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## Über dieses Buch

The two parts of the present volume contain extended conference abstracts corresponding to selected talks given by participants at the "Conference on Hamiltonian Systems and Celestial Mechanics 2014" (HAMSYS2014) (15 abstracts) and at the "Workshop on Virus Dynamics and Evolution" (12 abstracts), both held at the Centre de Recerca Matemàtica (CRM) in Barcelona from June 2nd to 6th, 2014, and from June 23th to 27th, 2014, respectively. Most of them are brief articles, containing preliminary presentations of new results not yet published in regular research journals. The articles are the result of a direct collaboration between active researchers in the area after working in a dynamic and productive atmosphere.

The first part is about Central Configurations, Periodic Orbits and Hamiltonian Systems with applications to Celestial Mechanics – a very modern and active field of research. The second part is dedicated to mathematical methods applied to viral dynamics and evolution. Mathematical modelling of biological evolution currently attracts the interest of both mathematicians and biologists. This material offers a variety of new exciting problems to mathematicians and reasonably inexpensive mathematical methods to evolutionary biologists. It will be of scientific interest to both communities.

The book is intended for established researchers, as well as for PhD and postdoctoral students who want to learn more about the latest advances in these highly active areas of research.

## Inhaltsverzeichnis

### On the Force Fields Which Are Homogeneous of Degree −3

Abstract
Soon after establishing the famous properties of the 1∕r 2 law of force, Newton described a spiraling orbit of a particle under a central force in 1∕r 3. He also noticed that the addition of a force in 1∕r 3 to another force results in a kind of precession of the orbit, see [14, Book 1, Proposition 44]. In 1842, Jacobi [8] gave general results about the force fields which are homogeneous of degree − 3 and derived from a potential. More recently, Montgomery [12] gave an impressive description of the dynamics of the planar 3-body problem with a force in 1∕r 3. Such homogeneity of the force also appears in Appell’s projective dynamics, where the force is considered together with a constraint, see [2].
Alain Albouy

### Bifurcations of the Spatial Central Configurations in the 5-Body Problem

Abstract
A configuration of n particles is called central when the acceleration vector of each particle is a common scalar multiple of its position vector. One of the reasons why central configurations are interesting is that they allow us to obtain explicit homographic solutions of the n-body problem, that is, motions where the configuration of the system changes size but keeps its shape. Also, they are important in the study of total collisions.
Martha Álvarez-Ramírez, Motserrat Corbera, Jaume Llibre

### Convex Central Configurations of Two Twisted n-gons

Abstract
The simplest motions that can be found in the Newtonian N-body problem are the ones whose configuration is constant up to rotations and scaling, and every body follows a trajectory being a keplerian orbit. Such kind of solutions are called central configurations.
Esther Barrabés, Josep Maria Cors

### The Newtonian n-Body Problem in the Context of Curved Space

Abstract
The idea that geometry and physics are intimately related made its way in human thought during the early part of the nineteenth century. Gauss measured the angles of a triangle formed by three mountain peaks near Göttingen, Germany, apparently hoping to learn whether the universe has positive or negative curvature, but the inevitable observational errors rendered his results inconclusive [3]. In the 1830s, Bolyai and Lobachevsky took these investigations further. They independently addressed the connection between geometry and physics by seeking a natural extension of the gravitational law from Euclidean to hyperbolic space. Their idea led to the study of the Kepler problem and the 2-body problem in spaces of nonzero constant Gaussian curvature, κ ≠ 0, two fundamental problems that are not equivalent, unlike in Euclidean space. A detailed history of the results obtained in this direction since Bolyai and Lobachevsky can be found in [3, 5, 6].
Florin Diacu

### Poincaré Maps and Dynamics in Restricted Planar (n + 1)-Body Problems

Abstract
This work deals with the motion of an infinitesimal particle, the secondary, in a plane subject to the gravitational attraction of n particles of mass m = 1, the primaries, which are placed in the vertices of a regular polygon on n vertices. The primaries can be fixed or rotate with an uniform velocity around their center of mass. The first case is called the n-center problem, and the second the restricted (n + 1)-body problem. The last case has been studied in [1], in this note we will mainly study the first one.
Antonio Garcí;a

