Orlando Merino

INVARIANT MANIFOLDS FOR COMPETITIVE DISCRETE SYSTEMS IN THE PLANE

Let T be a competitive map on a rectangular region , and assume T is C1 in a neighborhood of a fixed point . The main results of this paper give conditions on T that guarantee the existence of an invariant curve emanating from when both eigenvalues of the Jacobian of T at are nonzero and at least one of them has absolute value less than one, and establish that is an increasing curve that separates into invariant regions. The results apply to many hyperbolic and nonhyperbolic cases, and can be effectively used to determine basins of attraction of fixed points of competitive maps, or equivalently, of equilibria of competitive systems of difference equations. These results, known in hyperbolic case, have been used to determine the basins of attraction of hyperbolic equilibrium points and to establish certain global bifurcation results when switching from competitive coexistence to competitive exclusion. The emphasis in applications in this paper is on planar systems of difference equations with nonhyperbolic equilibria, where we establish a precise description of the basins of attraction of finite or infinite number of equilibrium points.

Global attractivity of the equilibrium of a difference equation: An elementary proof assisted by computer algebra system

Let p and q be arbitrary positive numbers. It is shown that if , then all solutions to the difference equation converge to the positive equilibrium . The above result, taken together with the 1993 result of Kocić and Ladas for equation (E) with , gives global attractivity of the positive equilibrium of (E) for all positive values of the parameters, thus completing the proof of a conjecture of Ladas.

Stability analysis of Pielou's equation with period-two coefficient

We study global attractivity of the period-two coefficient version of the delay logistic difference equation, also known as Pielous equation, where We prove that for , zero is the unique equilibrium point. If , then zero is globally asymptotically stable, with basin of attraction given by the nonnegative quadrant of initial conditions. If , then zero is unstable, and a sequence converges to zero if and only if . If , then the sequence converges to the unique period-two solution where and are uniquely determined by the equations

Global Behavior of Solutions to Two Classes of Second-Order Rational Difference Equations

For nonnegative real numbers , , , , , and such that and , the difference equation , has a unique positive equilibrium. A proof is given here for the following statements: (1) For every choice of positive parameters, , , , , and , all solutions to the difference equation, converge to the positive equilibrium or to a prime period-two solution. (2) For every choice of positive parameters, , , , and , all solutions to the difference equation, converge to the positive equilibrium or to a prime period-two solution.

Global attractivity of the equilibrium of for q<p

We investigate the global attractivity of the equilibrium of second-order difference equation where the parameters p, q, q < p and initial conditions x − 1, x 0 are nonnegative for all n. We prove that the unique equilibrium of this equation is global attractor which gives the affirmative answer to a conjecture of Kulenović and Ladas. The method of proof is innovative, and it has the potential to be used in the proof of global attractivity of equilibria of many similar equations.

Global dynamics of certain competitive system in the plane

The competitive system of difference equations where parameters a, b, d, e are positive real numbers, and the initial conditions and are non-negative real numbers is considered. A complete classification of all possible dynamical behaviour scenarios according to all different parameter configurations is obtained.

A note on unbounded solutions of a class of second order rational difference equations

We investigate the unbounded solutions of the second order difference equation where all parameters and C and initial conditions are nonnegative and such that for all n. We give a characterization of unbounded solutions for this equation showing that whenever an unbounded solution exists the subsequence of even indexed (resp. odd) terms tends to and the subsequence of odd indexed (resp. even) terms tends to a nonnegative number. We also show that two sets in the plane of initial conditions corresponding to the two cases are separated by the global stable manifold of the unique positive equilibrium. Our result answers two open problems posed by Kulenović and Ladas (2001, Dynamics of Second Order Rational Difference Equations with Open Problems and Conjectures, Boca Raton/London: Chapman and Hall/CRC).

H∞ optimization with plant uncertainty and semidefinite programming

SUMMARY The fundamental H= problem of control is that of finding the stable frequency response function that best fits worst case frequency domain specifications. This is a non-smooth optimization problem that underlies the frequency domain formulation of the H= problem of control; it is the main optimization problem in qualitative feedback theory for example. It is shown in this article how the fundamental H= optimization problem of control can be naturally treated with modern primal—dual interior point (PDIP) methods. The theory introduced here generalizes and unifies approaches to solving large classes of optimization problems involving matrix-valued functions, a subclass of which are commonly treated with linear matrix inequalities techniques. Also, in this article new optimality conditions for H= optimization problems over matrix-valued functions are proved, and numerical experience on natural (PDIP) algorithms for these problems is reported. In experiments we find the algorithms exhibit (local) quadratic convergence rate in many instances. Finally, H= optimization problems with an uncertainty parameter are considered. It is shown how to apply the theory developed here to obtain optimality conditions and derive algorithms. Numerical tests on simple examples are reported. ( 1998 John Wiley & Sons, Ltd.

Global dynamics of a Leslie host-parasite model

Abstract We consider the system of difference equations where are positive real numbers. This system was formulated by P. H. Leslie in 1948 and the present manuscript provides the most complete dynamical analysis to date. A boundedness and persistence result along with global attractivity results for various parameter regions are established.

Invariant curves for planar competitive and cooperative maps

Abstract In this paper we present results on the existence of invariant curves for planar maps that are monotone with respect to either the south-east or north-east ordering. Some of these curves are the stable or unstable manifolds of hyperbolic fixed points (saddle points) or non-hyperbolic fixed points, and are also the boundary of basins of attraction of such points.