A Lie-based approach to the general framework of chaotic synchronization

"Diverese phenomena have been reported on the synchornization of chaotic systems. Therefore, the generalized framework of the chaotic synchronization is an actual scienti¯c debate. Here, a Lie-based geometrical approach is presented to remark some geometrical properties of the nonlinear (chaotic) systems toward their synchronization. That is, we address the general problem of ¯nding the conditions for the existence of the synchronization function y = ¸(x). The contribution is focused on the 2 and 3 dimensional (unidirectionally coupled) systems. Illustrative examples are provided along the text.

A note on optimization of chaos suppression via robust asymptotic feedback: accounting control cost

"This paper deals with the chaos suppression for oscillators in canonical form. The underlying idea is to optimize the robust chaos supression by accounting the control cost. The robust chaos supression is attained by the robust asymptotic feedback while the optimization is solved via Riccatti ecuations. A finite horizon is arbitrarily settled and the suppression is achieved at this time by means of optimal control problem. This scheme allows to take into account the energy that is wasted by the controller and the closed-loop performance of states. Some experimental results show the features of the approach when a High-Gain observer is added in order to have available the complete state vector.

Grahp and automata in arterial vascular tree of the kidney

"The renal vascular development is not known and occurs through two mechanisms that sometimes overlap: vasculogenesis and angiogenesis. Here we only consider growth through angiogenesis, i.e., the arterial vascular tree of the kidney. There are two types of vascular angiogenesis in development: sprouting and splitting an- giogenesis. We study these processes through mathematical tools. The graphs and automatas allow modeling the vascular growth, can generated tree structures by incorporating the physiological laws of the arterial branching. That is, the graph prescribes topology of the vascular tree and the automaton can include the rule of dynamics in the phenomena of vascularization.

Feedback induction of limit cycle in a bioreactor: controlling towards scale-down

"The feedback stabilization of periodic orbits (induction of limit cycle) via PI-like control is proposed as plausible tool for scale-down studies. An isothermal continuous stirred tank bioreactor (CSTB), with nonideal mixing, is studied. Kinetics is assumed to be governed by Haldane law. The Ready-to-use equations for selecting the control gains are given. Thus, osccilatory behavior with arbitrary frequency and amplitude can be induced into the PI-controlled CSTB.

Comparing performance on chaos control via adaptive output-feedback

"Performance of four controllers is experimentally compared and evaluated in context of chaos suppression. Four output-feedback controllers are used in experi- ments for comparison. First three schemes utilize an adaptive observer to estimate the states and parameter required for feeding back and with different techniques, which are: (i) feedback linearization, (ii) backstepping, and (iii) sliding mode. The fourth scheme is a (low-parameterized) robust adaptive feedback. A simple class of dynamical systems that exhibit chaotic behavior, called P-class, is considered as benchmark due to involves distinct chaotic systems. The need of comparison is motivated to ask: What is the suitable adaptive scheme to suppress chaos in an specific implementation? Results show a trend on different applications, are illustrated experimentally by means circuits, and are discussed in terms of control effort. This comparative study is important to select a feedback scheme in specific implementations; for example, synchronization of complex networks.

On Brusselator and Oregonator as chemical reaction networks: a graph approach

"This paper shows graph similarities between Brusselator and Oregonator reaction mechanisms, using the jacobian matrix in convex coordinates as an adjacency matrix which defines a weighted directed pseudograph. A linear transformation is defined for the task of mapping the weights of the three dimensional system onto a two dimensional one where the Oregonator´s pseudograph is isomorphic to Brusselator´s.

On the emergence of chaos in dynamical networks

"We investigate how changes of specific topological features result on transitions among different bounded behaviours in dynamical networks. In particular, we focus on networks with identical dynamical systems, synchronised to a common equilibrium point, then a transition into chaotic behaviour is observed as the number of nodes and the strength of their coupling changes. We analyse the network's transverse Lyapunov exponents (tLes) to derive conditions for the emergence of bounded complex behaviour on different basic network models. We find that, for networks with a given number of nodes, chaotic behaviour emerges when the coupling strength is within a specific bounded interval; this interval is reduced as the number of nodes increases. Furthermore, the endpoints the emergence interval depend on the coupling structure of network. We also find that networks with homogeneous connectivity, such as regular lattices and small-world networks are more conducive to the emergence of chaos than networks with heterogeneous connectivity like scale-free and star-connected graphs. Our results are illustrated with numerical simulations of the chaotic benchmark Lorenz systems, and to underline their potential applicability to real-world systems, our results are used to establish conditions for the chaotic activation of a network of electrically coupled pancreatic β-cell models.

Steady state equivalence among autocatalytic peroxidase-oxidase reactions

"Peroxidase-oxidase is an enzymatic reaction that can exhibit dynamical scenarios such as bistability, sustained oscillations, and Shilnikov chaos. In this work, we apply the chemical reaction network theory approach to find kinetic constants such that the associated mass action kinetics ordinary differential equations induced by three four dimensional structurally different enzymatic reaction systems can support the same steady states for several chemical species despite differences in their chemical nature.

Robust sliding mode‐based extremum‐seeking controller for reaction systems via uncertainty estimation approach

"This paper deals with the design of a robust sliding mode‐based extremum‐seeking controller aimed at the online optimization of a class of uncertain reaction systems. The design methodology is based on an input–output linearizing method with variable‐structure feedback, such that the closed‐loop system converges to a neighborhood of the optimal set point with sliding mode motion. In contrast with previous extremum‐seeking control algorithms, the control scheme includes a dynamic modelling‐error estimator to compensate for unknown terms related with model uncertainties and unmeasured disturbances. The proposed online optimization scheme does not make use of a dither signal or a gradient‐based optimization algorithm. Practical stabilizability for the closed‐loop system around to the unknown optimal set point is analyzed. Numerical experiments for two nonlinear processes illustrate the effectiveness of the proposed robust control scheme.

On friction effects as a mechanism to induce complex dynamical behavior in earthquakes

"In this paper we analyze a nonlinear dynamical system that describes the kinetic mechanism between tectonic plates on the crust´s earth undergoing stick slip movement. The analysis includes friction effects and an empirical friction law of granite rocks. The phenomena involved in the analyzed model are Stribeck´s effect; Dieterich- Ruina´s law; and properties of media as a presence of fluids and deformation. Outcomes arise from analysis of the system, which is conceived by a single slider block of one degree of freedom over a roughness and lubricated surface and formulated by space-state model through a differential equation system. We describe the oscillatory behavior for both continuous and switched conditions in terms of the mathematical solutions. Periodic and aperiodic orbits exist under a driven force and even more complex behavior. A relationship is given between the stability of the switched system and the parameter related with the oscillation frequency associated to characteristic longitude of displacement of slider. A necessary condition for stability in an unstable regime is deduced, under certain conditions in terms of frictional and seismic parameters of the analyzed model. Thus, we show the stationary and aperiodic solutions that describe the friction mechanism inducing earthquakes with a complex and nonlinear behavior.