On Computer-Assisted Proving The Existence Of Periodic And Bounded Orbits

We announce a new result on determining the Conley index of the Poincaré map for a time-periodic non-autonomous ordinary differential equation. The index is computed using some singular cycles related to an index pair of a small-step discretization of the equation. We indicate how the result can be applied to computer-assisted proofs of the existence of bounded and periodic solutions. We provide also some comments on computer-assisted proving in dynamics.

On Solving Systems of Autonomous Ordinary Differential Equations by Reduction to a Variable of an Algebra

A new technique for solving a certain class of systems of autonomous ordinary differential equations over𝕂nis introduced (𝕂being the real or complex field). The technique is based on two observations: (1), if𝕂nhas the structure of certain normed, associative, commutative, and with a unit, algebras𝔸over𝕂, then there is a scheme for reducing the system of differential equations to an autonomous ordinary differential equation on one variable of the algebra; (2) a technique, previously introduced for solving differential equations overℂ, is shown to work on the class mentioned in the previous paragraph. In particular it is shown that the algebras in question include algebras linearly equivalent to the tensor product of matrix algebras with certain normal forms.

On the third order boundary value problems with asymmetric nonlinearity

The author considers two point third order boundary value problem with asymmetric nonlinearity. The structure and oscillatory properties of solutions of the third order nonlinear autonomous ordinary differential equation are discussed. Results on the estimation of the number of solutions to boundary value problem are provided. An illustrative example is given.

A CONTINUOUS TIME APPROXIMATION OF AN EVOLUTIONARY STOCK MARKET MODEL

We derive a continuous time approximation of the evolutionary market selection model of Blume and Easley (1992). Conditions on the payoff structure of the assets are identified that guarantee convergence. We show that the continuous time approximation equals the solution of an integral equation in a random environment. For constant asset returns, the integral equation reduces to an autonomous ordinary differential equation. We analyze its long-run asymptotic behavior using techniques related to Lyapunov functions, and compare our results to the benchmark of profit-maximizing investors.

ON THE STRUCTURE OF THE SOLUTIONS OF A TWO-PARAMETER FAMILY OF THREE-DIMENSIONAL ORDINARY DIFFERENTIAL EQUATIONS

We analyze the global structure of the solutions of a three-dimensional, autonomous ordinary differential equation which depends on two parameters. We use graphical, heuristic, and rigorous arguments to show that as the parameters vary, a wide range of dynamical behavior is displayed.

Some modified Michaelis–Menten equations having stable closed trajectories

A 3-dimensional autonomous ordinary differential equation is studied which models certain cellular biochemical reactions. Extended Poincaré-Bendixson theory is used to obtain algebraic conditions on the parameters which are sufficient for the existence of at least one stable closed trajectory. Similar conditions are also obtained for the absence of chaos and for the global convergence of solutions to a critical point.