The newton filtration and d-determination of bifurcation problems related to C0 contact equivalence

In this paper, from the Newton filtration’s point of view, we construct the singular Riemannian metric and use the method in singular theory to study the bifurcation problems, and give the sufficient condition of d-determination of bifurcation problems with respect to C ⁰ contact equivalence. The special cases of the main result in this paper are the results of Sun Weizhi and Zou Jiancheng.     

The recognition of a class of bifurcation problems

In bifurcation theory there are two recognition problems concerning a given normal form, the recognition for the normal form and the recognition for universal unfoldings of bifurcation problems which are equivalent to the normal form. The two recognition problems for the normal forms εx²+δλ^k were only partially solved. In this paper we give a complete solution of the two problems for all k?1 uniformly.     

Bifurcations of limit cycles in a Z 4-equivariant quintic planar vector field

In this paper, a Z4-equivariant quintic planar vector field is studied. The Hopf bifurcation method and polycycle bifurcation method are combined to study the limit cycles bifurcated from the compounded cycle with 4 hyperbolic saddle points. It is found that this special quintic planar polynomial system has at least four large limit cycles which surround all singular points. By applying the double homoclinic loops bifurcation method and Hopf bifurcation method, we conclude that 28 limit cycles with two different configurations exist in this special planar polynomial system. The results acquired in this paper are useful for studying the weakened 16th Hilbert＇s Problem.     

Bifurcations in a predator–prey model with general logistic growth and exponential fading memory

A multiparameter predator–prey system generalizing the model introduced in [6] is considered. The system studied in this paper corresponds to the type of models with exponential fading memory where the logistic per capita rate growth of the prey is given by an arbitrary function of class C^k, k ≥ 3. We prove that the model has a Hopf bifurcation and that there exist open sets in the parameter space such that the system exhibits singular attractors and asymptotically stable limit cycles. A numerical simulation is conducted in order to show the existence of critical attractor elements.As pointed out by Ayala et al. in [14], the Lotka–Volterra model of interspecific competition, which is based on the logistic theory of population growth and assumes that the intra and interspecific competitive interactions between species are linear, does not explain satisfactorily the population dynamics of some species. This is due to fact that the model does not take into account some important features of the population, which affect its dynamics. The model introduced in this paper provides independent conditions of these facts, for the existence of a Hopf bifurcation and the asymptotically stable limit cycles.     

Fourth order singular p-Laplacian differential equations with integral boundary conditions

In this paper, we investigate the existence of positive solutions for fourth order singular p-Laplacian differential equations with integral boundary conditions and non-monotonic function terms. Firstly, we establish a comparison theorem, then we define a partial ordering in E ₀ and construct lower and upper solutions to give a necessary and sufficient condition for the existence of C ²[0,1] as well as pseudo-C ³[0,1] positive solutions. Our nonlinearity f(t,x,y) may be singular at x=0, y=0, t=0 and t=1. Finally, we give some the dual results for the other cases of fourth order singular integral boundary value problems and an example to demonstrate the corresponding main results.     

Bifurcation set and distribution of limit cycles for a class of cubic Hamiltonian system with higher-order perturbed terms

A class of cubic Hamiltonion system with the higher-order perturbed term of degree n=5, 7, 9, 11, 13 is investigated. We find that there exist at least 13 limit cycles with the distribution C¹₉⊃2[C²₃⊃2C²₂] (let C_m^k denote a nest of limit cycles which encloses m singular points, and the symbol `⊂' is used to show the enclosing relations between limit cycles, while the sign `+' is used to divide limit cycles enclosing different critical points. Denote simply C_m^k+C_m^k=2C_m^k, etc.) in the Hamiltonian system under the perturbed term of degree 7, and give the complete bifurcation diagrams and classification of the phase portraits by using bifurcation theory and qualitative method and numerical simulations. These results in this paper are useful for the study of the weaken Hilbert 16th problem.     

