Bifurcation analysis of the Hénon map.
Bifurcations and Attractors in Bogdanov Map
Blowout bifurcation of chaotic saddles.
Breaking the continuity of a piecewise linear map
Knowledge about the behavior of discontinuous piecewise-linear maps is important for a wide range of applications. An efficient way to investigate the bifurcation structure in 2D parameter spaces of such maps is to detect specific codimension-2 bifurcation points, called organizing centers, and to describe the bifurcation structure in their neighborhood. In this work, we present the organizing centers in the 1D discontinuous piecewise-linear map...
Coexisting cycles in a class of 3-D discrete maps
In this paper we consider the class of three-dimensional discrete maps M (x, y, z) = [φ(y), φ(z), φ(x)], where φ : ℝ → ℝ is an endomorphism. We show that all the cycles of the 3-D map M can be obtained by those of φ(x), as well as their local bifurcations. In particular we obtain that any local bifurcation is of co-dimension 3, that is three eigenvalues cross simultaneously the unit circle. As the map M exhibits coexistence...
Doubling bifurcation of a closed invariant curve in 3D maps
The object of the present paper is to give a qualitative description of the bifurcation mechanisms associated with a closed invariant curve in three-dimensional maps, leading to its doubling, not related to a standard doubling of tori. We propose an explanation on how a closed invariant attracting curve, born via Neimark-Sacker bifurcation, can be transformed into a repelling one giving birth to a new attracting closed invariant curve which has doubled...
Epidemiology of Dengue Fever: A Model with Temporary Cross-Immunity and Possible Secondary Infection Shows Bifurcations and Chaotic Behaviour in Wide Parameter Regions
Basic models suitable to explain the epidemiology of dengue fever have previously shown the possibility of deterministically chaotic attractors, which might explain the observed fluctuations found in empiric outbreak data. However, the region of bifurcations and chaos require strong enhanced infectivity on secondary infection, motivated by experimental findings of antibody-dependent-enhancement. Including temporary cross-immunity in such models, which is common knowledge among field researchers...
Fractal analysis of Hopf bifurcation for a class of completely integrable nonlinear Schrödinger Cauchy problems.
Newton, Chebyshev, and Halley Basins of Attraction; A Complete Geometric Approach
Nonhyperbolic persistent attractors near the Morse–Smale boundary
On the Origin of Chaos in the Belousov-Zhabotinsky Reaction in Closed and Unstirred Reactors
We investigate the origin of deterministic chaos in the Belousov–Zhabotinsky (BZ) reaction carried out in closed and unstirred reactors (CURs). In detail, we develop a model on the idea that hydrodynamic instabilities play a driving role in the transition to chaotic dynamics. A set of partial differential equations were derived by coupling the two variable Oregonator–diffusion system to the Navier–Stokes equations. This approach allows us to shed light on the correlation between chemical oscillations...
Organizing centers in parameter space of discontinuous 1D maps. The case of increasing/decreasing branches
This work contributes to classify the dynamic behaviors of piecewise smooth systems in which border collision bifurcations characterize the qualitative changes in the dynamics. A central point of our investigation is the intersection of two border collision bifurcation curves in a parameter plane. This problem is also associated with the continuity breaking in a fixed point of a piecewise smooth map. We will relax the hypothesis needed in [4] where...
Pattern Formation of Competing Microorganisms in Sediments
We present a three species model describing the degradation of substrate by two competing populations of microorganisms in a marine sediment. Considering diffusion to be the main transport process, we obtain a reaction diffusion system (RDS) which we study in terms of spontaneous pattern formation. We find that the conditions for patterns to evolve are likely to be fulfilled in the sediment. Additionally, we present simulations that are consistent with experimental data from the literature. We...
Solution manifolds for systems of differential equations.
Statistical stability of geometric Lorenz attractors
We consider the robust family of geometric Lorenz attractors. These attractors are chaotic, in the sense that they are transitive and have sensitive dependence on initial conditions. Moreover, they support SRB measures whose ergodic basins cover a full Lebesgue measure subset of points in the topological basin of attraction. Here we prove that the SRB measures depend continuously on the dynamics in the weak* topology.
Tangencies for real and complex Hénon maps: An analytic method.
Unstable Orbits and Milnor Attractors in the Discontinuous Flat Top Tent Map
In this work we consider the discontinuous flat top tent map which represents an example for discontinuous piecewise-smooth maps, whereby the system function is constant on some interval. Such maps show several characteristics caused by this constant value which are still insufficiently investigated. In this work we demonstrate that in the discontinuous flat top tent map every unstable periodic orbit may become a Milnor attractor. Moreover, it turns...