Suppose r is a heteroclinic orbit of a Ck functional differential equation i(t) =f(x,) with a-limit set a(T) and o-limit set w(T) being either hyperbolic equilibrium points or periodic orbits. Necessary and sufficient conditions are given for the existence of an 7 close to f in Ck with the property that i(t) = 3(x,) has a heteroclinic orbit p close to f. The orbits p are obtained from the zeros...

In living cells, chemical reactions form a complex network. Complicated dynamics arising from such networks are the origins of biological functions. We propose a novel mathematical method to analyze bifurcation behaviors of a reaction system from the network structure alone. The whole network is decomposed into subnetworks based on “buffering structures”. For each subnetwork, the bifurcation co...

Ming Song, Zhengrong Liu, Anjan Biswas Department of Mathematics, Shaoxing University Shaoxing 312000, China Department of Mathematics, Yuxi Normal University Yuxi 653100, China [email protected] Department of Mathematics, South China University of Technology Guangzhou 510640, China [email protected] Department of Mathematical Sciences, Delaware State University Dover, DE 19901-2277, USA D...

It is becoming increasingly clear that bistability (or, more generally, multistability) is an important recurring theme in cell signaling. Bistability may be of particular relevance to biological systems that switch between discrete states, generate oscillatory responses, or "remember" transitory stimuli. Standard mathematical methods allow the detection of bistability in some very simple feedb...

What Is a Bifurcation? The word bifurcate is commonly used to denote a split, as in “up ahead, the road bifurcates into two parts.” The use of the term in this context implies that a driver traveling on this road can make a realtime choice of being able to take one or the other branch of the road when the bifurcation point is reached. In mathematics, and in its application to neuroscience, the ...

Global bifurcation analysis of a class of general predator–prey models with a strong Allee effect in prey population is given in details. We show the existence of a point-to-point heteroclinic orbit loop, consider the Hopf bifurcation, and prove the existence/uniqueness and the nonexistence of limit cycle for appropriate range of parameters. For a unique parameter value, a threshold curve separ...

The feedback control of border collision bifurcations is considered for two-dimensional discrete-time systems. These are bifurcations that can occur when a fixed point of a piecewise smooth system crosses the border between two regions of smooth operation. The goal of the control effort is to modify the bifurcation so that the bifurcated steady state is locally unique and locally attracting. In...

Emily Stone Department of Mathematical Sciences, The University of Montana, Missoula, MT 59812 U.S.A. Abstract In this paper we present stability analysis of a nonlinear model for chatter vibration in a drilling operation. The results build our previous work [1, 2], where the model was developed and the nonlinear stability of the vibration modes as cutting width is varied was presented. Here we...