of nonlinear dynamics. But what happens when the forcing is not stationary but performs a transition from one regime to another? In this talk I will discuss joint work with Sebastian Wieczorek and Clare Perryman systems undergoing a “parameter shift”, namely a (nonautonomous and nonstationary) change between different asymptotically autonomous systems. By examining properties of local pullback attractors for the associated nonautonomous system we can characterise bifurcation- and rate-dependent tipping in a rigorous way. In particular, rate-dependent tipping is associated with cases where a local pullback attractor is not a forwards attractor. (For details see arXiv:1506.07734).

**Serge Aubry** (Laboratoire Léon Brillouin, Saclay, France)

*KAM tori in the 1D Random DNLS model and Absence of Diffusion of a Wave-packet?*

Numerical investigations shows that a finite norm initially localized wave-packet in a system with linear Anderson localization and extra nonlinearity, may generate two kinds of trajectories which both are obtained with non vanishing probability. The first kind of wave-packets consists of almost periodic KAM trajectories which are recurrent and do not spread. The second kind consists of trajectories which look initially chaotic and often spread over long times. However it can be proven in several special models that complete spreading is impossible despite apparent initial chaos. It has been also proved recently in 1D Quasiperiodic DNLS model in the regime of linear localization, that most initially localized wave packet remains dynamically localized as KAM tori up to some perturbation parameter (J.Geng, J. You, Z. Zhao, Geom. Funct. Analysis. 24 (2014) 116-158)

Thus, it is still questionable whether diffusion may persist forever and under which conditions. In case where complete diffusion is impossible, the nature of the limit state of initially chaotic trajectories is still unknown.

**Christian Beck** (Queen Mary University of London, London, UK)

*Statistics of Lagrangian quantum turbulence*

Many complex driven nonequilibrium systems are effectively described by a superposition of several statistics on well-separated time scales, in short a 'superstatistics'. A simple example is a Brownian particle moving in a spatially inhomogeneous medium with fluctuations of a suitable parameter on a large time scale, but the concept is much more general. Superstatistical systems typically have marginal distributions that exhibit fat tails, for example power law tails or stretched exponentials.

In this talk I will concentrate on some superstatistical model building to better understand the statistics of tracer particles embedded in a) classical fully developed turbulent flows and b) turbulent quantum liquids. Different statistics arises in the quantum liquid due to constraints by chaotically moving vortex lines.

**Sergey Bolotin** (University of Wisconsin, Madison, USA, and Steklov Mathematical Institute, Moscow, Russian Federation)

*Degenerate builliards*

In an ordinary billiard trajectories of a Hamiltonian system

are elastically reflected when colliding with a hypersurface (scatterer).

If the scatterer is a submanifold of codimension more than one, then collisions are rare.

We study trajectories of such degenerate billiards with infinite number of collisions.

Degenerate billiards appear as limits of ordinary billiards or as limits of systems with singularities in celestial mechanics.

The main tool is the method of anti-integrable limit.

**Tassos Bountis** (University of Patras, Patras, Greece)

*Invertible Maps, the Parametrization Method and Breathers in KG Lattices*

To honor Robert MacKay on his 60th birthday, I will speak about recent work which combines three topics that have always been dear to Robert and to which he has contributed significantly: Invertible maps, homoclinic orbits and discrete breathers. I will weave them together using another one of Robert’s favorites: analytical approximations of invariant manifolds. Let me recall first that initial conditions for breathers in 1-Dimensional (1D) Hamiltonian lattices can be well approximated by homoclinic orbits of invertible maps. In fact, for a 1D lattice of the Klein-Gordon (KG) type, it suffices to use a 2D map with a cubic nonlinearity. In this talk, I will first prove the existence of a hyperbolic invariant set of this map, in the conservative case (dissipation parameter δ=1), generated by a transverse homoclinic point which we locate analytically using series expansions of the Parametrization Method (PM). Then I will vary δ < 1 and accurately predict the δ = δc at homoclinic tangency where these intersections cease to exist. I will then extend these results to the case of a 4D cubic map approximating initial conditions for breathers in two linearly coupled KG particle chains. Though more cumbersome, the PM still applies and yields accurate homoclinic points and precise estimates of homoclinic tangency in 4D.

(With Stavros Anastassiou and Arnd Baecker)

**Phil Boyland** (University of Florida, Gainesville, USA)

*When Topology Forces Dynamics*

Sometimes topology data about a dynamical system can dictate a lower bound on its complexity. Because the topological data required is often very coarse, such results are robust and lend themselves to physical applications. After covering the basics we present several theorems of this type including one on the growth of vorticity in certain periodically stirred two-dimensional Euler flows and another on the dynamics of time-periodic Lagrangian systems on hyperbolic manifolds. Applications will include viscous fluid mixing and the dynamics of mechanical linkages. A main theme in the talk is the analogy, pointed out by MacKay in 1988, between Morse's classical theorem on geodesics on higher genus surfaces and the isotopy stability of the dynamics of Thurston's pseudoAnosov maps.