### A Methodology for Obtaining Asymptotic Estimates for the Exponentially Small Splitting of Separatrices to Whiskered Tori with Quadratic Frequencies

Abstract
The aim of this work is to provide asymptotic estimates for the splitting of separatrices in a perturbed 3-degree-of-freedom Hamiltonian system, associated to a two-dimensional whiskered torus (invariant hyperbolic torus) whose frequency ratio is a quadratic irrational number. We show that the dependence of the asymptotic estimates on the perturbation parameter is described by some functions which satisfy a periodicity property, and whose behavior depends strongly on the arithmetic properties of the frequencies.
Amadeu Delshams, Marina Gonchenko, Pere Gutiérrez

### Homoclinic and Heteroclinic Orbits for a Class of Singular Planar Newtonian Systems

Abstract
The study of existence and multiplicity of solutions of differential equations possessing a variational nature is a problem of great meaning since most of them derives from mechanics and physics. In particular, this relates to Hamiltonian systems including Newtonian ones. During the past 30 years there has been a great deal of progress in the use of variational methods to find periodic, homoclinic and heteroclinic solutions of Hamiltonian systems. Hamiltonian systems with singular potentials, i.e., potentials that become infinite at a point or a larger subset of $$\mathbb{R}^{n}$$, are among those of the greatest interest. Let us remark that such potentials arise in celestial mechanics. For example, the Kepler problem with
$$\displaystyle{V (q) = - \frac{1} {\vert q -\xi \vert }}$$
has a point singularity at $$\xi$$ ($$q \in \mathbb{R}^{n}\setminus \{\xi \}$$). In physics, the gradient ∇V of the gravitational potential is called a weak force.
Joanna Janczewska

### Transport Dynamics: From the Bicircular to the Real Solar System Problem

Abstract
The main goal is to give an explanation of transport in the Solar System based in dynamical systems theory. More concretely, we consider as an approximation of the Solar System, a chain of independent Bicircular problems in order to get a first insight of transport in this simplified case. Each bicircular problem (BP) consists of the Sun (S), Jupiter (J), a planet and an infinitesimal mass.
Mercè Ollé, Esther Barrabés, Gerard Gómez, Josep Maria Mondelo

### Quasi-Periodic Almost-Collision Motions in the Spatial Three-Body Problem

Abstract
We deal with the spatial three-body problem in the various regimes where the Hamiltonian is split as the sum of two Keplerian systems plus a small perturbation. This is a region of the phase space $$T^{{\ast}}\mathbb{R}^{6}$$ where the perturbation is small [3], the so called perturbing region $$\mathcal{P}_{\varepsilon,n}$$. In particular, we prove the existence of quasi-periodic motions where the inner particles describe bounded near-rectilinear trajectories whereas the outer particle follows an orbit lying near the invariable plane. These motions fill in five-dimensional invariant tori. Moreover, the inner particles move in orbits either near an axis perpendicular to the invariable plane or near the invariable plane.
Jesús F. Palacián, Flora Sayas, Patricia Yanguas

### Generalized Discrete Nonlinear Schrödinger as a Normal Form at the Thermodynamic Limit for the Klein–Gordon Chain

Abstract
A still open challenge in Hamiltonian dynamics is the development of a perturbation theory for Hamiltonian systems with an arbitrarily large number of degrees of freedom and, in particular, in the thermodynamic limit. Indeed, motivated by the problems arising in the foundations of Statistical Mechanics, it is relevant to consider large systems (e.g., for a model of a crystal the number of particles should be of the order of the Avogadro number) with non vanishing energy per particle (which corresponds to a non zero temperature in the physical model).
Simone Paleari, Tiziano Penati

