Claudio Lazzari

Exponential dichotomies, heteroclinic orbits, and Melnikov functions☆

The interest in the behaviour of the solutions to perturbed autonomous systems with bounded perturbative term, like

On the pseudo-steady-state approximation and Tikhonov theorem for general enzyme systems

Abstract The effectiveness of the Tikhonov theorem for justifying the so-called pseudo-steady-state approximation (PSSA) in general closed enzyme systems is proved under a widely satisfied sufficient condition on the ligand stoichiometry, which also allows us to state that all slow variables of the degenerate system are driven by a single scalar function (the pseudo- steady-state rate equation). Finally, remarks on the perturbative small parameter and open enzyme systems are made.

Bounded solutions to singularly perturbed systems of O.D.E.

Abstract We consider existence and stability properties of bounded solutions to singular systems of O.D.E. The method we suggest considers the concept of singular maps between scales of Banach spaces and their properties. This theory is then applied to obtain existence and transversality of bounded solutions to singular systems of O.D.E.

PERTURBING TWO-DIMENSIONAL MAPS HAVING CRITICAL HOMOCLINIC ORBITS

Discrete planar maps such as xn+1=f(xn)+μ g(xn, μ), xn∈ℝ2, n∈ℤ, μ∈ℝ, are studied under the assumption that the unperturbed map xn+1=f(xn) has a critical homoclinic orbit to a hyperbolic fixed point. We give either necessary and sufficient conditions for a bifurcation from zero to two homoclinic orbits as the small parameter μ crosses zero. These conditions are stated in terms of geometrical objects such as critical lines, stable and unstable manifolds.

Boundedness and stable oscillations in two-dimensional enzyme reduced systems

Abstract Sufficient criteria for the boundedness of the solutions to planar system resulting from the application of the pseudo-steady-state hypothesis to many interesting enzyme reaction networks, and instability conditions for the equilibrium, are given. Owing to the analytic nature of such systems, the existence of at least one asymptotically orbitally stable periodic orbit is then assured. Some remarks concerning the application of the Hopf bifurcation theorem are made, and relationships with previous results are presented. Finally, domains of the parameter space for two significant biochemical models are found, where stable oscillations arise.

Cyclic Chemical Systems: An Asymptotic Stability Criterion

In previous papers (Beretta et al, 1979; Beretta and Lazzari, 1983) we obtained asymptotic stability criteria concerning positive equilibria of chemical networks with mass action kinetics.