Limit cycle bifurcations near a double homoclinic loop with a nilpotent saddle of order m

In this paper, we study the explicit expansion of the first order Melnikov function near a double homoclinic loop passing through a nilpotent saddle of order m in a near-Hamiltonian system. For any positive integer $m(m\ge 1)$, we derive the formulas of the coefficients in the expansion, which can be used to study the limit cycle bifurcations for near-Hamiltonian systems. In particular, for $m=2$, we use the coefficients to consider the limit cycle bifurcations of general near-Hamiltonian systems and give the existence conditions for 10, 11, 13, 15 and 16 (11, 13 and 16, respectively) limit cycles in the case that the homoclinic loop is of cuspidal type (smooth type, respectively) and their distributions. As an application, we consider a near-Hamiltonian system with a nilpotent saddle of order 2 and obtain the lower bounds of the maximal number of limit cycles.     

On the complexity of extending the convergence ball of Wang’s method for finding a zero of a derivative

Ball convergence results are very important, since they demonstrate the complexity in choosing initial points for iterative methods. One of the most important problems in the study of iterative methods is to determine the convergence ball. This ball is small in general restricting the choice of initial points. We address this problem in the case of Wang’s method utilized to determine a zero of a derivative. Finding such a zero has many applications in computational fields, especially in function optimization. In particular, we find the convergence ball of Wang’s method using hypotheses up to the second derivative in contrast to earlier studies using hypotheses up to the fourth derivative. This way, we also extend the applicability of Wang’s method. Numerical experiments used to test the convergence criteria complete this study.     

Rational solutions for a combined (3 + 1)-dimensional generalized BKP equation

Nonlinear Dynamics - In this paper, we focus on the rational solutions for a combined (3 + 1)-dimensional generalized BKP equation. By using the symbolic computation with Maple,...     

Photodeposition of CoOx nanoparticles on BiFeO3 nanodisk for efficiently piezocatalytic degradation of rhodamine B by utilizing ultrasonic vibration energy

Piezoelectric materials have received much attention due to their great potential in environmental remediation by utilizing vibrational energy. In this paper, a novel piezoelectric catalyst, CoO_x nanoparticles anchored BiFeO₃ nanodisk composite, was intentionally synthesized via a photodeposition method and applied in piezocatalytic degradation of rhodamine B (RhB) under ultrasonic vibration. The as-synthesized CoO_x/BiFeO₃ composite presents high piezocatalytic efficiency and stability. The RhB degradation rate is determined to be 1.29 h⁻¹, which is 2.38 folds higher than that of pure BiFeO₃. Via optimizing the reaction conditions, the piezocatalytic degradation rate of the CoO_x/BiFeO₃ can be further increased to 3.20 h⁻¹. A thorough characterization was implemented to investigate the structure, piezoelectric property, and charge separation efficiency of the CoO_x/BiFeO₃ to reveal the nature behind the high piezocatalytic activity. It is found that the CoO_x nanoparticles are tightly adhered and uniformly dispersed on the surface of the BiFeO₃ nanodisks. Strong interaction between CoO_x and BiFeO₃ triggers the formation of a heterojunction structure, which further induces the migration of the piezoinduced holes on the BiFeO₃ to CoO_x nanoparticles. The recombination of electron-hole pairs is retarded, thereby increasing the piezocatalytic performance greatly. This work may offer a new paradigm for the design of high-efficiency piezoelectric catalysts.