Cyclical strategies in two-dimensional optimal control models: Necessary conditions and existence

This paper derives necessary conditions such that cyclical policies may be optimal in concave, two state variable (economic) control problems. These conditions identify four different routes. One major implication is that two of these four conditions may be met by separable models. This possibility has been overlooked so far. Therefore, even separable and structurally very simple models may be characterized by optimal cyclical policies. Indeed, it will be shown that stable limit cycles exist for concave and separable control problems.     

Pathways to hopf bifurcations in dynamic continuous-time optimization problems

The purpose of this paper is to characterize pathways to Hopf bifurcation in continuous time, concave, two-dimensional optimal control models. It is shown that essentially two pathways exist: control-state interaction and growth. The knowledge of such pathways provides a criterion at the stage of modelling on the potential complexity of optimal trajectories.The author knowledges the many discussions with Professor Gustav Feichtinger.     

On the Bifurcations of a Hamiltonian Having Three Homoclinic Loops under Z3 Invariant Quintic Perturbations

A cubic system having three homoclinic loops perturbed by Z3 invariant quintic polynomials is considered. By applying the qualitative method of differential equations and the numeric computing method, the Hopf bifurcation, homoclinic loop bifurcation and heteroclinic loop bifurcation of the above perturbed system are studied. It is found that the above system has at least 12 limit cycles and the distributions of limit cycles are also given.     

Cubic Lienard Equations with Quadratic Damping (Ⅱ)

Abstract Applying Hopf bifurcation theory and qualitative theory, we show that the general cubic Lienardequations with quadratic damping have at most three limit cycles. This implies that the guess in which thesystem has at most two limit cycles is false. We give the sufficient conditions for the system has at most threelimit cycles or two limit cycles. We present two examples with three limit cycles or two limit cycles by usingnumerical simulation.     

Cubic Lienard equation with quadradic damping (I)

1. IntroductionLienard equationdZx dx~ f(.)g g(x) = 0 (l.0)dtZ dthas been extensively studied with particular emphasis on the ekistence and uniqueness oflimit cycles (see e.g. [l--4] and references there in). The number of limit cycles of (l.0) hasbeen also investigated by several authors (see e.g. [5--8]).In the present paper we study the general cubic Lienard equation, namelydx da~ = y ~ F(x), Z ~ ~g(x) (1.1)dt' dtwhereF(x) = ale a,x: a,x', (l.2)g(x) = blx b,x' b,x'. (1.3)Clea…     

BIFURCATION OF CUBIC INTEGRABLE SYSTEM UNDER CUBIC PERTURBATION

In this paper, we consider the bifurcation for a class of cubic integrable system under cubic perturbation. Using bifurcation theory and qualitative analysis, we obtain a complete bifurcation diagram of the system in a neighbourhood of the origin for parameter plane.     

A concrete example with multiple limit cycles for three dimensional Lotka–Volterra systems

The number of limit cycles for three dimensional Lotka–Volterra systems is an open problem. Recently, Yu et al. (2016) constructed some examples with the possibility of the existence of four limit cycles. Unfortunately, multiple limit cycles are not visible by numerical simulations, because all of them are very close to the interior equilibrium and extremely small. We present a concrete example with multiple limit cycles for three dimensional Lotka–Volterra systems which we can confirm them by numerical simulations. First we prepare the modified formula to compute coefficients of the normal form for the generalized Hopf bifurcation. Applying this formula to three dimensional Lotka–Volterra competitive systems with the aid of the computer algebra system, we derive the critical parameter values explicitly such that the interior equilibrium is exactly an unstable weak focus. Also we show that the heteroclinic cycle on the boundary of ${\mathbb{R}}_{+}^3$ is repelling. This implies that there exists a stable limit cycle by the Poincare–Bendixson theorem. Then, adding some suitable perturbations to parameters, we generate additional two limit cycles near the interior equilibrium by the generalized Hopf bifurcation. Finally we confirm that there exist three limit cycles by numerical simulations.     