### Stability of Euler-Type Relative Equilibria in the Curved Three Body Problem

Abstract
We consider three point particles of masses m 1, m 2, m 3 moving on a two-dimensional surface of constant curvature k. It is well known that, locally, these surfaces are characterized by the sign of the curvature k. If k > 0, the surface is the two dimensional sphere S 2 of radius R = 1∕k embedded in the Euclidian space $$\mathbb{R}^{3}$$.
Ernesto Pérez-Chavela, Juan Manuel Sánchez Cerritos

### Two-Dimensional Symplectic Return Maps and Applications

Abstract
The goal of this extended abstract is to show how return maps, even in simple cases, can provide accurate information in some dynamical aspects.
Regina Martínez, Carles Simó

### Central Configurations of an Isosceles Trapezoidal Five-Body Problem

Abstract
The study of central configurations is very popular for producing the simplest solutions of the planar n-body problems (cf., [1, 2, 4]). In this paper, we study the central configuration of the isosceles trapezoidal five-body problem where four of the masses are placed at the vertices of the isosceles trapezoid and the fifth body can take various positions on the axis of symmetry. We identify regions in the phase space where it is possible to choose positive masses which will make the configuration central. A similar approach was adopted by Shoaib et al. in [3] for the rhomboidal five-body problem.
Abdulrehman Kashif, Muhammad Shoaib, Anoop Sivasankaran

### The Discrete Hamiltonian–Hopf Bifurcation for 4D Symplectic Maps

Abstract
We consider a family of real-analytic symplectic four-dimensional maps $$F_{\tilde{\nu }}$$, $$\tilde{\nu }\in \mathbb{R}^{p}$$, p ≥ 1, with respect to the standard symplectic two-form $$\Omega = dx_{1} \wedge dy_{1} + dx_{2} \wedge dy_{2}$$, where (x 1, x 2, y 1, y 2) denote the Cartesian coordinates.
Ernest Fontich, Carles Simó, Arturo Vieiro

### Moment Map of the Action of $${SO(3)}$$ on $$\mathbb{R}^{3} \times \mathbb{R}^{3}$$

Abstract
The aim of this extended abstract is to expose the main results of the moment map of the action of SO(3) on the cotangent bundle of $$\mathbb{R}^{3}$$.
José Antonio Villa Morales

### Modelling Infection Dynamics and Evolution of Viruses in Plant Populations

Abstract
Mathematical models have been used extensively to analyse and/or predict the dynamics of pathogen infection in host populations, as well as the evolution of key pathogen traits, notably infectivity and virulence. Model analyses have been very useful in identifying factors that affect infection dynamics and pathogen evolution, and in predicting their effects under different scenarios. However, a serious shortcoming of theoretical analyses is that often there is not enough information on how realistic the underlying assumptions are, and very often there is a serious lack of information on the range of values of key model parameters. An example is the classical susceptible-infected-recovered (SIR) model, first proposed by Kermack and McKendrick [8] in 1927, and becoming the basis to predict virulence evolution. A central assumption of this model is that both virulence, defined as the effect of infection on host mortality, and the rate of transmission to new hosts, are positively correlated with the within-host multiplication rate of the pathogen, so that a trade-off between virulence and transmission is established to optimize the intrinsic reproduction value. Interestingly, a positive correlation between virulence and within-host multiplication has been demonstrated in few host-parasite systems, and seems not to be the rule for the whole classes of parasites, including plant viruses [10], which has not discouraged the use (and the utility) of SIR-based evolutionary models.
Aurora Fraile, Fernando García-Arenal

### The Spread of Two Viral Strains on a Plant Leaf

Abstract
Our objective is to construct a mathematical model for the spread of two subtypes (a wild type and a mutant) of a virus on a plant leaf.
Juan Carlos Cantero-Guardeño, Vladimir Sobolev, Andrei Korobeinikov

