Scaling law in Saddle-node bifurcations for one-dimensional maps: A complex variable approach

The study of transient dynamical phenomena near bifurcation thresholds has attracted the interest of many researchers due to the relevance of bifurcations in different physical or biological systems. In the context of saddle-node bifurcations, where two or more fixed points collide annihilating each other, it is known that the dynamics can suffer the so-called delayed transition. This phenomenon emerges when the system spends a lot of time before reaching the remaining stable equilibrium, found after the bifurcation, because of the presence of a saddle-remnant in phase space. Some works have analytically tackled this phenomenon, especially in time-continuous dynamical systems, showing that the time delay, tau, scales according to an inverse square-root power law, tau similar to (mu-mu (c) )(-1/2), as the bifurcation parameter mu, is driven further away from its critical value, mu (c) . In this work, we first characterize analytically this scaling law using complex variable techniques for a family of one-dimensional maps, called the normal form for the saddle-node bifurcation. We then apply our general analytic results to a single-species ecological model with harvesting given by a unimodal map, characterizing the delayed transition and the scaling law arising due to the constant of harvesting. For both analyzed systems, we show that the numerical results are in perfect agreement with the analytical solutions we are providing. The procedure presented in this work can be used to characterize the scaling laws of one-dimensional discrete dynamical systems with saddle-node bifurcations.

Controling delayed transitions with applications to prevent single species extinctions

A new method is proposed to control delayed transitions towards extinction in single population theoretical models with discrete time undergoing saddle-node bifurcations. The control method takes advantage of the delaying properties of the saddle remnant arising after the bifurcation, and allows to sustain populations indefinitely. Our method, which is shown to work for deterministic and stochastic systems, could generally be applied to avoid transitions tied to one-dimensional maps after saddle-node bifurcations.

On chaos, transient chaos and ghosts in single populations models with allee effects

Density-dependent effects, both positive or negative, can have an important impact on the population dynamics of species by modifying their population per-capita growth rates. An important type of such density-dependent factors is given by the so-called Allee effects, widely studied in theoretical and field population biology. In this study, we analyze two discrete single population models with overcompensating density-dependence and Allee effects due to predator saturation and mating limitation using symbolic dynamics theory. We focus on the scenarios of persistence and bistability, in which the species dynamics can be chaotic. For the chaotic regimes, we compute the topological entropy as well as the Lyapunov exponent under ecological key parameters and different initial conditions. We also provide co-dimension two bifurcation diagrams for both systems computing the periods of the orbits, also characterizing the period-ordering routes toward the boundary crisis responsible for species extinction via transient chaos. Our results show that the topological entropy increases as we approach to the parametric regions involving transient chaos, being maximum when the full shift R(L)(infinity) occurs, and the system enters into the essential extinction regime. Finally, we characterize analytically, using a complex variable approach, and numerically the inverse square-root scaling law arising in the vicinity of a saddle-node bifurcation responsible for the extinction scenario in the two studied models. The results are discussed in the context of species fragility under differential Allee effects. (C) 2011 Elsevier Ltd. All rights reserved.

Equilibrium self-assembly of colloids with distinct interaction sites: Thermodynamics, percolation, and cluster distribution functions

We calculate the equilibrium thermodynamic properties, percolation threshold, and cluster distribution functions for a model of associating colloids, which consists of hard spherical particles having on their surfaces three short-ranged attractive sites (sticky spots) of two different types, A and B. The thermodynamic properties are calculated using Wertheim's perturbation theory of associating fluids. This also allows us to find the onset of self-assembly, which can be quantified by the maxima of the specific heat at constant volume. The percolation threshold is derived, under the no-loop assumption, for the correlated bond model: In all cases it is two percolated phases that become identical at a critical point, when one exists. Finally, the cluster size distributions are calculated by mapping the model onto an effective model, characterized by a-state-dependent-functionality (f) over bar and unique bonding probability (p) over bar. The mapping is based on the asymptotic limit of the cluster distributions functions of the generic model and the effective parameters are defined through the requirement that the equilibrium cluster distributions of the true and effective models have the same number-averaged and weight-averaged sizes at all densities and temperatures. We also study the model numerically in the case where BB interactions are missing. In this limit, AB bonds either provide branching between A-chains (Y-junctions) if epsilon(AB)/epsilon(AA) is small, or drive the formation of a hyperbranched polymer if epsilon(AB)/epsilon(AA) is large. We find that the theoretical predictions describe quite accurately the numerical data, especially in the region where Y-junctions are present. There is fairly good agreement between theoretical and numerical results both for the thermodynamic (number of bonds and phase coexistence) and the connectivity properties of the model (cluster size distributions and percolation locus).

