Hiroshi Kokubu

Instability of Hes7 protein is crucial for the somite segmentation clock

During somitogenesis, a pair of somites buds off from the presomitic mesoderm every 2 hours in mouse embryos, suggesting that somite segmentation is controlled by a biological clock with a 2-hour cycle. Expression of the basic helix-loop-helix factor Hes7, an effector of Notch signaling, follows a 2-hour oscillatory cycle controlled by negative feedback; this is proposed to be the molecular basis for the somite segmentation clock. If the proposal is correct, this clock should depend crucially on the short lifetime of Hes7. To address the biological importance of Hes7 instability, we generated mice expressing mutant Hes7 with a longer half-life (∼30 min compared with ∼22 min for wild-type Hes7) but normal repressor activity. In these mice, somite segmentation and oscillatory expression became severely disorganized after a few normal cycles of segmentation. We simulated this effect mathematically using a direct autorepression model. Thus, instability of Hes7 is essential for sustained oscillation and for its function as a segmentation clock.

Bifurcations toN-homoclinic orbits andN-periodic orbits in vector fields

We study bifurcations of two types of homoclinic orbits—a homoclinic orbit with resonant eigenvalues and an inclination-flip homoclinic orbit. For the former, we prove thatN-homoclinic orbits (N⩾3) never bifurcate from the original homoclinic orbit. This answers a problem raised by Chow-Deng-Fiedler (J. Dynam. Diff. Eq.2, 177–244, 1990). For the latter, we investigate mainlyN-homoclinic orbits andN-periodic orbits forN=1, 2 and determine whether they bifurcate or not under an additional condition on the eigenvalues of the linearized vector field around the equilibrium point.

A Database Schema for the Analysis of Global Dynamics of Multiparameter Systems

A generally applicable, automatic method for the efficient computation of a database of global dynamics of a multiparameter dynamical system is introduced. An outer approximation of the dynamics for each subset of the parameter range is computed using rigorous numerical methods and is represented by means of a directed graph. The dynamics is then decomposed into the recurrent and gradient-like parts by fast combinatorial algorithms and is classified via Morse decom- positions. These Morse decompositions are compared at adjacent parameter sets via continuation to detect possible changes in the dynamics. The Conley index is used to study the structure of isolated invariant sets associated with the computed Morse decompositions and to detect the ex- istence of certain types of dynamics. The power of the developed method is illustrated with an application to the two-dimensional density-dependent Leslie population model. An interactive vi- sualization of the results of computations discussed in the paper can be accessed at the Web site http://chomp.rutgers.edu/database/, and the source code of the software used to obtain these results has also been made freely available.

THE CUSP HORSESHOE AND ITS BIFURCATIONS IN THE UNFOLDING OF AN INCLINATION-FLIP HOMOCLINIC ORBIT

Deng has demonstrated a mechanism through which a perturbation of a vector field having an inclination-flip homoclinic orbit would have a Smale horseshoe. In this article we prove that if the eigenvalues of the saddle to which the homoclinic orbit is asymptotic satisfy the condition 2lambda(u) <min{lambda(s), lambda(uu)} then there are arbitrarily small perturbations of the vector field which possess a Smale horseshoe. Moreover we analyze a sequence of bifurcations leading to the annihilation of the horseshoe. This sequence contains, in particular, the points of existence of n-homoclinic orbits with arbitrary n.

Homoclinic and heteroclinic bifurcations of Vector fields

We study a bifurcation of homoclinic and heteroclinic orbits in a two or more parameter family of autonomous ODEs, where the unperturbed system has two heteroclinic orbits joined at a common saddle point. Under some assumptions on eigenvalues of the linearized equation at equilibrium points and on a non-degeneracy condition for the system, we can show that heteroclinic orbits of new type appear besides the persistent ones of the unperturbed system. A bifurcation diagram is given for such families. Some homoclinic bifurcations are also treated including the one producing a twice-rounding homoclinic orbit.

Combinatorial-topological framework for the analysis of global dynamics

We discuss an algorithmic framework based on efficient graph algorithms and algebraic-topological computational tools. The framework is aimed at automatic computation of a database of global dynamics of a given m-parameter semidynamical system with discrete time on a bounded subset of the n-dimensional phase space. We introduce the mathematical background, which is based upon Conleys topological approach to dynamics, describe the algorithms for the analysis of the dynamics using rectangular grids both in phase space and parameter space, and show two sample applications.

