Living on the edge of chaos: minimally nonlinear models of genetic regulatory dynamics

Linearized catalytic reaction equations modeling e.g. the dynamics of genetic regulatory networks under the constraint that expression levels, i.e. molecular concentrations of nucleic material are positive, exhibit nontrivial dynamical properties, which depend on the average connectivity of the reaction network. In these systems the inflation of the edge of chaos and multi-stability have been demonstrated to exist. The positivity constraint introduces a nonlinearity which makes chaotic dynamics possible. Despite the simplicity of such minimally nonlinear systems, their basic properties allow to understand fundamental dynamical properties of complex biological reaction networks. We analyze the Lyapunov spectrum, determine the probability to find stationary oscillating solutions, demonstrate the effect of the nonlinearity on the effective in- and out-degree of the active interaction network and study how the frequency distributions of oscillatory modes of such system depend on the average connectivity.Comment: 11 pages, 5 figure

The Nikolaevskiy equation with dispersion

The Nikolaevskiy equation was originally proposed as a model for seismic waves and is also a model for a wide variety of systems incorporating a neutral, Goldstone mode, including electroconvection and reaction-diffusion systems. It is known to exhibit chaotic dynamics at the onset of pattern formation, at least when the dispersive terms in the equation are suppressed, as is commonly the practice in previous analyses. In this paper, the effects of reinstating the dispersive terms are examined. It is shown that such terms can stabilise some of the spatially periodic traveling waves; this allows us to study the loss of stability and transition to chaos of the waves. The secondary stability diagram (Busse balloon) for the traveling waves can be remarkably complicated.Comment: 24 pages; accepted for publication in Phys. Rev.

Invariant manifolds and the geometry of front propagation in fluid flows

Recent theoretical and experimental work has demonstrated the existence of one-sided, invariant barriers to the propagation of reaction-diffusion fronts in quasi-two-dimensional periodically-driven fluid flows. These barriers were called burning invariant manifolds (BIMs). We provide a detailed theoretical analysis of BIMs, providing criteria for their existence, a classification of their stability, a formalization of their barrier property, and mechanisms by which the barriers can be circumvented. This analysis assumes the sharp front limit and negligible feedback of the front on the fluid velocity. A low-dimensional dynamical systems analysis provides the core of our results.Comment: 14 pages, 11 figures. To appear in Chaos Focus Issue: Chemo-Hydrodynamic Patterns and Instabilities (2012

Competition between transport phenomena in a Reaction-Diffusion-Convection system

This doctoral dissertation consists of three main parts. In part one, a general overview of the basic concepts of nonlinear science, nonlinear analysis and non-equilibrium thermodynamics is presented. Kinetics of chemical oscillations and the well known Belousov-Zhabotinsky reaction are also illustrated. In part two, a Reaction-Diffusion-Convection (RDC) model is introduced as a convenient framework for studying instability scenarios by which chemical oscillators are driven to chaos, along with its translation to an opportune code for numerical simulations. In part three, we report the methods and the data obtained. We observe that distinct bifurcation points are found in the oscillating patterns as Diu-sion coecients (di) or Grashof numbers (Gri) vary. Singularly there emerge peculiar bifurcation paths, inscribed in a general scenario of the RTN type, in which quasi{periodicity transmutes into a period-doubling sequence to chemical chaos. The opposite influence exhibited by the two parameters in these transitions clearly indicate that diusion of active species and natural convection are in `competition` for the stability of ordered dynamics. Moreover, a mirrored behavior between chemical oscillations and spatio-temporal dynamics is observed, suggesting that the emergence of the two observables are a manifestation of the same phenomenon. The interplay between chemical and transport phenomena instabilities is at the general origin of chaos for these systems. Further, a molecular dynamics study has been carried out for the calculation of diusion coecients of active species in the Belousov-Zhabotinsky reaction, namely HBrO2 and Ce(III), by means of mean square displacement and velocity autocorrelation function. These data have been used for a deeper comprehension of the hydrodynamic competition observed between diusion and convective motions for the stability of the system.</br

Controlling spatiotemporal chaos in oscillatory reaction-diffusion systems by time-delay autosynchronization

Diffusion-induced turbulence in spatially extended oscillatory media near a supercritical Hopf bifurcation can be controlled by applying global time-delay autosynchronization. We consider the complex Ginzburg-Landau equation in the Benjamin-Feir unstable regime and analytically investigate the stability of uniform oscillations depending on the feedback parameters. We show that a noninvasive stabilization of uniform oscillations is not possible in this type of systems. The synchronization diagram in the plane spanned by the feedback parameters is derived. Numerical simulations confirm the analytical results and give additional information on the spatiotemporal dynamics of the system close to complete synchronization.Comment: 19 pages, 10 figures submitted to Physica

Stoichiometric network analysis as mathematical method for examinations of instability region and oscillatory dynamics

Reaction systems in chemistry, physical chemistry, and biochemistry, which can be described by true or pseudo-stoichiometric relationships between species, and, therefore, represented with stoichiometric models, are usually very complex. For the analysis of the models of these complex nonlinear reaction systems with more than three variables, which can be in different dynamic states like multistability, oscillatority or chaos, some general mathematical methods such as the Stoichiometric network analysis (SNA) must be used. Although the SNA is a powerful method for systematic examination of complex reaction systems, identification of underlying reaction pathways, and stability analysis of dynamic states, this method is practically unknown among mathematicians. Therefore, a simple application of SNA to one five-dimensional model is given here

Non-linear effects on Turing patterns: time oscillations and chaos.

We show that a model reaction-diffusion system with two species in a monostable regime and over a large region of parameter space, produces Turing patterns coexisting with a limit cycle which cannot be discerned from the linear analysis. As a consequence, Turing patterns oscillate in time, a phenomenon which is expected to occur only in a three morphogen system. When varying a single parameter, a series of bifurcations lead to period doubling, quasi-periodic and chaotic oscillations without modifying the underlying Turing pattern. A Ruelle-Takens-Newhouse route to chaos is identified. We also examined the Turing conditions for obtaining a diffusion driven instability and discovered that the patterns obtained are not necessarily stationary for certain values of the diffusion coefficients. All this results demonstrates the limitations of the linear analysis for reaction-diffusion systems

Belousov-Zhabotinsky type reactions: the non-linear behavior of chemical systems

Chemical oscillators are open systems characterized by periodic variations of some reaction species concentration due to complex physico-chemical phenomena that may cause bistability, rise of limit cycle attractors, birth of spiral waves and Turing patterns and finally deterministic chaos. Specifically, the Belousov-Zhabotinsky reaction is a noteworthy example of non-linear behavior of chemical systems occurring in homogenous media. This reaction can take place in several variants and may offer an overview on chemical oscillators, owing to its simplicity of mathematical handling and several more complex deriving phenomena. This work provides an overview of Belousov-Zhabotinsky-type reactions, focusing on modeling under different operating conditions, from the most simple to the most widely applicable models presented during the years. In particular, the stability of simplified models as a function of bifurcation parameters is studied as causes of several complex behaviors. Rise of waves and fronts is mathematically explained as well as birth and evolution issues of the chaotic ODEs system describing the Györgyi-Field model of the Belousov-Zhabotinsky reaction. This review provides not only the general information about oscillatory reactions, but also provides the mathematical solutions in order to be used in future biochemical reactions and reactor designs