Joseph Páez Chávez

DISCRETIZING BIFURCATION DIAGRAMS NEAR CODIMENSION TWO SINGULARITIES

We consider parameter-dependent, continuous-time dynamical systems under discretizations. It is shown that fold-Hopf singularities are O(hp)-shifted and turned into fold-Neimark–Sacker points by one-step methods of order p. Then we analyze the effect of discretizations methods on the local bifurcation diagram near Bogdanov–Takens and fold-Hopf singularities. In particular, we prove that the discretized codimension one curves intersect at the singularities in a generic manner. The results are illustrated by a numerical example.

An SIR-Dengue transmission model with seasonal effects and impulsive control

In recent decades, Dengue fever and its deadly complications, such as Dengue hemorrhagic fever, have become one of the major mosquito-transmitted diseases, with an estimate of 390 million cases occurring annually in over 100 tropical and subtropical countries, most of which belonging to the developing world. Empirical evidence indicates that the most effective mechanism to reduce Dengue infections is to combat the disease-carrying vector, which is often implemented via chemical pesticides to destroy mosquitoes in their adult or larval stages. The present paper considers an SIR epidemiological model describing the vector-to-host and host-to-vector transmission dynamics. The model includes pesticide control represented in terms of periodic impulsive perturbations, as well as seasonal fluctuations of the vector growth and transmission rates of the disease. The effectiveness of the control strategy is studied numerically in detail by means of path-following techniques for non-smooth dynamical systems. Special attention is given to determining the optimal timing of the pesticide applications, in such a way that the number of infections and the required amount of pesticide are minimized.

Discretizing dynamical systems with generalized Hopf bifurcations

We consider the discretizations of parameter-dependent, continuous-time dynamical systems. We show that the general one-step methods shift a generalized Hopf bifurcation and turn it into a generalized Neimark–Sacker point. We analyze the effect of discretization methods on the emanating Hopf curve. In particular, we obtain estimates for the eigenvalues of the discretized system along this curve. A detailed analysis of the discretized first Lyapunov coefficient is also given. The results are illustrated by a numerical example. Dynamical consequences are discussed.We consider the discretizations of parameter-dependent, continuous-time dynamical systems. We show that the general one-step methods shift a generalized Hopf bifurcation and turn it into a generalized Neimark–Sacker point. We analyze the effect of discretization methods on the emanating Hopf curve. In particular, we obtain estimates for the eigenvalues of the discretized system along this curve. A detailed analysis of the discretized first Lyapunov coefficient is also given. The results are illustrated by a numerical example. Dynamical consequences are discussed.

Numerical and experimental studies of stick–slip oscillations in drill-strings

The cyclic nature of the stick–slip phenomenon may cause catastrophic failures in drill-strings or at the very least could lead to the wear of expensive equipment. Therefore, it is important to study the drilling parameters which can lead to stick–slip, in order to develop appropriate control methods for suppression. This paper studies the stick–slip oscillations encountered in drill-strings from both numerical and experimental points of view. The numerical part is carried out based on path-following methods for non-smooth dynamical systems, with a special focus on the multistability in drill-strings. Our analysis shows that, under a certain parameter window, the multistability can be used to steer the response of the drill-strings from a sticking equilibrium or stick–slip oscillation to an equilibrium with constant drill-bit rotation. In addition, a small-scale downhole drilling rig was implemented to conduct a parametric study of the stick–slip phenomenon. The parametric study involves the use of two flexible shafts with varying mechanical properties to observe the effects that would have on stick–slip during operation. Our experimental results demonstrate that varying some of the mechanical properties of the drill-string could in fact control the nature of stick–slip oscillations.

Finite-time Lyapunov exponents and metabolic control coefficients for threshold detection of stimulus–response curves

ABSTRACT In biochemical networks transient dynamics plays a fundamental role, since the activation of signalling pathways is determined by thresholds encountered during the transition from an initial state (e.g. an initial concentration of a certain protein) to a steady-state. These thresholds can be defined in terms of the inflection points of the stimulus–response curves associated to the activation processes in the biochemical network. In the present work, we present a rigorous discussion as to the suitability of finite-time Lyapunov exponents and metabolic control coefficients for the detection of inflection points of stimulus–response curves with sigmoidal shape.

Starting homoclinic tangencies near 1:1 resonances

We construct a theory-based numerical method for starting the continuation of homoclinic tangencies near 1 : 1 resonances, for systems with arbitrary dimension ≥ 2. The core of the method is numerical center manifold reduction and flow approximation. The reduction is implemented by means of the homological equation. The starting procedure is applied in numerical examples.

Analysis and control of the dynamical response of a higher order drifting oscillator

This paper studies a position feedback control strategy for controlling a higher order drifting oscillator which could be used in modelling vibro-impact drilling. Special attention is given to two control issues, eliminating bistability and suppressing chaos, which may cause inefficient and unstable drilling. Numerical continuation methods implemented via the continuation platform COCO are adopted to investigate the dynamical response of the system. Our analyses show that the proposed controller is capable of eliminating coexisting attractors and mitigating chaotic behaviour of the system, providing that its feedback control gain is chosen properly. Our investigations also reveal that, when the slider’s property modelling the drilled formation changes, the rate of penetration for the controlled drilling can be significantly improved.

A comparative study of integrated pest management strategies based on impulsive control

ABSTRACT The paper presents a comprehensive numerical study of mathematical models used to describe complex biological systems in the framework of integrated pest management. Our study considers two specific ecosystems that describe the application of control mechanisms based on pesticides and natural enemies, implemented in an impulsive and periodic manner, due to which the considered models belong to the class of impulsive differential equations. The present work proposes a numerical approach to study such type of models in detail, via the application of path-following (continuation) techniques for nonsmooth dynamical systems, via the novel continuation platform COCO (Dankowicz and Schilder). In this way, a detailed study focusing on the influence of selected system parameters on the effectiveness of the pest control scheme is carried out for both ecological scenarios. Furthermore, a comparative study is presented, with special emphasis on the mechanisms upon which a pest outbreak can occur in the considered ecosystems. Our study reveals that such outbreaks are determined by the presence of a branching point found during the continuation analysis. The numerical investigation concludes with an in-depth study of the state-dependent pesticide mortality considered in one of the ecological scenarios.

Modeling and Analysis of Integrated Pest Control Strategies via Impulsive Differential Equations

The paper is concerned with the development and numerical analysis of mathematical models used to describe complex biological systems in the framework of Integrated Pest Management (IPM). Established in the late 1950s, IPM is a pest management paradigm that involves the combination of different pest control methods in ways that complement one another, so as to reduce excessive use of pesticides and minimize environmental impact. Since the introduction of the IPM concept, a rich set of mathematical models has emerged, and the present work discusses the development in this area in recent years. Furthermore, a comprehensive parametric study of an IPM-based impulsive control scheme is carried out via path-following techniques. The analysis addresses practical questions, such as how to determine the parameter values of the system yielding an optimal pest control, in terms of operation costs and environmental damage. The numerical study concludes with an exploration of the dynamical features of the impulsive model, which reveals the presence of codimension-1 bifurcations of limit cycles, hysteretic effects, and period-doubling cascades, which is a precursor to the onset of chaos.