Exponentially small splitting of separatrices in a weakly hyperbolic case

We validate the Poincaré-Melnikov method in the singular case of high-frequency periodic perturbations of the Hamiltonian h₀(x,y)=(1/2)y²-x³+x⁴ under appropriate conditions, which among other things, imply that we are considering the bifurcation case when the character of the fixed point changes from parabolic in the unperturbed case to hyperbolic in the perturbed one. The splitting is exponentially small.     

The bifurcation of limit cycles in Zn-equivariant vector fields

In this paper, we study the bifurcation of limit cycles from fine focus in Z_n-equivariant vector fields. An approach for investigating bifurcation was obtained. In order to show our work is efficacious, an example on bifurcations behavior is given, namely five order singular points values are given in the seventh degree Z₈-equivariant systems. We discuss their bifurcation behavior of limit cycles, and show that there are eight fine focuses of five order and five small amplitude limit cycles can bifurcate from each. So 40 small amplitude limit cycles can bifurcate from eight fine focuses under a certain condition. In terms of the number of limit cycles for seventh degree Z₈-equivariant systems, our results are good and interesting.     

Positive solutions of singular Caputo fractional differential equations with integral boundary conditions

In this paper, we investigate the existence of positive solutions of singular super-linear (or sub-linear) integral boundary value problems for fractional differential equation involving Caputo fractional derivative. Necessary and sufficient conditions for the existence of C³[0, 1] positive solutions are given by means of the fixed point theorems on cones. Our nonlinearity f(t, x) may be singular at t = 0 and/or t = 1.     

Singular perturbations and vanishing passage through a turning point

The paper deals with planar slow-fast cycles containing a unique generic turning point. We address the question on how to study canard cycles when the slow dynamics can be singular at the turning point. We more precisely accept a generic saddle-node bifurcation to pass through the turning point. It reveals that in this case the slow divergence integral is no longer the good tool to use, but its derivative with respect to the layer variable still is. We provide general results as well as a number of applications. We show how to treat the open problems presented in Artés et al. (2009) [1] and Dumortier and Rousseau (2009) [13], dealing respectively with the graphics DI2a and DF1a from Dumortier et al. (1994) [14].     

Extremal solutions of Cauchy problems for abstract fractional differential equations

In this paper, we study the extremal solutions of Cauchy problems for abstract fractional differential equations. Some definitions such as L ¹-Lipschitz-like, L ¹-Carathéodory-like and L ¹-Chandrabhan-like are introduced. By virtue of the singular integral inequalities with several nonlinearities due to Medved’, the properties of solutions are given. By using a hybrid fixed point theorem due to Dhage, existence results for extremal solutions are established. Finally, we present an example to illustrate our main results.     

Local analysis near a folded saddle-node singularity

Folded saddle-nodes occur generically in one parameter families of singularly perturbed systems with two slow variables. We show that these folded singularities are the organizing centers for two main delay phenomena in singular perturbation problems: canards and delayed Hopf bifurcations. We combine techniques from geometric singular perturbation theory—the blow-up technique—and from delayed Hopf bifurcation theory—complex time path analysis—to analyze the flow near such folded saddle-nodes. In particular, we show the existence of canards as intersections of stable and unstable slow manifolds. To derive these canard results, we extend the singularly perturbed vector field into the complex domain and study it along elliptic paths. This enables us to extend the invariant slow manifolds beyond points where normal hyperbolicity is lost. Furthermore, we define a way-in/way-out function describing the maximal delay expected for generic solutions passing through a folded saddle-node singularity. Branch points associated with the change from a complex to a real eigenvalue structure in the variational equation along the critical (slow) manifold make our analysis significantly different from the classical delayed Hopf bifurcation analysis where these eigenvalues are complex only.     

On the existence of positive solutions for a class of singular boundary value problems

In this paper, by the use of a fixed point theorem, many new necessary and sufficient conditions for the existence of positive solutions in C[0,1]∩C¹[0,1]∩C²(0,1) or C[0,1]∩C²(0,1) are presented for singular superlinear and sublinear second-order boundary value problems. Singularities at t=0, t=1 will be discussed.     

