A degenerate Hopf–saddle-node bifurcation analysis in a family of electronic circuits

This paper presents a study of a three-parameter unfolding of a degenerate case in the Hopf--saddle-node singularity. This analysis shows that this nonlinear degeneracy is a source of interesting bifurcations of periodic orbits as well as global bifurcations of equilibria. The results achieved are applied to the study of a simple autonomous electronic circuit, which has just only one nonlinearity. The numerical results include the analysis of interesting resonance behaviors.     

On the dynamics of European two-axle railway freight wagons

The standardized design of European two-axle freight wagons has been used since the 1890s. The basic construction is a carbody suspended on leaf springs by a system of links working as pendulums. The internationally approved suspension system for these vehicles is the UIC standard suspension. The design is advantageous because it is simple, cheap, and robust. Despite the simple design, its influence on the dynamic performance of the vehicle is highly intricate due to dry friction damping and motion delimiters causing impacts. This paper deals with the dynamic behavior of European two-axle freight wagons analyzed through the use of a multibody model developed specifically for this purpose. The modeling of the vehicle and the integration of the dynamic equations of motion are far from trivial due to the complicated nature of the interacting forces. The model exhibits interesting bifurcation patterns emphasizing the complexity of the dynamic behavior of these wagons.     

Chaotic behavior analysis based on corner bifurcations

In this paper, a mathematical analysis in order to generate a chaotic behavior for bounded piecewise smooth systems of dimension three submitted to one of its specific bifurcations, namely the corner one, is proposed. This study is based on period doubling method.     

Dynamical survey of a generalized-Zakharov equation and its exact travelling wave solutions

We investigate the dynamical behavior of a generalized-Zakharov equation for the complex envelope of the high-frequency wave and the real low-frequency field by analyzing its phase portraits. Following a dynamical system approach, in different parameter regions, we depict phase portraits of a travelling wave system. Through discussing the bifurcation of phase portraits, we unwrap explicit miscellaneous travelling waves including localized and periodic ones.     

Hopf bifurcation at infinity in 3D symmetric piecewise linear systems. Application to a Bonhoeffer–van der Pol oscillator

In this work, a Hopf bifurcation at infinity in three-dimensional symmetric continuous piecewise linear systems with three zones is analyzed. By adapting the so-called closing equations method, which constitutes a suitable technique to detect limit cycles bifurcation in piecewise linear systems, we give for the first time a complete characterization of the existence and stability of the limit cycle of large amplitude that bifurcates from the point at infinity. Analytical expressions for the period and amplitude of the bifurcating limit cycles are obtained. As an application of these results, we study the appearance of a large amplitude limit cycle in a Bonhoeffer–van der Pol oscillator.     

On certain generalizations of the Hopf bifurcation

New sufficient conditions for the existence of generalized Hopf bifurcations are given in the context of asymptotically compact dynamical or semidynamical systems on a metric space. These conditions weaken the hypotheses of previous contributions to the subject. Mathematics Subject Classification (2000) 34K18, 74H60     

The set of periodic scalar differential equations with cubic nonlinearities

We study the structure induced by the number of periodic solutions on the set of differential equations x^′=f(t,x) where f∈C³(R²) is T-periodic in t, fx³(t,x)<0 for every (t,x)∈R², and f(t,x)→?∞ as x→∞, uniformly on t. We find that the set of differential equations with a singular periodic solution is a codimension-one submanifold, which divides the space into two components: equations with one periodic solution and equations with three periodic solutions. Moreover, the set of differential equations with exactly one periodic singular solution and no other periodic solution is a codimension-two submanifold.     

Comment on noise and bifurcations

We calculate in exact form the first correction in a parameter measuring the strength of the noise to the effective potential for one-variable diffusion processes. The use of this potential to study transitions is discussed.