Chapters authored
Parrondian Games in Discrete Dynamic Systems
An interesting problem in nonlinear dynamics is the stabilization of chaotic trajectories, assuming that such chaotic behavior is undesirable. The method described in this chapter is based on the Parrondo’s paradox, where two losing games can be alternated, yielding a winning game. The idea of alternating parameter values has been used in chemical systems, but for these systems, the undesirable behavior is not chaotic. In contrast, ecological relevant map in one and two dimensions, most of the time, can sustain chaotic trajectories, which we consider as undesirable behaviors. Therefore, we analyze several of such ecological relevant maps by constructing bifurcation diagrams and finding intervals in parameter space that satisfy the conditions to yield a desirable behavior by alternating two undesirable behaviors. The relevance of the work relies on the apparent generality of method that establishes a dynamic pattern of behavior that allows us to state a simple conjecture for two-dimensional maps. Our results are applicable to models of seasonality for 2-D ecological maps, and it can also be used as a stabilization method to control chaotic dynamics.
Part of the book: Fractal Analysis
Complex Dynamics of Competitive First Order Chemical Self-Replication
In most experimental conditions, the initial concentrations of a chemical system are at stoichiometric proportions, allowing us to eliminate at least one variable from the mathematical analysis. Under different initial conditions, we need to consider other manifolds defined by stoichiometry and the principle of conservation of mass. Therefore, a given set of initial conditions defines a dynamic manifold and the system, a tall times, has to satisfy a particular relation of its concentrations. To illustrate the relevance of the initial conditions in a dynamic analysis, we consider a chemical system consisting of two first-order self-replicating peptides competing for a common nucleophile in a semi-batch reactor. For the symmetric case, we find different complex oscillations for a given set of parameter values but different initial conditions.
Part of the book: Chaos Theory