Bifurcation Theory and Spatio-Temporal Pattern Formation

At the 1900 International Congress of Mathematicians, held that year in Paris, the German mathematician David Hilbert put forth a list of 23 problems that he saw as being the greatest challenges for twentieth-century mathematics. Hilbert's 10th problem, to find a method (what we now call an algorithm) for deciding whether a Diophantine equation has an integral solution, was solved by Yuri Matiyasevich in 1970. Proving the undecidability of Hilbert's 10th problem is clearly one of the great mathematical results of the century.This book presents the full, self-contained negative solution of Hilbert's 10th problem. In addition it contains a number of diverse, often striking applications of the technique developed for that solution (scattered previously in journals), describes the many improvements and modifications of the original proof - since the problem was unsolved 20 years ago, and adds several new, previously unpublished proofs.Included are numerous exercises that range in difficulty from the elementary to small research problems, open questions,and problems. Each chapter concludes with a commentary providing a historical view of its contents. And an extensive bibliography contains references to all of the main publications directed to the negative solution of Hilbert's 10th problem as well as the majority of the publications dealing with applications of the solution.Intended for young mathematicians, Hilbert's 10th Problem requires only a modest mathematical background. A few less well known number-theoretical results are presented in the appendixes. No knowledge of recursion theory is presupposed. All necessary notions are introduced and defined in the book, making it suitable for the first acquaintance with this fascinating subject.Yuri Matiyasevich is Head of the Laboratory of Mathematical Logic, Steklov Institute of Mathematics, Russian Academy of Sciences, Saint Petersburg.

In the present paper, we study a new kind of p-adic measures for q + 1-state Potts model, called p-adic quasi Gibbs measure. For such a model, we derive a recursive relations with respect to boundary conditions. Note that we consider two mode of interactions: ferromagnetic and antiferromagnetic. In both cases, we investigate a phase transition phenomena from the associated dynamical system point of view. Namely, using the derived recursive relations we define a fractional p-adic dynamical system. In ferromagnetic case, we establish that if q is divisible by p, then such a dynamical system has two repelling and one attractive fixed points. We find basin of attraction of the fixed point. This allows us to describe all solutions of the nonlinear recursive equations. Moreover, in that case there exists the strong phase transition. If q is not divisible by p, then the fixed points are neutral, and this yields that the existence of the quasi phase transition. In antiferromagnetic case, there are two attractive fixed points, and we find basins of attraction of both fixed points, and describe solutions of the nonlinear recursive equation. In this case, we prove the existence of a quasi phase transition.

Hysteresis bifurcation and application to delayed FitzHugh-Nagumo neural systems

Takagi-Sugeno (TS) fuzzy models have been the popular choice for multi-model based nonlinear system identification for controller design. However, the interpretation of the model from a dynamical systems point of view is still unclear. In this paper, we investigate the interpretation of the TS fuzzy models in terms of local dynamics of nonlinear systems in state space. The investigation is based on the comparison between the constant-affine recursive i/o equation and constant-affine state space representations. We show that the true local dynamics is not identifiable by TS fuzzy models by analysis and example. For the identification of constant-affine state space models, a subspace identification algorithm is proposed to identify the true local dynamics with zero initial conditions.

Small limit cycles bifurcating from fine focus points in quartic order Z 3-equivariant vector fields

In this chapter we deal with abelian integrals. They play a key role in the infinitesimal version of the 16th Hilbert problem. Recall that 16th Hilbert problem and its ramifications is one of the principal research subject of Christiane Rousseau and of the first author. We recall briefly the definition and explain the role of abelian integrals in 16th Hilbert problem. We also give a simple well-known proof of a property of abelian integrals. The reason for presenting it here is that it serves as a model for more complicated and more original treatment of abelian integrals in the study of Hamiltonian monodromy of fully integrable systems, which is the main subject of this chapter. We treat in particular the simplest example presenting non-trivial Hamiltonian monodromy: the spherical pendulum.

Predicting foreign exchange rates and stock market indices have been a well researched topic in the field of financial engineering. However, most methods suffer from serious drawback due to the inherent uncertainty in the data acquisition process. Here, we have analyzed the very nature of the time series data from a pure dynamic system point of view and explored the deterministic chaotic characteristic in it. In this research, the concept of chaos has been analyzed thoroughly and the relationships among chaos, stability and order have been explained with respect to the concept of time. A method of predicting time series data based on deterministic dynamically system has been presented in this monograph. The present research revolves around the concepts of embedding and fuzzy reconstruction. In this regard, the necessary and sufficient condition for this reconstruction of the state space of the dynamic system in a multi-dimensional Euclidean space has been substantiated in accordance to Theory of embedding. Finally, a fuzzy reconstruction method based on fuzzy multiple regression analysis method has been used to predict the foreign exchange rates with accuracy.