**Tom Bridges** (University of Surrey, Guilford, UK)

*Baesens and MacKay (1992): theme and variations*

The first part of the talk will review the seminal 1992 paper of Baesens & MacKay on water waves, highlighting its key points and history, especially the discovery of a limit point in the flow force. The second part will discuss a new view on how and why the KdV equation emerges as a model PDE, showing that the conventional wisdom, particularly in the theory of water waves, is wrong (right equation wrong mechanism!). Limit points are the key. Dispersion relations, shallow water assumptions, etc, are symptoms rather than causes. An added bonus is that coefficient of nonlinearity in KdV is a curvature which comes for free. Connecting these two parts of the talk, we see that the fold point discovered in BM92 can be used to infer that the KdV equation can appear as a model in deep water, showing that shallow water is neither necessary nor sufficient for KdV in water waves. Other, more tractable, examples of KdV in deep water will also be shown. Time permitting, some of the background "geometric modulation theory", as well as generalizations to 2+1, and how to generate new modulation equations, will be discussed.

**Henk Broer** (University of Groningen, Groningen, The Netherlands)

*Classification of constrained differential equations embedded in the theory of slow fast systems*

(with H. Jardón-Kojakhmetov and R. Roussarie) We review the theory of Constrained Differential Equations as set up by E.C. Zeeman and F. Takens, extending the topological classification of singularities. This theory provides local polynomial models under topological equivalence, which play an important role in the study of Slow-Fast systems. Applying geometric desingularisation, as developed by F. Dumortier, R. Roussarie and others, and a normal form approach by E. Lombardi and L. Stolovitch, we obtain formal normal forms, which gives useful geometric and asymptotic results.

**Alessandra Celletti** (University of Rome - Tor Vergata, Rome, Italy)

*Breakdown of invariant attractors in dissipative (conformally symplectic) systems.*

We compute numerically the breakdown threshold of invariant attractors in conformally symplectic systems by two different techniques.

First: A recent proof of KAM theorem for conformally symplectic systems leads to very efficient algorithms (See R. de la Llave's talk) to compute the embedding and the drift. This also predicts that the analytic attractors cease to exist when and only when the Sobolev norms blow up. This method does not rely on periodic orbits and also leads to rigorous estimates, which are 99.8% of the true value.

Second, we formulate a variant of Greene's method for conformally symplectic systems, we provide a justification and implement it.

The actual computations reveal that the breakdown presents remarkable scaling relations; it was noticed that ([Calleja-Figueras 2012]) the breakdown of normal hyperbolicity can only happen because the stable direction becomes close to the tangent (the exponents remain constant).

Joint work with R. Calleja and R. de la Llave.

**Yi-Chiuan Chen** (Academia Sinica, Taipei, Taiwan)

*Transitions for multi-hole Denjoy minimal sets*

*Following the terminology of the 1993, 1994 papers of Baesens and MacKay on Denjoy minimal sets, we call a pair of distinct points of a Denjoy minimal set whose forward and backward orbits converge together a "gap". The gaps come in orbits, which we call "holes". Some results on transitions for Denjoy minimal sets of different number of holes will be discussed.*

**Thierry Dauxois** (CNRS & ENS de Lyon, Lyon, France)

*Energy cascade in internal-wave attractors*

We will discuss the physics of internal gravity waves, these being hydrodynamic waves beneath the ocean surface, which have unexpected properties for a physicist. They are linked to one of the pivotal questions in the dynamics of the oceans: the cascade of mechanical energy in the abyss and its contribution to mixing. More precisely, we propose internal-wave attractors in the large-amplitude regime as a unique self-consistent experimental and numerical setup that models a cascade of triadic interactions transferring energy from large-scale monochromatic input to multi-scale internal-wave motion. We also provide signatures of a discrete wave turbulence framework for internal waves. Finally, we show how, beyond this regime, we have a clear transition to a regime of small-scale high-vorticity events which induce mixing.

**Gianne Derks** (University of Surrey, Guilford, UK)

*Sleep-wake dynamics: an interplay of maps, gaps and naps*

Sleep-wake dynamics are often modelled by multi-oscillator systems. The interaction of two oscillatory processes play a dominant role: the circadian oscillation of the body clock, and a relaxation oscillator known as the 'sleep homeostat' that results in a sleep pressure that increases during wake and decreases during sleep. The resulting two-process model has been very successful, providing the language which frames most sleep research. Yet, surprisingly, there has been very little mathematical analysis of the two-process model.

In the last decade, advances in physiology have led to more sophisticated models incorporating neuronal interactions. In this talk we will discuss how some of these models can be reduced to two-process model. And in its turn, the two-process can be represented as a one-dimensional map with discontinuities. Transitions between different sleep-wake cycles with differing numbers of daily sleep episodes can thus be understood as the result of grazing (tangent) bifurcations. This could provide an explanation for observed changes in sleep patterns of babies and young children.

This is joint work with Anne Skeldon, Matthew Bailey and Derk-Jan Dijk.

**Marina Diakonova** (QMUL)

*Irreducibility of a Multiplex Network: the Case of the Voter Model*

The multiplex has been a major focus of investigation for the scientific community in the recent years. In the context of opinion-dynamics, this framework of coupled networks is a perfect setting to model the combined effect of the presence of different communities on individual opinion formation. We let the dynamically evolving state of a network node stand for some internal opinion of an agent, and implement the voter model on a two-layer multiplex network. We show how the purely multiplex structural parameters affect the general system phenomenology, and, crucially, compare the numerical results with the predicted behaviour of several equivalent single-layer aggregates of the original system. The existence of differences at 'realistic' parameter values suggests that the multiplex is, at least trivially, irreducible, which we collaborate further by inverting the observed trends to arrive at nonlinear functional forms of some topological properties of the ideal aggregate. We conclude that if the effect of the multiplex structure is to be reproduced on a single network, then it is not obvious how to obtain such a network from the original multilayer system.