The Sticky Information Macro Model: Beyond Perfect Foresight

Sticky information monetary models have been used in the macroeconomic literature to explain some of the observed features regarding inflation dynamics. In this paper, we explore the consequences of relaxing the rational expectations assumption usually taken in this type of model; in particular, by considering expectations formed through adaptive learning, it is possible to arrive to results other than the trivial convergence to a fixed point long-term equilibrium. The results involve the possibility of endogenous cyclical motion (periodic and a-periodic), which emerges essentially in scenarios of hyperinflation. In low inflation settings, the introduction of learning implies a less severe impact of monetary shocks that, nevertheless, tend to last for additional time periods relative to the pure perfect foresight setup.

Grid structure impact in sparse point representation of derivatives

In the Sparse Point Representation (SPR) method the principle is to retain the function data indicated by significant interpolatory wavelet coefficients, which are defined as interpolation errors by means of an interpolating subdivision scheme. Typically, a SPR grid is coarse in smooth regions, and refined close to irregularities. Furthermore, the computation of partial derivatives of a function from the information of its SPR content is performed in two steps. The first one is a refinement procedure to extend the SPR by the inclusion of new interpolated point values in a security zone. Then, for points in the refined grid, such derivatives are approximated by uniform finite differences, using a step size proportional to each point local scale. If required neighboring stencils are not present in the grid, the corresponding missing point values are approximated from coarser scales using the interpolating subdivision scheme. Using the cubic interpolation subdivision scheme, we demonstrate that such adaptive finite differences can be formulated in terms of a collocation scheme based on the wavelet expansion associated to the SPR. For this purpose, we prove some results concerning the local behavior of such wavelet reconstruction operators, which stand for SPR grids having appropriate structures. This statement implies that the adaptive finite difference scheme and the one using the step size of the finest level produce the same result at SPR grid points. Consequently, in addition to the refinement strategy, our analysis indicates that some care must be taken concerning the grid structure, in order to keep the truncation error under a certain accuracy limit. Illustrating results are presented for 2D Maxwell's equation numerical solutions.

Dynamic equilibrium through reinforcement learning

Reinforcement Learning is an area of Machine Learning that deals with how an agent should take actions in an environment such as to maximize the notion of accumulated reward. This type of learning is inspired by the way humans learn and has led to the creation of various algorithms for reinforcement learning. These algorithms focus on the way in which an agent’s behaviour can be improved, assuming independence as to their surroundings. The current work studies the application of reinforcement learning methods to solve the inverted pendulum problem. The importance of the variability of the environment (factors that are external to the agent) on the execution of reinforcement learning agents is studied by using a model that seeks to obtain equilibrium (stability) through dynamism – a Cart-Pole system or inverted pendulum. We sought to improve the behaviour of the autonomous agents by changing the information passed to them, while maintaining the agent’s internal parameters constant (learning rate, discount factors, decay rate, etc.), instead of the classical approach of tuning the agent’s internal parameters. The influence of changes on the state set and the action set on an agent’s capability to solve the Cart-pole problem was studied. We have studied typical behaviour of reinforcement learning agents applied to the classic BOXES model and a new form of characterizing the environment was proposed using the notion of convergence towards a reference value. We demonstrate the gain in performance of this new method applied to a Q-Learning agent.