Commentary to “pharmacokinetics from a dynamical systems point of view”

The 120-year old so-called Painleve paradox involves the loss of determinism in models of planar rigid bodies in point contact with a rigid surface, subject to Coulomb-like dry friction. The phenomenon occurs due to coupling between normal and rotational degrees-of-freedom such that the effective normal force becomes attractive rather than repulsive. Despite a rich literature, the forward evolution problem remains unsolved other than in certain restricted cases in 2D with single contact points. Various practical consequences of the theory are revisited, including models for robotic manipulators, and the strange behaviour of chalk when pushed rather than dragged across a blackboard. Reviewing recent theory, a general formulation is proposed, including a Poisson or energetic impact law. The general problem in 2D with a single point of contact is discussed and cases or inconsistency or indeterminacy enumerated. Strategies to resolve the paradox via contact regularisation are discussed from a dynamical systems point of view. By passing to the infinite stiffness limit and allowing impact without collision, inconsistent and indeterminate cases are shown to be resolvable for all open sets of conditions. However, two unavoidable ambiguities that can be reached in finite time are discussed in detail, so called dynamic jam and reverse chatter. A partial review is given of 2D cases with two points of contact showing how a greater complexity of inconsistency and indeterminacy can arise. Extension to fully three-dimensional analysis is briefly considered and shown to lead to further possible singularities. In conclusion, the ubiquity of the Painleve paradox is highlighted and open problems are discussed.

In their pioneering work BUSSE & HEIKES [2] looked at the classical Benard problem in a rotating frame. They observed rolls with a certain axial direction which seem to be stable, ie they remain unchanged for long periods of time, but suddenly the behavior changes: new rolls appear which are rotated with respect to the original rolls by approximately 60 degrees. GUCKENHEIMER & HOLMES [15] looked at the Busse-Heikes problem from a theoretical point of view. They derived a three dimensional ODE exhibiting heteroclinic cycles. Thereafter many papers have dealt with various aspects of cycles: existence, stability, bifurcations and structural stability under certain settings: [1, 3, 4, 5, 7, 8, 16, 17, 18, 20, 21, 22, 25, 23, 24, 26, 27, 28, 29]. Due to the fact that solutions which pass near steady states remain there for a long time heteroclinic cycles serve as a model for metastable behavior. We see such a metastable behavior if we look at the polarity reversals of the magnetic field of the Earth, see for example GHIL & CHILDRESS [13]. Since the origin of the magnetic field and the mechanism of its reversals are unknown we try to look at it from a dynamical systems point of view and ask ourselves whether there are heteroclinic cycles in this problem. Here, we only look at the fluid mechanical part and not at full MHD-equations. In this paper we emphasize the role of rotation and the question what happens to heteroclinic cycles in the presence of rotations. Of course this approach dictates to look at the non rotating case first and treat the rotating case as a perturbation of the non rotating one. It is not clear whether such a approach is reasonable for studying the Earth’ field but on the other hand numerical computations indicate that the region of validity of the results which are presented here exceeds the marginal speeds of rotation allowed by the usual perturbation methods.

We investigate the effect and the impact of predator-prey interactions, diffusivity and chemotaxis on the ability of survival of multiple consumer levels in a predator-prey microbial food chain. We aim at answering the question of how many consumer levels can survive from a dynamical system point of view. To solve this standing issue on food-chain length, first we construct a chemotactic food chain model. A priori bounds of the steady state populations are obtained. Then under certain sufficient conditions combining the effect of conversion efficiency, diffusivity and chemotaxis parameters, we derive the co-survival of all consumer levels, thus obtaining the food chain length of our model. Numerical simulations not only confirm our theoretical results, but also demonstrate the impact of conversion efficiency, diffusivity and chemotaxis behavior on the survival and stability of various consumer levels.

We construct stable manifolds for a class of singular evolution equations including the steady Boltzmann equation, establishing in the process exponential decay of associated kinetic shock and boundary layers to their limiting equilibrium states. Our analysis is from a classical dynamical systems point of view, but with a number of interesting modifications to accomodate ill-posedness with respect to the Cauchy problem of the underlying evolution equation.

Uncertainty arises in all stages of the visualization pipeline. However, the majority of flow visualization applications convey no uncertainty information to the user. In tools where uncertainty is conveyed, the focus is generally on data, such as error that stems from numerical methods used to generate a simulation or on uncertainty associated with mapping visualiza-tion primitives to data. Our work is aimed at another source of uncertainty - that associated with user-controlled input param-eters. The navigation and stability analysis of user-parameters has received increasing attention recently. This work presents an investigation of this topic for flow visualization, specifically for three-dimensional streamline and pathline seeding. From a dynamical systems point of view, seeding can be formulated as a predictability problem based on an initial condition. Small perturbations in the initial value may result in large changes in the streamline in regions of high unpredictability. Analyzing this predictability quantifies the perturbation a trajectory is subjugated to by the flow. In other words, some predictions are less certain than others as a function of initial conditions. We introduce novel techniques to visualize important user input parameters such as streamline and pathline seeding position in both space and time, seeding rake position and orientation, and inter-seed spacing. The implementation is based on a metric which quantifies similarity between stream and pathlines. This is important for Computational Fluid Dynamics (CFD) engineers as, even with the variety of seeding strategies available, manual seeding using a rake is ubiquitous. We present methods to quantify and visualize the effects that changes in user-controlled input parameters have on the resulting stream and pathlines. We also present various visualizations to help CFD scientists to intuitively and effectively navigate this parameter space. The reaction from a domain expert in fluid dynamics is also reported.

On the second part of Hilbert's 16th problem

PP-graphics with a nilpotent elliptic singularity in quadratic systems and Hilbert's 16th problem