**Raphael Douady** (University of Paris – Sorbonne, Paris, France)

*Nonlinear Factor Analysis by Polymodels*

Nonlinear polymodels is a factor-based statistical analysis technique which can be applied in a broad range of fields. Inpired by pattern recognition methods used in DNA analysis or in hand writing recognition, it is particularly adapted to situations where space-dependent changes of regime modify the relation between dependent variables and their independent drivers, making ot difficult for a single multi-factor model to fit all the possible situations that may potentially occur. We shall present in the particular context of financial modeling.

Traditional multi-factor analysis is essentially used in finance in a linear setting. Asset returns are replicated by a linear combination of factor returns. Not only it provides answers to questions related to the statistical behaviour of assets with respect to the market, but it is intellectually comfortable, as a portfolio is naturally represented as a reduced "portfolio" of risk factors. However, this representation sadly lacks of any predictive value, especially when we need ot the most, that is, when a crisis is coming. We shall show how nonlinear polymodels provide a reliable solution to the main questions factor analysis aims at addressing:

1) finding the probability distribution of individual asset returns (risk measurement)

2) assessing the impact of a given shift of risk factors (stress testing)

3) estimating the joint probability distribution of family of assets (portfolio risk and optimization)

We shall show how the nonlinear polymodel-based "Dominant Factorsô" methodology provides superior portfolio returns, simply thanks to a better control of the downside, without the pitfalls of traditional Markowitz and Black-Litterman methods.

**Jean-Pierre Eckmann** (University of Geneva, Geneva, Switzerland)

*Hamiltoninan chains with dissipation*

I plan to speak about a new result, with Noé Cuneo and Gene Wayne, which should remind Robert of times long gone: The idea is to see how

the dissipation at the end of a chain manifests itself in the interior, once one has transformed the system to canonical form. At the time of submitting this abstract, the ideas seem clear, and the role of almost-breathers comes out beautifully. But many details are still open. We will see how far we get when I will talk.

**Bastien Fernandez** (Université de Paris-Diderot, Paris, France)

*Breaking of Ergodicity in Expanding Systems of Globally Coupled Piecewise Affine Circle Maps*

To identify and to explain coupling-induced phase transitions in Coupled Map Lattices (CML) has been a lingering enigma for about two decades. In numerical simulations, this phenomenon has always been observed preceded by a lowering of the Lyapunov dimension, suggesting that the transition might require changes of linear stability. Yet, recent proofs of co-existence of several phases in specially designed models work in the expanding regime where all Lyapunov exponents remain positive.

In this talk, I will consider a family of CML composed by piecewise expanding individual maps, global interaction and finite number N of sites, in the weak coupling regime where the CML is uniformly expanding. I will show, mathematically for N=3 and numerically for N>3, that a transition in the asymptotic dynamics occurs as the coupling strength increases. The transition breaks the (Milnor) attractor into several chaotic pieces of positive Lebesgue measure, with distinct empiric averages. It goes along with various symmetry breaking, quantified by means of magnetization-type characteristics.

Despite that it only addresses finite-dimensional systems, to some extent, this result reconciles the previous ones as it shows that loss of ergodicity/symmetry breaking can occur in basic CML, independently of any decay in the Lyapunov dimension.

**Sergej Flach** (IBS PCS, Daejeon, South Korea)

*On FPU, and nonlinearity-induced destruction of Anderson localization*

The FPU paradox is about a linear eigenstate (mode) not sharing its energy with all other participating modes - i.e. about the absence

of equipartition, despite the presence of nonlinear mode-coupling which destroys integrability. In fact, with modern computers and long enough waiting times equipartition does recover, with equipartition time scales depending sensitively on the parameters of the initial state.

Wave packets in nonlinear disordered lattices evolve along similar lines - the linear eigenstates are Anderson localized, yet the

nonlinear wave equations produce destruction of Anderson localization, thermalization inside the spreading wave packet, and universal

exponents which characterize the evolution. I will briefly touch the main results and discuss open problems.

**Paul Glendinning** (University of Manchester, Manchester, UK)

*Creation of discontinuities in model mappings*

We describe how discontinuities are created in maps of the circle derived from biological threshold models. This leads to a class of maps with discontinuities and infinite derivatives. We contrast this with the creation of discontinuities in return maps of Cherry flows. Some of the analysis uses classic results of Mackay and Tresser from the 1980s, and the problem is also related to more recent work of Baesens and MacKay describing flows on the 2-torus.

This is a joint work with Gianne Derks and Anne Skeldon (University of Surrey)

**Gabriela Gomes** (Liverpool School of Tropical Medicine, Liverpool, UK and Research Centre in Biodiversity and Genetic Resources, University of Porto, Portugal)

*A double-edged sword in the treatment of high-risk groups*

Having started a research career in non-linear dynamics, symmetries and patterns, I was easily intrigued by the complexity of risk factors and processes that determine the patterns of disease in populations. After some initial years studying fluids and chemical reactions, my attention drifted to the dynamics of microbes in interaction with their hosts, and resulting infectious diseases.