### Tracking the Population Dynamics of Plant Virus Escape Mutants

Abstract
One of the most challenging problems in agronomy is to obtain plants that are resistant to the infection of pathogens. Not only this, but also that resistance must be as durable as possible. Unfortunately, most, if not all, strategies to generate such resistant plants have been overcome by the tremendous evolutionary potential of viral pathogens. In recent years, a new strategy based on the transgenic expression of artificial micro-RNAs (amiRs), designed to target viral genomes and induce their degradation, has been developed. This resistance has proven to be highly effective and sequence-specific against several plant viruses infecting Arabidopsis thaliana [7]. However, before these transgenic plants can be deployed in the field, it was important to evaluate the likelihood of the emergence of resistance-breaking mutants [2, 5]. Two issues were of particular interest: (1) whether such mutants can arise in non-transgenic plants that may act as reservoirs for the viral populations and (2) whether a suboptimal expression level of the transgene, resulting in sub-inhibitory concentrations of the amiR, would favor the emergence of escape mutants.
Santiago F. Elena

### Evolutionary Escape in Populations with Genotype-Phenotype Structure

Abstract
Evolutionary escape is the process whereby a population under sudden changes in the selective pressures acting upon it try to evade extinction by evolving from previously well-adapted phenotypes to those that are favoured by the new selective pressure. This evolutionary process is driven by gene mutations. Some examples are: (1) viruses evading anti-microbial therapy, (2) cancer cells escaping from chemotherapy, (3) parasite infecting a new host, and also (4) species trying to invade a new ecological niche.
Esther Ibáñez-Marcelo, Tomás Alarcón

### Evolution of Stalk/Spore Ratio in a Social Amoeba: Cell-to-Cell Interaction via a Signaling Chemical Shaped by Cheating Risk

Abstract
The cellular slime mold, or social amoeba, exists as a unicellular form that divides and multiplies rapidly. When food is depleted, cells aggregate to form a fruiting body within which cells differentiate into spores and stalks. Some spores succeed in dispersing to a new micro-habitat with plenty of food and in resuming a unicellular phase with fast population growth. In contrast, stalk cells lift spores to aid in their dispersal and then die, see Fig. 1 (left). Becoming a stalk cell is an altruistic behavior, see [2]. This system provides an ideal system for studying the maintenance of altruism.
Yoh Iwasa

### Within-Host Viral Evolution Model with Cross-Immunity

Abstract
In the last 20 years, a considerable amount of research has been done in order to mathematically study the dynamics of viruses and immunity, and HIV in particular. A mathematical model of within-host dynamics of HIV, which incorporates random mutations modelled by diffusion in a continuous one dimensional strain space and postulates that immune response is phenotype specific, shows that for a fast evolving virus the phenotype specific immune response is not able to clear HIV from its host. In this contribution, we develop the model further by including a cross-immunity, rather than specific immunity, modelled by a weight function which represents the broadness of cross-immunity. Numerical simulations show that if the cross-immunity is sufficiently broad, cell mediated immune response is able to clear a virus from its host.
Narani van Laarhoven, Andrei Korobeinikov

### Modelling Viral Evolution and Adaptation

Abstract
Viral populations are extremely plastic [5]. They maintain and steadily generate high levels of genotypic and phenotypic diversity that result in the coexistence of several different viral types in quasi-species, and eventually constitute a powerful tool to deploy different adaptive strategies. The interest in understanding and formally describing viral populations has steadily increased. At present, there are major unknown factors that difficult the construction of realistic models of viral evolution, as the way in which mutations affect fitness [19] or, in a broader scenario, which is the statistical nature of viral fitness landscapes. Our understanding of viral complexity is however improving thanks to new techniques as deep sequencing [17] or massive computation, and to systematic laboratory assays that reveal that, as other complex biological systems (e.g., cancer or ecosystems) the term virus embraces a dissimilar collection of populations with a remarkable ensemble of evolutionary strategies. New empirical data and improved models of viral dynamics are clearing up the role played by neutral networks of genotypes [21], by defective and cooperative interactions among viral mutants [13], by co-evolution with immune systems [22], or by changes in host populations [10], to cite but a few examples. Models of viral evolution are steadily improving their accuracy and becoming more competent from a conceptual and a predictive viewpoint [11, 12]. Here, we review some examples were well-motivated models of viral evolution succeed at capturing experimentally described features of those populations. Such are the relationship between intra-species competition and the geometry of the propagating substrate of a viral infection [3], the origin of bipartite viral genomes [8], and the adaptation to multi-drug therapies [9, 16].
Susanna Manrubia

