Bifurcation theory for finitely smooth planar autonomous differential systems

In this paper we establish bifurcation theory of limit cycles for planar $C^k$ smooth autonomous differential systems, with $k\in \mathbb{N}$. The key point is to study the smoothness of bifurcation functions which are basic and important tool on the study of Hopf bifurcation at a fine focus or a center, and of Poincaré bifurcation in a period annulus. We especially study the smoothness of the first order Melnikov function in degenerate Hopf bifurcation at an elementary center. As we know, the smoothness problem was solved for analytic and $C^{\infty }$ differential systems, but it was not tackled for finitely smooth differential systems. Here, we present their optimal regularity of these bifurcation functions and their asymptotic expressions in the finite smooth case.     

A criticality result for polycycles in a family of quadratic reversible centers

We consider the family of dehomogenized Loud's centers $X_{\mu }=y(x?1){?}_x+(x+D{x}^2+F{y}^2){?}_y$, where $\mu =(D,F)\in {\mathbb{R}}^2$, and we study the number of critical periodic orbits that emerge or disappear from the polycycle at the boundary of the period annulus. This number is defined exactly the same way as the well-known notion of cyclicity of a limit periodic set and we call it criticality. The previous results on the issue for the family $\{X_{\mu },\mu \in {\mathbb{R}}^2\}$ distinguish between parameters with criticality equal to zero (regular parameters) and those with criticality greater than zero (bifurcation parameters). A challenging problem not tackled so far is the computation of the criticality of the bifurcation parameters, which form a set ${\mathrm{\Gamma }}_B$ of codimension 1 in ${\mathbb{R}}^2$. In the present paper we succeed in proving that a subset of ${\mathrm{\Gamma }}_B$ has criticality equal to one.     

Multi-resurgence and connection-to-Stokes formulæ for some linear meromorphic differential systems

In this article, we consider a linear meromorphic differential system with several levels $r_1<...<r_p$. For any k, we prove that the Borel transforms of its $r_k$-reduced formal solutions are resurgent and we give the general form of all their singularities. Next, under some convenient hypotheses on the geometric configuration of singular points, we display exact formulæ to express some Stokes multipliers of level $r_k$ of initial system in terms of connection constants in the Borel plane, generalizing thus formulæ already obtained by M. Loday-Richaud and the author for systems with a single level. As an illustration, we develop one numerical example.     

Small noise and long time phase diffusion in stochastic limit cycle oscillators

We study the effect of additive Brownian noise on an ODE system that has a stable hyperbolic limit cycle, for initial data that are attracted to the limit cycle. The analysis is performed in the limit of small noise – that is, we modulate the noise by a factor $\epsilon \searrow 0$ – and on a long time horizon. We prove explicit estimates on the proximity of the noisy trajectory and the limit cycle up to times $\exp ?\left(c{\epsilon }^{?2}\right)$, $c>0$, and we show both that on the time scale ${\epsilon }^{?2}$ the dephasing (i.e., the difference between noiseless and noisy system measured in a natural coordinate system that involves a phase) is close to a Brownian motion with constant drift, and that on longer time scales the dephasing dynamics is dominated by the drift. The natural choice of coordinates, that reduces the dynamics in a neighborhood of the cycle to a rotation, plays a central role and makes the connection with the applied science literature in which noisy limit cycle dynamics are often reduced to a diffusion model for the phase of the limit cycle.     

Topological invariants for semigroups of holomorphic self-maps of the unit disk

Let $({\phi }_t)$, $({?}_t)$ be two one-parameter semigroups of holomorphic self-maps of the unit disk $\mathbb{D}?\mathbb{C}$. Let $f:\mathbb{D}\to \mathbb{D}$ be a homeomorphism. We prove that, if $f°{?}_t={\phi }_t°f$ for all $t\ge 0$, then f extends to a homeomorphism of $\stackrel{￣}{\mathbb{D}}$ outside exceptional maximal contact arcs (in particular, for elliptic semigroups, f extends to a homeomorphism of $\stackrel{￣}{\mathbb{D}}$). Using this result, we study topological invariants for one-parameter semigroups of holomorphic self-maps of the unit disk.     

