We present a modified version of the well-known geometric Lorenz attractor. It consists of a $C^1$ open set ${\mathcal O}$ of vector fields in ${\mathbb R}^3$ having an attracting region ${\mathcal U}$ satisfying three properties. Namely, a unique singularity $\sigma $; a unique attractor $\Lambda $ including the singular point and the maximal invariant in ${\mathcal U}$ has at most two chain recurrence classes, which are $\Lambda $ and (at most) one hyperbolic horseshoe. The horseshoe and the singular attractor have a collision along with the union of $2$ codimension $1$ submanifolds which split ${\mathcal O}$ into three regions. By crossing this collision locus, the attractor and the horseshoe may merge into a two-sided Lorenz attractor, or they may exchange their nature: the Lorenz attractor expels the singular point $\sigma $ and becomes a horseshoe, and the horseshoe absorbs $\sigma $ becoming a Lorenz attractor.

We consider a certain two-parameter family of automorphisms of the affine plane over a complete, locally compact non-Archimedean field. Each of these automorphisms admits a chaotic attractor on which it is topologically conjugate to a full two-sided shift map, and the attractor supports a unit Borel measure which describes the distribution of the forward orbit of Haar-almost all points in the basin of attraction. We also compute the Hausdorff dimension of the attractor, which is non-integral.

We study dynamics and bifurcations of three-dimensional diffeomorphisms with nontransversal heteroclinic cycles. We show that bifurcations under consideration lead to the birth of Lorenz-like attractors. They can be viewed as attractors in the Poincare map for periodically perturbed classical Lorenz attractors and hence they can allow for the existence of homoclinic tangencies and wild hyperbolic sets.

Economic theory indicates the need for nonlinear structural models to study medium-term and long-run dynamic behavior of an economy. This paper argues that economic systems can be better specified and estimated using differential-equation rather than difference-equation systems and briefly reviews the estimators of continuous models. This approach of specifying structural models on the basis of economic theory and institutional structure explicitly and then testing the underlying hypothesis to verify the structural form is contrasted with a general-to-specific approach of successively more restricted VARMAX processes. Previous analyses of stability about the steady state or fixed point in phase space are extended to more general attractors to allow an investigation of complexity in economic systems. The critical dependence of some attractors, and particularly strange attractors, on parameter values emphasizes the need for consistent, efficient estimation. A structural approach provides a rigorous alternative to using single time series to determine whether economic systems exhibit aperiodic or chaotic dynamical behavior.