This paper is concerned with bifurcation of limit cycles in a fourth-order near-Hamiltonian system with quartic perturbations. By bifurcation theory, proper perturbations are given to show that the system may have 20, 21 or 23 limit cycles with different distributions. This shows thatH(4) ≥ 20, whereH(n) is the Hilbert number for the second part of Hilbert’s 16th problem. It is well known that ...

We introduce analogues of the Hopf algebra of Free quasi-symmetric functions with bases labelled by colored permutations. As applications, we recover in a simple way the descent algebras associated with wreath products Γ ≀ Sn and the corresponding generalizations of quasi-symmetric functions. Finally, we obtain Hopf algebras of colored parking functions, colored non-crossing partitions and park...

In a recent dynamical systems model of platform adaptation in spatial voting models, Miller and Stadler (J. Econ. Dyn. & Control 1998, in press) showed that, assuming concave voter utility functions and complete participation, there is a globally stable equilibrium located at the mean voter position. Here we show that abstention depending on the distances between voters and platforms may lead t...

In consideration of attention as a means for goal-directed behavior in non-stationary environments, we argue that the dynamics of attention should satisfy two opposing demands: long-term maintenance and quick transition. These two characteristics are contradictory within the linear domain. We propose the near saddlenode bifurcation behavior of a sigmoidal unit with self-connection as a candidat...

The center problem for polynomial differential equations in the plane is one of the celebrated and longstanding problems in the qualitative theory of planar differential equations (see e.g. the works [Dulac, 1908], [Kapteyn, 1911], [Kapteyn, 1912], [Bautin, 1954], [Vulpe & Sibirskĭı, 1988], [Schlomiuk, 1993], [Żo la̧dek, 1994a] and [Żo la̧dek, 1994b]). It asks to distinguish a linear center betwe...

In a turning process modeled using delay differential equations (DDEs), we investigate the stability of the regenerative machine tool chatter problem. An approach using the matrix Lambert W function for the analytical solution to systems of delay differential equations is applied to this problem and compared with the result obtained using a bifurcation analysis. The Lambert W function, known to...

In this paper, we study the cubic three-parameter autonomous planar system ẋ1 = k1 + k2x1 − x1 − x2, ẋ2 = k3x1 − x2, where k2, k3 > 0. Our goal is to obtain a bifurcation diagram; i.e., to divide the parameter space into regions within which the system has topologically equivalent phase portraits and to describe how these portraits are transformed at the bifurcation boundaries. Results may be a...

• Beading of soft tubes under surface tension is studied as a localization problem. Bifurcation conditions are presented for general strain-energy function. Further insight into the post-bifurcation behaviour provided. Theoretical predictions verified by numerical simulations. We investigate localised bulging or necking in an incompressible, hyperelastic cylindrical tube axial stretching and te...

In this paper, we introduce a two-strain SIR epidemic model with viral mutation and vaccine administration. We discuss analyze the existence stability of equilibrium points. This has three types points, namely disease-free equilibrium, dominance point strain two, coexistence endemic point. The local two are verified by using Routh--Hurwitz criteria, while for global used suitable Lyapunov funct...

The phase reduction approach has manifested its efficacy in investigating synchronization behaviors limit-cycle oscillators. However, spatial distributions of the value on limit cycle may lead to illusions synchronizations for oscillators close bifurcations. In this paper, we compared sensitivity function domain and time saddle-node homoclinic (SNH) bifurcation, also known as bifurcation an inv...