Lyapunov exponents of partially hyperbolic volume-preserving maps with 2-dimensional center bundle

We consider the set of partially hyperbolic symplectic diffeomorphisms which are accessible, have 2-dimensional center bundle and satisfy some pinching and bunching conditions. In this set, we prove that the non-uniformly hyperbolic maps are $C^r$ open and there exists a $C^r$ open and dense subset of continuity points for the center Lyapunov exponents. We also generalize these results to volume-preserving systems.     

Denjoy minimal sets and Birkhoff periodic orbits for non-exact monotone twist maps

A non-exact monotone twist map ${\overline{\phi }}_F$ is a composition of an exact monotone twist map $\overline{\phi }$ with a generating function H and a vertical translation $V_F$ with $V_F((x,y))=(x,y?F)$. We show in this paper that for each $\omega \in \mathbb{R}$, there exists a critical value $F_d(\omega )\ge 0$ depending on H and ω such that for $0\le F\le F_d(\omega )$, the non-exact twist map ${\overline{\phi }}_F$ has an invariant Denjoy minimal set with irrational rotation number ω lying on a Lipschitz graph, or Birkhoff $(p,q)$-periodic orbits for rational $\omega =p/q$. Like the Aubry–Mather theory, we also construct heteroclinic orbits connecting Birkhoff periodic orbits, and show that quasi-periodic orbits in these Denjoy minimal sets can be approximated by periodic orbits. In particular, we demonstrate that at the critical value $F=F_d(\omega )$, the Denjoy minimal set is not uniformly hyperbolic and can be approximated by smooth curves.     

Classification and evolution of bifurcation curves for the one-dimensional Minkowski-curvature problem and its applications

In this paper, we study the classification and evolution of bifurcation curves of positive solutions for the one-dimensional Minkowski-curvature problem

$\{\begin{array}{c}?{(u^{\prime }/\sqrt{1?u^{\prime 2}})}^{\prime }=\lambda f(u),\text{ in }(?L,L),\hfill \\ u(?L)=u(L)=0,\hfill \end{array}$

where $\lambda ,L>0$, $f\in C[0,\infty )\cap C^2(0,\infty )$ and $f(u)>0$ for $u\ge 0$. Furthermore, we show that, for sufficiently large $L>0$, the bifurcation curve $S_L$ may have arbitrarily many turning points. Finally, we apply these results to obtain the global bifurcation diagrams for Ambrosetti–Brezis–Cerami problem, Liouville–Bratu–Gelfand problem and perturbed Gelfand problem with the Minkowski-curvature operator, respectively. Moreover, we will make two lists which show the different properties of bifurcation curves for Minkowski-curvature problems, corresponding semilinear problems and corresponding prescribed curvature problems.     

Vaught's conjecture for quite o-minimal theories

We study Vaught's problem for quite o-minimal theories. Quite o-minimal theories form a subclass of the class of weakly o-minimal theories preserving a series of properties of o-minimal theories. We investigate quite o-minimal theories having fewer than $2^{\omega }$ countable models and prove that the Exchange Principle for algebraic closure holds in any model of such a theory and also we prove binarity of these theories. The main result of the paper is that any quite o-minimal theory has either $2^{\omega }$ countable models or $6^a{3}^b$ countable models, where a and b are natural numbers.     

Generic expansion and Skolemization in NSOP1 theories

We study expansions of ${\mathrm{NSOP}}_1$ theories that preserve ${\mathrm{NSOP}}_1$. We prove that if T is a model complete ${\mathrm{NSOP}}_1$ theory eliminating the quantifier ${?}^{\infty }$, then the generic expansion of T by arbitrary constant, function, and relation symbols is still ${\mathrm{NSOP}}_1$. We give a detailed analysis of the special case of the theory of the generic L-structure, the model companion of the empty theory in an arbitrary language L. Under the same hypotheses, we show that T may be generically expanded to an ${\mathrm{NSOP}}_1$ theory with built-in Skolem functions. In order to obtain these results, we establish strengthenings of several properties of Kim-independence in ${\mathrm{NSOP}}_1$ theories, adding instances of algebraic independence to their conclusions.     