Confident that infectious disease epidemiology would be all about specific population processes, differing essentially from non-communicable diseases, I spent over a decade developing mathematical models of infectious disease transmission. These models may come in the simplest forms that compartmentalise the population into a minimal set of groups - such as susceptible (S) and infected (I) - or in more detailed implementations, going all the way to representing hosts at the level of individuals, connected by a network of contacts, exposed to a diversity of evolving pathogens, in a dynamic environment. Now, I realise that when it actually comes to quantify and interpret measurements of disease, whether observationally or experimentally, most valuable insights result from processes that are common to both infectious and non-communicable diseases. Most remarkably, any heterogeneous population under pressure by any disease, undergoes a process of cohort selection whereby high-risk individuals are affected preferentially. This can easily overrule the complexities of transmission dynamics and feedbacks in the particular case of infectious diseases, and turn interventions into double-edged swords. Applying this general population process, we can answer outstanding questions, such as:

1. Why do vaccines appear less efficacious in higher-incidence settings?

2. Why do transmission models over-predict the impact of interventions to control infectious diseases?

3. What does cohort selection have to say about “irreproducible research”?

**Celso Grebogi** (University of Aberdeen, Aberdeen, UK)

*Compressive Sensing Based Prediction of Complex Dynamics and Complex Networks*

In the fields of complex dynamics and complex networks, the reverse engineering, systems identification, or inverse problem is generally regarded as hard and extremely challenging mathematically as complex dynamical systems and networks consists of a large number of interacting units. However, our ideas based on compressive sensing, in combination with innovative approaches, generates a new paradigm that offers the possibility to address the fundamental inverse problem in complex dynamics and networks. In particular, in this talk, I will argue that evolutionary games model a common type of interactions in a variety of complex, networked, natural systems and social systems. Given such a system, uncovering the interacting structure of the underlying network is key to understanding its collective dynamics. Based on compressive sensing, we develop an efficient approach to reconstructing complex networks under game-based interactions from small amounts of data. The method is validated by using a variety of model networks and by conducting an actual experiment to reconstruct a social network. While most existing methods in this area assume oscillator networks that generate continuous-time data, our work successfully demonstrates that the extremely challenging problem of reverse engineering of complex networks can also be addressed even when the underlying dynamical processes are governed by realistic, evolutionary-game type of interactions in discrete time. I will also touch on the issue of detecting hidden nodes, on how to ascertain its existence and its location in the network, this being highly relevant to metabolic networks.

Network reconstruction based on evolutionary-game data via compressive sensing, W.-X. Wang, Y.-C. Lai, C. Grebogi, and J. Ye, Phys. Rev. X 1, 021021 (2011);

Predicting catastrophe in nonlinear dynamical systems by compressive sensing, W.-X. Wang, R. Yang, Y.-C. Lai, V. Kovanis, and C. Grebogi, Phys. Rev. Lett. 106, 154101 (2011);

Forecasting the future: Is it possible for adiabatically time-varying nonlinear dynamical systems? R. Yang, Y.-C. Lai, and C. Grebogi, Chaos 22, 033119 (2012);

Optimizing controllability of complex networks by minimum structural perturbations, W.-X. Wang, X. Ni, Y.-C. Lai, and C. Grebogi, Phys. Rev. E 85, 026115 (2012);

Uncovering hidden nodes in complex networks in the presence of noise, R.-Q. Su, Y.-C. Lai, X. Wang, and Y. Do, Nature Sci. Rep. 4, 3944 (2014)

**John Guaschi** (University of Caen, Caen, France)

*Braid groups and configuration spaces of surfaces*

Braid groups and configuration spaces play an important rôle in low-dimensional dynamics, the former having been used to study the forcing problem for periodic orbits of homeomorphisms of the disc. They are interesting objects in their own right, and admit a number of generalisations that have been widely studied from algebraic and topological points of view. In this talk, we discuss the existence and classification of the torsion of surface braid groups. We show that the finite subgroups of the 2-sphere and the real projective plane are closely related to the symmetry groups of the Platonic solids. This is joint work with D. Gonçalves (São Paulo).

**Guillaume James** (INRIA Grenoble Rhône-Alpes, Grenoble, France)

*Solitary waves in the Burridge-Knopoff model*

The Burridge-Knopoff model is a lattice differential equation describing a chain of blocks connected by springs and pulled over a surface. This model incorporates a nonlinear velocity-dependent friction force between the blocks and the fixed surface. For some classes of non-monotonic friction forces, this system displays a large response to perturbations above a threshold, which is characteristic of excitable dynamics. In some parameter regimes, this response corresponds to the propagation of a solitary wave. We introduce a simplified piecewise linear friction law (reminiscent of the McKean nonlinearity in spiking neuron models) which allows us to prove the existence of large amplitude solitary waves and study their qualitative properties.