Flat deformations of ℙ n

In this paper we study projective flat deformations of ? ⁿ . We prove that the singular fibers of a projective flat deformation of ? ⁿ appear either in codimension 1 or over singular points of the base. We also describe projective flat deformations of ? ⁿ with smooth total space, and discuss flatness criteria.     

The method of lower and upper solutions for fourth order singular m-point boundary value problems

This paper investigates the existence of positive solutions for fourth order singular m-point boundary value problems. Firstly, we establish a comparison theorem, then we define a partial ordering in C²[0,1]∩C⁴(0,1) and construct lower and upper solutions to give a necessary and sufficient condition for the existence of C²[0,1] as well as C³[0,1] positive solutions. Our nonlinearity f(t,x,y) may be singular at x, y, t=0 and/or t=1.     

NONEXISTENCE AND EXISTENCE OF POSITIVE SOLUTIONS FOR 2nTH-ORDER SINGULAR SUPERLINEAR PROBLEMS WITH STRUM-LIOUVILLE BOUNDARY CONDITIONS

This paper investigates a class of 2nth-order singular superlinear problems with Strum-Liouville boundary conditions. We obtain a necessary and sufficient condition for the existence of C 2 n- 2 [0, 1] positive solutions, and a sufficient condition, a necessary condition for the existence of C 2 n-1 [0, 1] positive solutions. Relations between the positive solutions and the Green’s functions are depicted. The results are used to judge nonexistence or existence of positive solutions for given boundary value problems.     

Necessary and sufficient conditions for the existence of positive solutions of fourth order multi-point boundary value problems

This paper investigates the existence of positive solutions of singular multi-point boundary value problems of fourth order ordinary differential equation with p-Laplacian. A necessary and sufficient condition for the existence of C²[0,1] positive solution as well as pseudo-C³[0,1] positive solution is given by means of the fixed point theorems on cones.     

Entire solutions for a semilinear fourth order elliptic problem with exponential nonlinearity

We investigate entire radial solutions of the semilinear biharmonic equation Δ²u=λexp(u) in Rn, n?5, λ>0 being a parameter. We show that singular radial solutions of the corresponding Dirichlet problem in the unit ball cannot be extended as solutions of the equation to the whole of Rn. In particular, they cannot be expanded as power series in the natural variable s=log|x|. Next, we prove the existence of infinitely many entire regular radial solutions. They all diverge to −∞ as |x|→∞ and we specify their asymptotic behaviour. As in the case with power-type nonlinearities [F. Gazzola, H.-Ch. Grunau, Radial entire solutions for supercritical biharmonic equations, Math. Ann. 334 (2006) 905-936], the entire singular solution x?−4log|x| plays the role of a separatrix in the bifurcation picture. Finally, a technique for the computer assisted study of a broad class of equations is developed. It is applied to obtain a computer assisted proof of the underlying dynamical behaviour for the bifurcation diagram of a corresponding autonomous system of ODEs, in the case n=5.     

Convergence and superconvergence analysis of an anisotropic nonconforming finite element methods for singularly perturbed reaction-diffusion problems

The numerical approximation by a lower order anisotropic nonconforming finite element on appropriately graded meshes are considered for solving singular perturbation problems. The quasi-optimal order error estimates are proved in the ε-weighted H¹-norm valid uniformly, up to a logarithmic factor, in the singular perturbation parameter. By using the interpolation postprocessing technique, the global superconvergent error estimates in ε-weighted H¹-norm are obtained. Numerical experiments are given to demonstrate validity of our theoretical analysis.     

The Newton filtration and d-determination of bifurcation problems related to C~0 contact equivalence

In this paper, from the Newton filtration's point of view, we construct the singular Riemannian metric and use the method in singular theory to study the bifurcation problems, and give the sufficient condition of d-determination of bifurcation problems with respect to C0 contact equivalence. The special cases of the main result in this paper are the results of Sun Weizhi and Zou Jiancheng.