A Step-up mu-Power Converter for Solar Energy Harvesting Applications, using Hill Climbing Maximum Power Point Tracking

This paper presents a step-up micro-power converter for solar energy harvesting applications. The circuit uses a SC voltage tripler architecture, controlled by an MPPT circuit based on the Hill Climbing algorithm. This circuit was designed in a 0.13 mu m CMOS technology in order to work with an a-Si PV cell. The circuit has a local power supply voltage, created using a scaled down SC voltage tripler, controlled by the same MPPT circuit, to make the circuit robust to load and illumination variations. The SC circuits use a combination of PMOS and NMOS transistors to reduce the occupied area. A charge re-use scheme is used to compensate the large parasitic capacitors associated to the MOS transistors. The simulation results show that the circuit can deliver a power of 1266 mu W to the load using 1712 mu W of power from the PV cell, corresponding to an efficiency as high as 73.91%. The simulations also show that the circuit is capable of starting up with only 19% of the maximum illumination level.

Three-dimensional patchy lattice model for empty fluids

The phase diagram of a simple model with two patches of type A and ten patches of type B (2A10B) on the face centred cubic lattice has been calculated by simulations and theory. Assuming that there is no interaction between the B patches the behavior of the system can be described in terms of the ratio of the AB and AA interactions, r. Our results show that, similarly to what happens for related off-lattice and two-dimensional lattice models, the liquid-vapor phase equilibria exhibit reentrant behavior for some values of the interaction parameters. However, for the model studied here the liquid-vapor phase equilibria occur for values of r lower than 1/3, a threshold value which was previously thought to be universal for 2AnB models. In addition, the theory predicts that below r = 1/3 (and above a new condensation threshold which is < 1/3) the reentrant liquid-vapor equilibria are so extreme that it exhibits a closed loop with a lower critical point, a very unusual behavior in single-component systems. An order-disorder transition is also observed at higher densities than the liquid-vapor equilibria, which shows that the liquid-vapor reentrancy occurs in an equilibrium region of the phase diagram. These findings may have implications in the understanding of the condensation of dipolar hard spheres given the analogy between that system and the 2AnB models considered here. (C) 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4771591]

Sensing Cloud Optimization to Solve ED of Units with Valve-Point Effects and Multi-fuels

In this paper a solution to an highly constrained and non-convex economical dispatch (ED) problem with a meta-heuristic technique named Sensing Cloud Optimization (SCO) is presented. The proposed meta-heuristic is based on a cloud of particles whose central point represents the objective function value and the remaining particles act as sensors "to fill" the search space and "guide" the central particle so it moves into the best direction. To demonstrate its performance, a case study with multi-fuel units and valve- point effects is presented.

Colloidal dipolar interactions in 2D smectic-C films

We use a two-dimensional (2D) elastic free energy to calculate the effective interaction between two circular disks immersed in smectic-C films. For strong homeotropic anchoring, the distortion of the director field caused by the disks generates topological defects that induce an effective interaction between the disks. We use finite elements, with adaptive meshing, to minimize the 2D elastic free energy. The method is shown to be accurate and efficient for inhomogeneities on the length scales set by the disks and the defects, that differ by up to 3 orders of magnitude. We compute the effective interaction between two disk-defect pairs in a simple (linear) configuration. For large disk separations, D, the elastic free energy scales as similar to D-2, confirming the dipolar character of the long-range effective interaction. For small D the energy exhibits a pronounced minimum. The lowest energy corresponds to a symmetrical configuration of the disk-defect pairs, with the inner defect at the mid-point between the disks. The disks are separated by a distance that, is twice the distance of the outer defect from the nearest disk. The latter is identical to the equilibrium distance of a defect nucleated by an isolated disk.