(With Jose E. Morales and Arnaud Tonnelier)

**Jeffrey Johnson** (Open University, Milton Keynes, UK)

*New dimensions for network science*

A system is a set of interacting elements. Classically the interaction between two elements can be represented by a directed edge in a network, <a, b>. When there is more than one relation such as R and R’ on a set of vertices, the notation <a, b; R> != <a, b; R’> enables this to be discriminated, as required for multiplex networks. Also the notation opens up the possibility of relational algebra, e.g. <a, b; R > AND <a, b; R’> = <a, b; R AND R’ >. But why is network science so focused on the one-dimensional binary relation? There are many examples of n-ary relations, e.g. <a, b, c; R_proving_a_theorem>, <a, b, c, d; R_playing_bridge> and <a, b, c, … ; R_working_as_a_team>. The multidimensional generalization of the 1-dimensional network edge <a, b > is the simplex, where <a, b, c> is a triangle, <a, b, c, d> is a tetrahedron, and so on. The generalization of simplicial complexes and multiplex networks is the multidimensional hypernetwork which has elements hypersimplices, <a, b, c, … ; R>. These have multidimensional connectivity, generalizing the 0-dimensional connectivity of networks to q-dimensions. It will be shown that graphs, networks, multiplex networks, hypergraphs, simplicial complexes and hypernetworks form a coherent family of mathematical structures. The algebraic and topological structure of these qualitative structures supports and constrains patterns of numbers representing the quantitative system dynamics. Beyond this multidimensional generalization of networks, hypernetworks allow the definition of multilevel systems and provide a formal representation of the coupled dynamics within and between levels. In this representation the dynamics are information-closed at a given level if the phenomenology of a system requires no information below that level. Many social systems are heterogeneous at all levels, and they are only information-closed at the microlevel of interactions between individual people. Hence the need for multilevel modelling at all levels between micro and macro. This talk will sketch these mathematical structures and suggest there is great potential for the science of complex systems in opening up these new dimensions in network science.

**Ralph Kenna** (Coventry University, Coventry, UK)

*Maths Meets Myths: Quantitative Investigations of Ancient Narratives*

In recent years, new techniques have been developed to study a wide variety of complex systems, including social networks. Here, after a very brief outline of the long relationship between statistical physics and the social sciences, we contextualise network theory as opening a new bridge to interdisciplinary collaborations with the humanities. We briefly explain the techniques involved and report on an application to study networks of characters appearing in four ancient narratives: the Anglo-Saxon Beowulf, the Greek Iliad, Njáls Saga from Iceland and the Irish Táin Bó Cúailnge. We determine a number of mathematical characteristics of the social network underlying the societies depicted in these narratives and quantitatively compare them to each other and to other networks, some real, some fictitious. In this manner, we introduce a new way to analyse old material and to develop new perspectives which may help shed mathematical light on our cultural inheritance.

**Konstantin Khanin** (University of Toronto, Toronto, Canada)

*On renormalization for a class of generalized interval exchange transformations*

We shall discuss the problem of renormalization and rigidity for circle maps with multiple break points. Such maps form a class of generalized interval exchange transformation. We shall present recent results and conjectures in this direction. We shall also discuss a connection with the problem considered by Robert 25 years ago.

**Greg King** (ICM, Barcelona, Spain)

*Turbulent cascades and intermittency in winds over the Tropical Pacific*

Under typical conditions, turbulent fluid motions are three-dimensional and energy cascades from large scales to small scales. However, in the atmosphere over the range of scales governing weather phenomena (the mesoscales: 2-2000km), geophysical constraints (stratification, rotation, thin atmosphere) decouple motions into layers. This quasi-two-dimensional flow motivates a picture of stratified turbulence with an upscale cascade (from small scales to large scales). To test this picture of turbulence requires an observational dataset of global winds -- an enormous undertaking. Attempts to provide a definitive answer on the cascade direction eluded investigators until Erik Lindborg (1999) proposed a test based on Kolmogorov's third-order structure

function law (the most rigorous result in turbulence theory). This test, when applied to a dataset of global upper troposphere winds, indicated, to great surprise, that the cascade was downscale.

In this talk I will describe the application of the third-order structure function test to the mesoscale winds over the Tropical Pacific Ocean. The winds we studied were measured from space by instruments (called scatterometers) carried on the NASA QuikSCAT satellite and the European MetOp-A satellite. Our analysis supplied further surprises: evidence for both upscale and downscale cascades, depending on geographical region and season. Our results show that turbulence models need to include information about air-sea interaction.

**Jacques Laskar** (CNRS, Observatoire de Paris, Paris, France)

*In search of new planets*

Until 1781 and the discovery of Uranus by Herschel, the only known planets were those visible to the naked eye: Mercury, Venus, Earth, Mars, Jupiter and Saturn. The discovery of Neptune by Le Verrier and Galle in 1846 was particularly striking because Le Verrier was able to predict the position of Neptune by only calculus. Many have then sought to renew this feat, but the discovery of Pluto was due to chance and, to the misfortune of its discoverers, Pluto has been downgraded to dwarf planet by the International Astronomical Union in 2006.

However, since 1995 and the discovery of 51 Pegb by Michel Mayor and Didier Queloz, new planets have been found by thousands around nearby stars.

Everything could stop there, but the discovery in 2014 of the 2012VP113 Kuiper belt object by Sheppard and Trujillo has raised the issue of an additional planet in the solar system, a possible Planet 9.

Indeed, 2012VP113 has orbital elements close to those of Sedna, another distant massive object, with a very eccentric orbit. The similarity of the orbits of Sedna and 2012VP113 led to assume the existence of an additional planet, hypothesis reinforced by the publication last January of a dynamical study making the scenario plausible. Since then, a part of the astronomical community has embarked on the hunt for Planet 9.

