Computational Study in Chaotic Dynamical Systems and Mechanisms for Pattern Generation in Three-Cell Networks

A computational technique is introduced to reveal the complex intrinsic structure of homoclinic and heteroclinic bifurcations in a chaotic dynamical system. This technique is applied to several Lorenz-like systems with a saddle at the center, including the Lorenz system, the Shimizu-Morioka model, the homoclinic garden model, and the laser model. A multi-fractal, self-similar organization of heteroclinic and homoclinic bifurcations of saddle singularities is explored on a bi-parametric plane of those dynamical systems. Also a great detail is explored in the Shimizu-Morioka model as an example. The technique is also applied to a re exion symmetric dynamical system with a saddle-focus at the center (Chua\u27s circuits). The layout of the homoclinic bifurcations near the primary one in such a system is studied theoretically, and a scalability ratio is proved. Another part of the dissertation explores the intrinsic mechanisms of escape in a reciprocally inhibitory FitzHugh-Nagumo type threecell network, using the phase-lag technique. The escape network can produce phase-locked states such as pace-makers, traveling-waves, and peristaltic patterns with recurrently phaselag varying

The Shilnikov Saddle-Node Bifurcation in a Monetary Policy with Endogenous Time Preference

We improve the analysis made in Chang et al (2011), by exploring the possibilities for the raise of global indeterminacy via a Shilnikov saddle-node bifurcation on an invariant circle. This allows us to better understand the determinants for the emergence of endogenous fluctuations in a monetary policy model, and to explain the existence of irregular patterns. Hence, the economy may start at some point to oscillate around the long run equilibrium, and eventually deviate from its saddle-path stable solution, thus locating the economy in a particular optimal converging path that could not coincide with the one corresponding to the lowest desired interest rate