How patchy can one get and still condense? The role of dissimilar patches in the interactions of colloidal particles

We investigate the influence of strong directional, or bonding, interactions on the phase diagram of complex fluids, and in particular on the liquid-vapour critical point. To this end we revisit a simple model and theory for associating fluids which consist of spherical particles having a hard-core repulsion, complemented by three short-ranged attractive sites on the surface (sticky spots). Two of the spots are of type A and one is of type B; the interactions between each pair of spots have strengths [image omitted], [image omitted] and [image omitted]. The theory is applied over the whole range of bonding strengths and results are interpreted in terms of the equilibrium cluster structures of the coexisting phases. In systems where unlike sites do not interact (i.e. where [image omitted]), the critical point exists all the way to [image omitted]. By contrast, when [image omitted], there is no critical point below a certain finite value of [image omitted]. These somewhat surprising results are rationalised in terms of the different network structures of the two systems: two long AA chains are linked by one BB bond (X-junction) in the former case, and by one AB bond (Y-junction) in the latter. The vapour-liquid transition may then be viewed as the condensation of these junctions and we find that X-junctions condense for any attractive [image omitted] (i.e. for any fraction of BB bonds), whereas condensation of the Y-junctions requires that [image omitted] be above a finite threshold (i.e. there must be a finite fraction of AB bonds).

Criticality of colloids with distinct interaction patches: The limits of linear chains, hyperbranched polymers, and dimers

We use a simple model of associating fluids which consists of spherical particles having a hard-core repulsion, complemented by three short-ranged attractive sites on the surface (sticky spots). Two of the spots are of type A and one is of type B; the bonding interactions between each pair of spots have strengths epsilon(AA), epsilon(BB), and epsilon(AB). The theory is applied over the whole range of bonding strengths and the results are interpreted in terms of the equilibrium cluster structures of the phases. In addition to our numerical results, we derive asymptotic expansions for the free energy in the limits for which there is no liquid-vapor critical point: linear chains (epsilon(AA)not equal 0, epsilon(AB)=epsilon(BB)=0), hyperbranched polymers (epsilon(AB)not equal 0, epsilon(AA)=epsilon(BB)=0), and dimers (epsilon(BB)not equal 0, epsilon(AA)=epsilon(AB)=0). These expansions also allow us to calculate the structure of the critical fluid by perturbing around the above limits, yielding three different types of condensation: of linear chains (AA clusters connected by a few AB or BB bonds); of hyperbranched polymers (AB clusters connected by AA bonds); or of dimers (BB clusters connected by AA bonds). Interestingly, there is no critical point when epsilon(AA) vanishes despite the fact that AA bonds alone cannot drive condensation.

Efficient Implementation Of A Single-Precision Floating-Point Arithmetic Unit on FPGA

This paper presents a single precision floating point arithmetic unit with support for multiplication, addition, fused multiply-add, reciprocal, square-root and inverse squareroot with high-performance and low resource usage. The design uses a piecewise 2nd order polynomial approximation to implement reciprocal, square-root and inverse square-root. The unit can be configured with any number of operations and is capable to calculate any function with a throughput of one operation per cycle. The floatingpoint multiplier of the unit is also used to implement the polynomial approximation and the fused multiply-add operation. We have compared our implementation with other state-of-the-art proposals, including the Xilinx Core-Gen operators, and conclude that the approach has a high relative performance/area efficiency. © 2014 Technical University of Munich (TUM).

Offshore wind turbine simulation: multibody drive train. Bacl-to-back NPC (Neutral Point Clamped) converters: Fractional-order control

This paper is on offshore wind energy conversion systems installed on the deep water and equipped with back-to-back neutral point clamped full-power converter, permanent magnet synchronous generator with an AC link. The model for the drive train is a five-mass model which incorporates the dynamic of the structure and the tower in order to emulate the effect of the moving surface. A three-level converter and a four-level converter are the two options with a fractional-order control strategy considered to equip the conversion system. Simulation studies are carried out to assess the quality of the energy injected into the electric grid. Finally, conclusions are presented. (C) 2014 Elsevier Ltd. All rights reserved.