**Roberto Livi** (Università degli Studi Firenze, Firenze, Italy)

*Discrete breathers and negative temperature state in the DNLS equation*

We discuss the statistical behaviour of the discrete nonlinear Schrodinger equation as a test bed for the observation of negative-temperature (i.e. above infinite temperature) states in Bose–Einstein condensates in optical lattices and arrays of optical waveguides. By monitoring the microcanonical temperature, we show that there exists a parameter region, where the system evolves towards a state characterized by a finite density of discrete breathers and a negative temperature.

**Rafael de la Llave** (Georgia Tech, Atlanta, USA)

*A posteriori approach in KAM theory: Rigorous results, efficient algorithms.*

KAM theory studies the persistence of quasi-periodic solutions.

We present some recent results stated in an a-posteriori format. The existence of quasi-periodic solutions is formulated as existence of solution of a functional equation. We show that if there exists an approximate solution, there is a true solution close by. The method of proof also leads to very efficient algorithms, which work even far from integrable systems. The a-posteriori format also allows to assess the reliability even close to the breakdown, where many false positives abound.

These algorithms have been implemented in the case of symplectic maps and "conformally symplectic maps" and lead to the observation of scaling properties (See A. Celletti's talk).

Joint work with R. Calleja and A. Celletti.

**Jaume Llibre** (Autonomous University of Barcelona, Bellaterra, Spain)

*On the real Jacobian conjecture
*Abstract

**João Lopes Dias** (University of Lisbon, Lisbon, Portugal)

*Renormalisation of vector fields with Gevrey regularity*

We will discuss the construction of renormalisation operators acting on the space of Gevrey vector fields on the torus. The convergence of the renormalisation implies the conjugacy to a constant vector of Brjuno-like arithmetical type. This method is based on original ideas by Robert MacKay in the 90’s.

**Stefano Luzzatto** (ICTP, Trieste, UK)

*Sinai-Ruelle-Bowen measures for nonuniformly hyperbolic surface diffeomorphisms.*

In the 1970’s, Sinai, Ruelle and Bowen, constructed Markov partitions for uniformly hyperbolic systems and used these to prove the existence of a special class of physically relevant measures, which we now call SRB measures. At the same time, Pesin introduced and studied the basic geometric properties of a much more general class of systems which we call nonuniformly hyperbolic systems, and a basic open question has been whether such systems also admit Markov partitions and SRB measures. A positive answer to this question has been conjectured by Viana in his ICM address in 1998.

In joint work with V. Climenhaga and Y. Pesin we prove that nonuniformly hyperbolic surface diffeomorphisms with some natural recurrence assumptions admit a Young tower, which naturally generalises Markov partitions, and that this can be used to prove the existence of SRB measures.

**Robert MacKay** (University of Warwick, Coventry, UK)

*Towards a spectral proof of Riemann’s hypothesis*

Riemann’s hypothesis is that all the zeroes of his ξ function are real. Because ξ is even, this is equivalent to saying that all the zeroes of Ξ(E) = ξ(2√E) are real and non-negative. Extending a suggestion attributed to Polya and Hilbert, we seek a Hermitian operator H such that the functional determinant of H-E is Ξ(E) for all complex E, which would prove Riemann’s hypothesis. I have not found it yet, but will outline arguments that the magnetic Laplacian on a surface of curvature -1 with magnetic field 9/4, a cusp of width 1 and a flux tube at i, comes close. It governs the quantum dynamics of a charged particle on the surface.

**James Meiss** (University of Colorado, Boulder, USA)

*Stickiness, Drift and Diffusion in Volume-Preserving Maps*

Most nearly integrable Hamiltonian systems and symplectic maps have many invariant tori, as shown by KAM theory, and their actions do not drift very far on exponentially long time scales, as shown by Nekhoroshev theory. How do these phenomena generalize to the incompressible/volume-preserving context?

In this situation, an integrable system most naturally corresponds to one in an action-angle form, though there need not be an equal number of actions and angles. Moreover, the existence of invariant actions is no longer tied directly to symmetries. Nevertheless, as was shown by Chen & Sun and Xia, KAM theory can apply to volume-preserving maps, though the tori may no longer be identified by their fixed frequencies. We will show

how tori can be computed for several examples, and the break-up of tori can be computed using singular values of the conjugacy.

Does Nekhoroshev theory apply for volume-preserving maps? We will show that even in the two-action, two-angle context, that it does not. The chaotic drift of action variables can often be characterized by a diffusion coefficient, though, as is well known in the area-preserving context, this motion can also be sub-diffusive or super-diffusive due to the stickiness of tori. We investigate this phenomena is the volume-preserving context.

**Ben Mestel** (Open University, MIlton Keynes, UK)

*The barrier billiard renormalization strange set: golden mean trajectory*

The analysis of correlations in symmetric barrier billiards for the golden mean trajectory

is characterized by the action of a renormalization operator on the space of piecewise

constant functions taking the values $\pm 1$. We describe a renormalization strange set for this operator

in function-pair space, dubbed the barrier billiard renormalization strange set.