Optimal consumption,training, working time,and leisure over the life cycle

In this paper, we analyze the joint determination of optimal consumption and allocation of time between learning (accumulation of human capital), working for wages (used for consumption and accumulation of financial capital), and leisure. Using Hopf bifurcation theory, we are able to show that cyclical training, working, leisuring, and consumption are optimal under certain constellations of parameters.This research was supported by the Austrian Science Foundation under Contract No. P6601.     

SAME DISTRIBUTION OF LIMIT CYCLES TO PERTURBED CUBIC HAMILTONIAN SYSTEMS WITH DIFFERENT PERTURBATIONS

Using qualitative analysis, we study perturbed Hamiltonian systems with different n-th order polynomial as perturbation terms. By numerical simulation, we show that these perturbed systems have the same distribution of limit cycles. Our results imply that these perturbed systems are equivalent in the sense of distribution of limit cycles. This is useful for studying limit cycles of perturbed systems.     

The 3-D Hopf bifurcation in bio-reactor when one competitor produces a toxin

In this paper, a model of competition in the bio-reactor of two competitors for a single nutrient where one of the competitors can produce toxin against its opponent is investigated. The conditions of the three dimensional Hopf bifurcation are obtained. The Hopf bifurcation implies the existence of limit cycles in the model that corresponds to the nonlinear oscillation in the reactor.     

DISTRIBUTIONS OF LIMIT CYCLES FOR POLYNOMIAL VECTOR FIELDS (P_2, Q_3)

The distributions of limit cycles of cubic vector fields (P2, Q3) are considered in this paper, where P2 and Q3 are polynomials of x and y of order two and three, respectively. It is possibly seven different distributions of limit cycles given in [1]. We now prove that in which three kinds of distributions are impossible and other four kinds all can be realized by concrete vector fields of (P2,Q3). Some related results are also given.     

QUALITATIVE STUDY FOR A CUBIC SYSTEM

This article analyses a non-Lienard type planar cubic system, and a complete qualitative analysis is given for the system, especially the conclusions for the non-existence, existence and uniqueness of limit cycles are obtained.     

NEW FAMILIES OF CENTERS AND LIMIT CYCLES FOR POLYNOMIAL DIFFERENTIAL SYSTEMS WITH HOMOGENEOUS NONLINEARITIES

We consider the class of polynomial differential equations x = -y+Pn(x,y), y = x + Qn(x, y), where Pn and Qn are homogeneous polynomials of degree n. Inside this class we identify a new subclass of systems having a center at the origin. We show that this subclass contains at least two subfamilies of isochro-nous centers. By using a method different from the classical ones, we study the limit cycles that bifurcate from the periodic orbits of such centers when we perturb them inside the class of all polynomial differential systems of the above form. In particular, we present a function whose simple zeros correspond to the limit cycles vvhich bifurcate from the periodic orbits of Hamiltonian systems.     

Bifurcations and distribution of limit cycles for near-Hamiltonian polynomial systems

This paper concerns with limit cycles through Hopf and homoclinic bifurcations for near-Hamiltonian systems. By using the coefficients appeared in Melnikov functions at the centers and homoclinic loops, some sufficient conditions are obtained to find limit cycles.     

THE HOPF BIFURCATION OF QUADRATIC SYSTEM (I)_(n=0)

δlm is the parameter space of quadratic system (I)n=0. A partition of parameters corresponding to the existence and nonexistence of the limit cycle of the system is given in detail. The Hopf bifurcation surfaces of (I)m=0 are obtained, and the sketch of Hopf bifurcation surfaces of (I)n=0 are drawn.     

KUKLES SYSTEM WITH TWO FINE FOCI

1IntroductionTheso-calledKuklessystemisacubicsystemintheformofwhereQ(x,y)isapolynomialofdegree3.ItiswellknownthatthefirstoneinvestigatingthecentreproblemofsuchasystemisI.S.Kukles[11.Kuklessystemisprobablyoneofthesimplestcubicsystem,butithasmanyimportantpracticalsignificance.NowadaysthemainproblemofKuklessystemistostudythenumberofitslimitcycles.Themodernapproachofinvestigatingthisprobemisbasedonbifurcationtheory--closedorbitsbifurcation,homoclinicbifurcationandHopfbifurcation.Perturbatingth…