Rigorous verification of cocoon bifurcations in the Michelson system

We prove the existence of cocoon bifurcations for the Michelson system where and is a parameter, based on the theory given in (Dumortier et al 2006 Nonlinearity 19 305–28). The main difficulty lies in the verification of the (topological) transversality of some invariant manifolds in the system. The proof is computer-assisted and combines topological tools including covering relations and the smooth ones using the cone conditions. These new techniques developed in this paper will have broader applicability to similar global bifurcation problems.

Cocoon bifurcation in three-dimensional reversible vector fields

The cocoon bifurcation is a set of rich bifurcation phenomena numerically observed by Lau (1992 Int. J. Bifurc. Chaos 2 543–58) in the Michelson system, a three-dimensional ODE system describing travelling waves of the Kuramoto–Sivashinsky equation. In this paper, we present an organizing centre of the principal part of the cocoon bifurcation in more general terms in the setting of reversible vector fields on . We prove that in a generic unfolding of an organizing centre called the cusp-transverse heteroclinic chain, there is a cascade of heteroclinic bifurcations with an increasing length close to the organizing centre, which resembles the principal part of the cocoon bifurcation.We also study a heteroclinic cycle called the reversible Bykov cycle. Such a cycle is believed to occur in the Michelson system, as well as in a model equation of a Josephson Junction (van den Berg et al 2003 Nonlinearity 16 707–17). We conjecture that a reversible Bykov cycle is, in its unfolding, an accumulation point of a sequence of cusp-transverse heteroclinic chains. As a first result in this direction, we show that a reversible Bykov cycle is an accumulation point of reversible generic saddle-node bifurcations of periodic orbits, the main ingredient of the cusp-transverse heteroclinic chain.

MULTIPLE HOMOCLINIC BIFURCATIONS FROM ORBIT-FLIP I: SUCCESSIVE HOMOCLINIC DOUBLINGS

The purpose of this and forthcoming papers is to obtain a better understanding of complicated bifurcations for multiple homoclinic orbits. We shall take one particular type of codimension two homoclinic orbits called orbit-flip and study bifurcations to multiple homoclinic orbits appearing in a tubular neighborhood of the original orbit-flip. The main interest of the present paper lies in the occurrence of successive homoclinic doubling bifurcations under an appropriate condition, which is a part of the entire bifurcation for multiple homoclinic orbits. Since this is a totally global bifurcation, we need the aid of numerical experiments for which we must choose a concrete set of ordinary differential equations that exhibits the desired bifurcation. In this paper we employ a family of continuous piecewise-linear vector fields for such a model equation. In order to explain the cascade of homoclinic doubling bifurcations theoretically, we also derive a two-parameter family of unimodal maps as a singular limit of the Poincare maps along homoclinic orbits. We locate bifurcation curves for this family of unimodal maps in the two-dimensional parameter space, which basically agree with those for the piecewise-linear vector fields. In particular, we show, using a standard technique from the theory of unimodal maps, that there exists an infinite sequence of doubling bifurcations which corresponds to the sequence of homoclinic doubling bifurcations for the piecewise-linear vector fields described above. Since our unimodal map has a singularity at a boundary point of its domain of definition, the doubling bifurcation is slightly different from that for standard quadratic unimodal maps, for instance the Feigenbaum constant associated with the accumulation of the doubling bifurcations is different from the standard value 4.6692.…

Heteroclinic and homoclinic bifurcations in bistable reaction diffusion systems

Abstract We apply a general heteroclinic and homoclinic bifurcation theory to the study of bifurcations of travelling waves of bistable reaction diffusion systems. Using the notion of separation, we first prove the existence of a cusp point of the set of travelling front solutions in the parameter space. This as well as the symmetry of the system yields a coexisting pair of front and back solutions which undergoes the homoclinic bifurcation producing a pulse solution. All the hypotheses imposed on the general heteroclinic and homoclinic bifurcation theorem are rigorously verified for a system of bistable reaction diffusion equations containing a small parameter e by using singular perturbation techniques, especially the SLEP method. A relation between the stability of front (or back) solutions and the intersecting manner of the stable and unstable manifolds is also given by means of the separation.