(joint work with Luke Adamson and Andrew Osbaldestin)

**Mauro Mobilia** (University of Leeds, Leeds, UK)

*Switching dynamics and non-equilibrium steady state in nonlinear q-voter models with zealotry*

*The importance of relating "micro-level" interactions with "macro-level" phenomena in modeling social dynamics is well established. The study of parsimonious models like the voter model and its variants, commonly used in statistical mechanics, has therefore received a growing interest in the last decades. In this talk, I will discuss the properties of two variants of the nonlinear q-voter model with inflexible zealots in a finite well-mixed population. I will consider a two-opinion model in which each individual supports one of two parties and is either a zealot or a susceptible voter of type q1 or q2. While zealots never change their opinion, a qi-susceptible voter (i=1,2) consults a group of qi neighbours at each time step, and adopts their opinion if all group members agree. When q1=q2 >1, the detailed balance is satisfied and the model is characterized by a bimodal distribution below a critical zealotry density [1]. The long-time dynamics is thus driven by fluctuations and after a characteristic time, most susceptibles become supporters of the party having more zealots (with an asymmetric opinion distribution). When the number of zealots of both parties is the same, susceptibles endlessly swing from the state where they all support one party to the opposite state. When q2 and q1 are not equal, the so-called 2qVZ model violates the detailed balance and its non-equilibrium stationary state is characterized by its probability distribution and currents in the distinct regimes of low and high zealotry density. In particular, the opinion distribution and the probability currents, as well as the unequal-time two-point correlations are computed [2]. This is a joint work with Andrew Mellor (Leeds) and Royce Zia (Virginia Tech). Refs.: [1] M. Mobilia, Physical Review E 92, 012803 (2015); e-print: http://arxiv.org/abs/1506.04911; [2] A. Mellor, M. Mobilia, R.K.P. Zia, EPL 113, 48001 (2016); e-print: http://arxiv.org/abs/1601.03766*

**Mark Muldoon** (University of Manchester, Manchester, UK)

*Scores, parameters and the single cell*

In this talk I will discuss a problem that arose out of the study of cell-to-cell signalling in the mammalian immune system. I’ll introduce the biological system

and then describe the methods invented by my collaborators to infer the parameters of an ODE model of the system. Their approach involves the study of certain “scores”: features extracted from beautiful experiments that involve following fluorescently-tagged molecules as they move about in single cells and I will finish by setting their approach in a more general mathematical framework that permits one to infer things about the distribution of biochemical properties across the whole population of cells.

**Diogo Pinheiro** (City University of New York, New York City, USA)

*Optimal Control of dynamical systems with multiple sources of uncertainty*

I will discuss optimal control problems associated with continuous-time stochastic dynamical systems containing multiple sources of randomness. The focus will be on the discussion of generalized dynamic programming principles and the corresponding Hamilton-Jacobi-Bellman equations.

**Alberto Pinto** (University of Porto, Porto, Portugal)

*Dynamics of Human decisions.*

We study a dichotomous decision model, where individuals can make the decision yes or no and can influence the decisions of others. We characterize all decisions that form Nash equilibria. Taking into account the way individuals influence the decisions of others, we construct the decision tilings where the axes reflect the personal preferences of the individuals for making the decision yes or no. These tilings characterize geometrically all the pure and mixed Nash equilibria. We show, in these tilings, that Nash equilibria form degenerated hysteresis with respect to the dynamics, with the property that the pure Nash equilibria are asymptotically stable and the strict mixed equilibria are unstable. These hysteresis can help to explain the sudden appearance of social, political and economic crises. We observe the existence of limit cycles for the dynamics associated to situations where the individuals keep changing their decisions along time, but exhibiting a periodic repetition in their decisions.

**Colin Rourke** (University of Warwick, Coventry, UK)

*How black holes feed and grow into spiral galaxies*

There is a basic problem with black holes "grazing" on a surrounding medium and thereby growing: angular momentum. If infalling gas or plasma has even a very small AM about the BH then, as it approaches the centre, conservation of AM causes an arbitrary increase in tangential velocity which limits the closest approach. A gas/plasma behaves like a single particle in this respect. Thus it seems that, contrary to natural intuition, BHs cannot grow by grazing on the medium which surounds them.

In this talk I suggest a mechanism whereby grazing can take place: if the BH is rotating at just the correct velocity to match the infalling gas, then dragging of inertial frames, caused by the rotation, changes the AM, which is measured in the local inertial frame, and cancels it out, allowing the gas to proceed into the BH. Moreover there is a natural feedback mechanism which locks the rotation of the BH to match the surrounding medium. This allows the BH to grow bigger until the process reaches a natural limit depending on the size of the BH and the density of the surrounding medium.

If the BH is too big or the surrounding medium too dense, the rate of accretion can exceed the amount of matter which is absorbed by the BH in this way and the surplus forms a toroidal accretion structure, similar to an accretion disc in conventional theory, which stores the excess AM. This accretion torus then grows until it becomes unstable and the resulting explosions radiate away some of the AM causing rotation of the BH to unlock from the local AM and thereby increasing the effective AM in the torus and feeding back into the energy stored in the torus. The explosions naturally occur at two antipodally opposite points and the resulting streams of matter form themselves into the familiar spiral arms. I have a Mathematica notebook which models spiral structure (and also the ubiquitous rotation curve) based on this model. Full details are on my webpage at: http://msp.warwick.ac.uk/~cpr/paradigm/

**David Sanders** (National Autonomous University of Mexico, Mexico City, Mexico, and MIT, Cambridge, USA)

*Rigorously finding periodic orbits of billiard models*

We show how interval arithmetic, and validated numerical techniques that build on it, may be used to rigorously find enclosures of periodic orbits of billiard models, including open systems in which trajectories may escape.