Abstract elementary classes stable in ℵ0

We study abstract elementary classes (AECs) that, in ${\mathrm{?}}_0$, have amalgamation, joint embedding, no maximal models and are stable (in terms of the number of orbital types). Assuming a locality property for types, we prove that such classes exhibit superstable-like behavior at ${\mathrm{?}}_0$. More precisely, there is a superlimit model of cardinality ${\mathrm{?}}_0$ and the class generated by this superlimit has a type-full good ${\mathrm{?}}_0$-frame (a local notion of nonforking independence) and a superlimit model of cardinality ${\mathrm{?}}_1$. We also give a supersimplicity condition under which the locality hypothesis follows from the rest.     

A classification of orbits admitting a unique invariant measure

We consider the space of countable structures with fixed underlying set in a given countable language. We show that the number of ergodic probability measures on this space that are $S_{\infty }$-invariant and concentrated on a single isomorphism class must be zero, or one, or continuum. Further, such an isomorphism class admits a unique $S_{\infty }$-invariant probability measure precisely when the structure is highly homogeneous; by a result of Peter J. Cameron, these are the structures that are interdefinable with one of the five reducts of the rational linear order $(\mathbb{Q},<)$.     

A concrete example with multiple limit cycles for three dimensional Lotka–Volterra systems

The number of limit cycles for three dimensional Lotka–Volterra systems is an open problem. Recently, Yu et al. (2016) constructed some examples with the possibility of the existence of four limit cycles. Unfortunately, multiple limit cycles are not visible by numerical simulations, because all of them are very close to the interior equilibrium and extremely small. We present a concrete example with multiple limit cycles for three dimensional Lotka–Volterra systems which we can confirm them by numerical simulations. First we prepare the modified formula to compute coefficients of the normal form for the generalized Hopf bifurcation. Applying this formula to three dimensional Lotka–Volterra competitive systems with the aid of the computer algebra system, we derive the critical parameter values explicitly such that the interior equilibrium is exactly an unstable weak focus. Also we show that the heteroclinic cycle on the boundary of ${\mathbb{R}}_{+}^3$ is repelling. This implies that there exists a stable limit cycle by the Poincare–Bendixson theorem. Then, adding some suitable perturbations to parameters, we generate additional two limit cycles near the interior equilibrium by the generalized Hopf bifurcation. Finally we confirm that there exist three limit cycles by numerical simulations.     

THE GLOBAL ATTRACTOR FOR A VISCOUS WEAKLY DISSIPATIVE GENERALIZED TWO-COMPONENT μ-HUNTER-SAXTON SYSTEM

This article is concerned with the existence of global attractor of a weakly dissipative generalized two-component μ-Hunter-Saxton(gμHS2) system with viscous terms.Under the period boundary conditions and with the help of the Galerkin procedure and compactness method, we first investigate the existence of global solution for the viscous weakly dissipative(gμHS2) system. On the basis of some uniformly prior estimates of the solution to the viscous weakly dissipative(gμHS2) system, we show that the semi-group of the solution operator {S(t)}t≥0 has a bounded absorbing set. Moreover, we prove that the dynamical system {S(t)}t≥0 possesses a global attractor in the Sobolev space H~2(S) × H~2(S).     

The Picard group of the forms of the affine line and of the additive group

We obtain an explicit upper bound on the torsion of the Picard group of the forms of ${\mathbb{A}}_k^1$ and their regular completions. We also obtain a sufficient condition for the Picard group of the forms of ${\mathbb{A}}_k^1$ to be nontrivial and we give examples of nontrivial forms of ${\mathbb{A}}_k^1$ with trivial Picard groups.     

A limit equation and bifurcation diagrams of semilinear elliptic equations with general supercritical growth

We study radial solutions of the semilinear elliptic equation

$\mathrm{\Delta }u+f(u)=0$

under rather general growth conditions on f. We construct a radial singular solution and study the intersection number between the singular solution and a regular solution. An application to bifurcation problems of elliptic Dirichlet problems is given. To this end, we derive a certain limit equation from the original equation at infinity, using a generalized similarity transformation. Through a generalized Cole–Hopf transformation, all the limit equations can be reduced into two typical cases, i.e., $\mathrm{\Delta }u+u^p=0$ and $\mathrm{\Delta }u+e^u=0$.