Geometric analysis of anharmonic downward distortion following paths

A mathematical aspect of the anharmonic downward distortion following (ADDF) path is discussed. The ADDF method is utilized as an automated reaction path search method, which can explore transition state geometries on a potential energy surface from a potential minimum. We show that the maximum number of the ADD stationary paths intersecting the potential minimum is 2^{f + 1} ? 2 , where f denotes the degree of freedom of the system. We also show that the bifurcation of the ADD stationary path is essential to detect all the transition states connected from a given minimum. The ADDF computation is demonstrated for a H₂O molecule in which pitchfork bifurcation is observed.     

Stability and bifurcation analysis in the delay-coupled van der Pol oscillators

In this paper, the dynamics of a system of two van der Pol equations with a finite delay are investigated. We show that there exist the stability switches and a sequence of Hopf bifurcations occur at the zero equilibrium when the delay varies. Using the theory of normal form and the center manifold theorem, the explicit expression for determining the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions are derived.     

Neural spike renormalization. Part I — Universal number 1

For a class of circuit models for neurons, it has been shown that the transmembrane electrical potentials in spike bursts have an inverse correlation with the intra-cellular energy conversion: the fewer spikes per burst the more energetic each spike is. Here we demonstrate that as the per-spike energy goes down to zero, a universal constant to the bifurcation of spike-bursts emerges in a similar way as Feigenbaum's constant does to the period-doubling bifurcation to chaos generation, and the new universal constant is the first natural number 1.     

Controlling bifurcations and chaos in discrete small-world networks

We propose an impulsive hybrid control method to control the period-doubling bifurcations and stabilize unstable periodic orbits embedded in a chaotic attractor of a small-world network. Simulation results show that the bifurcations can be delayed or completely eliminated. A periodic orbit of the system can be controlled to any desired periodic orbit by using this method.     

Equilibrium points and bifurcation control of a chaotic system

Based on the Routh-Hurwitz criterion, this paper investigates the stability of a new chaotic system. State feedback controllers are designed to control the chaotic system to the unsteady equilibrium points and limit cycle. Theoretical analyses give the range of value of control parameters to stabilize the unsteady equilibrium points of the chaotic system and its critical parameter for generating Hopf bifurcation. Certain nP periodic orbits can be stabilized by parameter adjustment. Numerical simulations indicate that the method can effectively guide the system trajectories to unsteady equilibrium points and periodic orbits.     

Periodic solutions and flip bifurcation in a linear impulsive system

In this paper, the dynamical behaviour of a linear impulsive system is discussed both theoretically and numerically. The existence and the stability of period-one solution are discussed by using a discrete map. The conditions of existence for flip bifurcation are derived by using the centre manifold theorem and bifurcation theorem. The bifurcation analysis shows that chaotic solutions appear via a cascade of period-doubling in some interval of parameters. Moreover, the periodic solutions, the bifurcation diagram, and the chaotic attractor, which show their consistence with the theoretical analyses, are given in an example.     

Different types of bursting in Chay neuronal model

Based on actual neuronal firing activities, bursting in the Chay neuronal model is considered, in which V _K, reversal potentials for K⁺, V _C, reversal potentials for Ca²⁺, time kinetic constant λ _n and an additional depolarized current I are considered as dynamical parameters. According to the number of the Hopf bifurcation points on the upper branch of the bifurcation curve of fast subsystem, which is associated with the stable limit cycle corresponding to spiking states, different types of bursting and their respective dynamical behavior are surveyed by means of fast-slow dynamical bifurcation analysis. Supported by the National Natural Science Foundation of China (Grant Nos. 10432010, 10526002 and 10702002)     

Small Periodic Solutions Generated by Sublinear Terms

The paper is concerned with Hopf bifurcations in systems of autonomous ordinary differential equations with a parameter. The principal distinction between usual theorems on Hopf bifurcations and our results is that here the linearized equation is degenerate and independent of the parameter. We present sufficient conditions for a parameter value to be a bifurcation point and analyze properties of small cycles arising in the vicinity of the equilibrium. Sublinear nonlinearities play the main role in the results obtained.     

The index of a critical point for densely defined operators of type in Banach spaces

The purpose of this paper is to demonstrate that it is possible to define and compute the index of an isolated critical point for densely defined operators of type acting from a real, reflexive and separable Banach space into This index is defined via a degree theory for such operators which has been recently developed by the authors. The calculation of the index is achieved by the introduction of a special linearization of the nonlinear operator at the critical point. This linearization is a new tool even for continuous everywhere defined operators which are not necessarily Fréchet differentiable. Various cases of operators are considered: unbounded nonlinear operators with unbounded linearization, bounded nonlinear operators with bounded linearization, and operators in Hilbert spaces. Examples and counterexamples are given in 2,$"> illustrating the main results. The associated bifurcation problem for a pair of operators is also considered. The main results of the paper are substantial extensions and improvements of the classical results of Leray and Schauder (for continuous operators of Leray-Schauder type) as well as the results of Skrypnik (for bounded demicontinuous mappings of type Applications to nonlinear Dirichlet problems have appeared elsewhere.