The algorithms are implemented in the Julia programming language, in particular in the ValidatedNumerics.jl package. Julia provides a unique combination of interactive usability and performance.

Joint work with Nikolay Kryukov (Faculty of Sciences, UNAM) and Luis Benet (Institute of Physical Sciences, UNAM)

**Jacques-Alexandre Sepulchre** (INLN, Nice, France)

*A discrete map approach to steady states of intracellular signaling cascades*

In living organisms, covalent modification cycles are one of the major intracellular signaling mechanisms, allowing to transduce physical or chemical stimuli of the external world into variations of activated biochemical species within the cell. Kinase cascades are sequences of such cycles, in which the activated protein in one tier promotes the activation of the protein in the next one.

Signaling cascades are generally characterized by their simulus-response curves, quantifying how activated proteins at steady state vary in function of the dose of some ligand activating the top of the cascade.

In this talk we present a novel approach to describe the behaviors of such stimulus-responses, by means of a discrete dynamical system. Indeed the tiers of a signaling cascade with n layers are coupled 3 by 3, allowing to make a correspondence between the steady states and a piece of orbit of a 2-D discrete map. The study of its phase portrait, and in particular of the contraction/expansion properties around the fixed points of this map, and their bifurcations, can yield biological interpretations. In particular this approach enables to highlight the idea, well known in the context of signaling pathways, that one biological role of signaling cascades is to amplify or to attenuate external signals. The method allows also to pinpoint some consequences of the retroactivity in signaling cascades, a feature that is generally overlooked in these systems.

**Siniša Slijepčević** (University of Zagreb, Zagreb, Croatia)

*Lyapunov functions on spaces of measures and phase transitions*

*We consider the problem of uniqueness of stable invariant measures in some deterministic and stochastic complex systems. Standard examples of Lyapunov functions on the spaces of measures on the phase space (implying stability of measures minimising or maximising them) include entropy for many systems, as well as the Foster-Lyapunov functions for Markov chains.*

*We give examples of new Lyapunov functions on the spaces of measures in some complex systems, closely related to the phase transitions observed in them. For dissipative Frenkel-Kontorova models (as well as for parabolic semilinear scalar partial differential equations on unbounded domains), the appropriate Lyapunov function is the average number of self-intersections with respect to a measure. This leads to a description of all space-time invariant measures, and a characterisation of dynamical Aubry transitions. For pulsating (or ratchet) Frenkel-Kontorova models, the variance of the mean spacing is a Lyapunov function in the "off-phase", with explicitly known decay. This enables obtaining lower bounds on the transport speed. We also give related examples in some probabilistic cellular automata.*

**Charles Tresser** (Aperio, Ramat Hahayal, Israel)

*On a question of B. Riemann, a recorded remark of Napoleon, and a simple interpretation of Quantum Mechanics*

PAM Dirac almost got the right way to interpret QM, and made real progresses along his career as physicist (before turning astrophysicist). Would he have known a question raised by B. Riemann in his Abilitation memoir (the part on geometry, translated into English in Spivak's volume on Riemannian Geometry (and into French before that) he would have probably been in a much better position. What Riemann asked is "does geometry make physical sense at small enough scale?". As reported by Etienne Geoffroy St. Hilaire, before Riemann (just before the end of the XVIIIth century), Napoleon stated to Gaspard Monge that microscopic physics would be different and more fundamental than Newtonian physics (and that this is what he would have taken care of had he not become a soldier).

Because Quantum Physics, even once on firm bases for the non-relativistic part which will be our main worry, is hard to "understand".

As a result, also because of some unfounded credos, too many people are pushing towards non-locality, multiverses, and other gates to easy drifts out of science. Lack of realism is too hard to swallow. I will make the case that once one takes seriously what is most probably an intuition of Riemann, (i.e., a "NO" answer to his question reported above), everything becomes very simple. The problems/questions/issues/etc. fall into two classes: those which can be understood, and those that one can understand as not understandable for a good reason. A new version of the myth of the cave (Plato'sRepublic (514a–520a)) where the philosopher also fails to see all the light but, contary to others, understands why. Lacking Riemann's question, even big Heros of non-realism have often occasionally fallen into the realism trap (enough to be confused). Examples will be versions of the double-slit experiment, including the basic phenomenology and a version proposed by Einstein to Bohr, effectively as launch of the "Welcher Weg" controversy: I will explain why, despite Einstein being wrong, Bohr's answer was not to the point. The solution is much simpler as I will explain. Among consequences of these views, many puzzles are easily solved, e.g., the moon is there even when you do not take you turn in guarding her in place, while nothing of that type works at the atomic scale and below. I will also explain why what was considered by Heisenberg as a the main philosophical issue raised by Quantum Mechanics (the need of macroscopic measurement apparatus that prevents Quantum Mechanics from being self-contained, except if one considers measurements as inessential, falling then back toward the worse of Descartes views) transforms into a non-issue with the help of Riemann's question. Many questions remain, but the overall view becomes rather simple. I will try to share my views on how geometry builds up at larger scales.