### Changes in Codon-Pair Bias of Human Immunodeficiency Virus Type 1 Affect Virus Replication

Abstract
The standard genetic code consists of 64 codons for a set of 20 amino acids and the stop signal, showing its redundancy (except for tryptophan and methionine) and implying that several synonymous triplets encode for the same amino acid. Usually, the position of such synonymous codons is not constant along the protein coding sequences and therefore, their properties are not entirely comparable. Relative amounts of tRNAs iso-acceptors have been associated to codon usage patterns in different organisms. Codon usage bias, the frequency of occurrence of synonymous codons in coding DNA, in human cells is limited by amino acids needed for protein structure/function and by genome signatures (dinucleotide relative abundances). In contrast, translational and transcriptional influences appear to play a minor role in human codon usage [5].
Miguel Ángel Martínez

### Competing Neutral Populations of Different Diffusivity

Abstract
The possibility of moving in space is fundamental for the survival of many biological organisms. While movement patterns can sometimes be complex, reflecting evolutionary strategies to search for food, in many settings movement can be mathematically described as Brownian motion. Macroscopically, this leads to a description in terms of a diffusion equation, or a Fisher equation [2, 3] if birth-death dynamics is also taken into account.
Simone Pigolotti

### Density-Dependent Diffusion and Epidemics on Heterogeneous Metapopulations

Abstract
Systems with many components (individuals or local populations as cities, or metropolitan areas, or regions, ) connected by non-trivial associations or relationships can be statistically described by means of the formalism of complex networks which is based on descriptors like degree distributions, degree-degree correlations, etc. In the last years, many researchers from different fields have been using different approaches to model processes taking place on complex networks.
Albert Avinyó, Marta Pellicer, Jordi Ripoll, Joan Saldaña

### Are Viral Blips in HIV-1-Infected Patients Clinically Relevant?

Abstract
HIV infection has a disturbing feature, after being treated by administration of highly active anti-retroviral therapy (HAART), plasma viral load decays below the detection threshold of standard clinical assays ($$\sim$$ 50 copies RNA∕mL) but appears to fail to completely eradicate the infection, a residual viral load (detectable only by supersensitive assays) persists in plasma. An evidence that the virus is not completely suppressed is the observation of the so-called vira blips, transient episodes of viremia where the viral load raises above the standard test detection limit for a brief period of time. The origin and clinical relevance of these blips remains unclear. There are several studies that have compiled evidence against viral blips being correlated with virological failure.
Daniel Sánchez-Taltavull, Tomás Alarcón

### Models of Developmental Plasticity and Cell Growth

Abstract
In this note we discuss the following topics:
1.
Epigenetics: How to alter your genes? This is evolution within a lifetime. Epigenetics is a relatively new scientific field; research only began in the mid nineties, and has only found traction in the wider scientific community in the last decade or so. We have long been told our genes are our destiny. But it is now thought a genotype’s expression (that is, its phenotype), can change during its lifetime by habit, lifestyle, even finances. What does this mean for our children? So we consider phenotype change:
(a)
firstly in a stochastic setting, where we consider the expected value of the mean fitness;

(b)
then we consider a Plastic Adaptive Response (PAR) in which the response to an environmental cue is initiated after a period of waiting;

(c)
finally, we consider the steady-fitness states, when the phenotype is modelled on a continuous scale providing a structured variable to quantify the phenotype state.

2.
Consider the steady-size distribution of an evolving cohort of cells and therein establish thresholds for growth or decay of the cohort.

Graeme Wake
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