Anticontrol of Chaos for a Class of Delay Difference Equations Based on Heteroclinic Cycles Connecting Repellers

This paper is concerned with anticontrol of chaos for a class of delay difference equations via the feedback control technique. The controlled system is first reformulated into a high-dimensional discrete dynamical system. Then, a chaotification theorem based on the heteroclinic cycles connecting repellers for maps is established. The controlled system is proved to be chaotic in the sense of both Devaney and Li-Yorke. An illustrative example is provided with computer simulations

Controllability of Cardiac Alternans

An arrhythmia is a disorder in the heart that occurs due to irregular or abnormal heartbeats. There are many types of arrhythmias, some of which are harmless, but some, including ventricular tachycardia and fibrillation, can be life-threatening. Certain arrhythmias are preceded by electrical alternans, which is a state characterized by beat-to-beat alternation in cellular action potential duration. Cardiac alternans may arise from instabilities in either voltage or intracellular calcium cycling. Although a number of techniques have been proposed to suppress alternans, most have focused on appropriately adding a new ionic current or adjusting the timing of pacing stimuli based on the difference between recent action potential durations, rather than affecting intracellular calcium directly. In addition, most of the methods proposed to suppress alternans have been tested using models that do not include calcium-driven alternans. Therefore, it is important to establish a theoretical basis for understanding how control methods may apply when alternans is driven by instabilities in calcium cycling. In this study, we apply controllability analysis to a discrete map of alternans dynamics in a cardiac cell. In particular, we compare three different controllability measures to determine to what extent different control strategies can suppress alternans. The modal controllability measure was found to be the most informative measure, with effective variables through which to apply control being action potential duration regardless of alternans mechanism along with sarcoplasmic reticulum calcium load in the calcium-driven alternans case. Moreover, we designed and compared three feedback controllers, with the aim of suppressing alternans, based on our controllability results. As expected, full state feedback methods, such as pole placement and the Linear Quadratic Regulator, were more successful in stabilizing unstable alternans modes compared with feedback based on a single variable. We also conducted preliminary work on analyzing controllability of a different model of cardiac alternans described by nonlinear differential equations. Our study provides insight into the feasibility of controlling alternans driven wholly or partially by voltage or intracellular calcium instabilities