Period two implies chaos for a class of ODEs: a dynamical system approach

The aim of this note is to set in the field of dynamical systems a recent theorem by Obersnel and Omari about the presence of periodic solutions of all periods for a class of scalar time-periodic first order differential equations without uniqueness, provided a subharmonic solution (and thus, for instance, a solution of period two) does exist. Indeed, making use of the Bebutov flow, we try to clarify in what sense the term "chaos" has to be understood and which dynamical features can be inferred for the system under analysis.Comment: 10 page

Chaotic dynamics in the Volterra predator-prey model via linked twist maps

We prove the existence of infinitely many periodic solutions and complicated dynamics, due to the presence of a topological horseshoe, for the classical Volterra predator--prey model with a periodic harvesting. The proof relies on some recent results about chaotic planar maps combined with the study of geometric features which are typical of linked twist maps.Comment: 24 pages, 4 figure

Some Remarks on Fixed Points for Maps which are Expansive along one Direction

We present some fixed point theorems for planar maps which satisfy a property of path–expansion along a certain direction. We also show some links between these fixed point theorems and other recent results about covering relations and topological horseshoes

Complex Dynamics in an Evolutionary General Equilibrium Model

We propose an exchange economy evolutionary model with discrete time, in which there are two utility-maximizing groups of agents which differ in the preference structure. Assuming an evolutionary mechanism based on the relative utility values realized by the two kinds of agents, we analytically and numerically investigate the existence of equilibria, their stability, and possible phenomena of coexistence between groups, mainly in terms of the heterogeneity degree in the preference structure. We find that our system has two trivial equilibria, at which just one of the two groups is present, and possibly a nontrivial equilibrium, characterized by the coexistence of the two groups of agents. Such nontrivial equilibrium may be stable, attracting all trajectories, or unstable. In the latter case, interesting, periodic, or chaotic, dynamics arise. We prove that the nontrivial equilibrium emerges via a transcritical bifurcation and loses stability via a flip bifurcation, after which the coexistence between groups is oscillatory in nature, presenting a regular or irregular behavior. In order to better investigate the role of the heterogeneity degree parameter, we perform a bifurcation analysis considering different scenarios, characterized by a balanced or unbalanced endowment distribution of the two goods

Market sentiment and heterogeneous fundamentalists in an evolutive financial market mode

We study a financial market populated by heterogeneous fundamentalists, whose decisions are driven by ``animal spirits''. Each agent may have optimistic or pessimistic beliefs about the fundamental value, which are selected from time to time on the basis of an evolutionary mechanism. The evolutionary selection depends on a weighted evaluation of the general market sentiment perceived by the agents and on a profitability measure of the existent strategies. As the relevance given to the sentiment index increases, a herding phenomenon in agents behavior may take place and the animal spirits can drive the market toward polarized economic regimes, which coexist and are characterized by persistent high or low levels of optimism and pessimism. This conduct is detectable from agents polarized shares and beliefs, which in turn influence the price level. Such polarized economic regimes can consist in stable steady states or can be characterized by endogenous complex dynamics, generating persistent alternating waves of optimism and pessimism, as well as return distributions displaying fat tails and excess volatility

Market sentiment and heterogeneous fundamentalists in an evolutive financial market mode

We study a financial market populated by heterogeneous fundamentalists, whose decisions are driven by ``animal spirits''. Each agent may have optimistic or pessimistic beliefs about the fundamental value, which are selected from time to time on the basis of an evolutionary mechanism. The evolutionary selection depends on a weighted evaluation of the general market sentiment perceived by the agents and on a profitability measure of the existent strategies. As the relevance given to the sentiment index increases, a herding phenomenon in agents behavior may take place and the animal spirits can drive the market toward polarized economic regimes, which coexist and are characterized by persistent high or low levels of optimism and pessimism. This conduct is detectable from agents polarized shares and beliefs, which in turn influence the price level. Such polarized economic regimes can consist in stable steady states or can be characterized by endogenous complex dynamics, generating persistent alternating waves of optimism and pessimism, as well as return distributions displaying fat tails and excess volatility