Bifurcations of piecewise smooth flows:perspectives, methodologies and open problems

In this paper, the theory of bifurcations in piecewise smooth flows is critically surveyed. The focus is on results that hold in arbitrarily (but finitely) many dimensions, highlighting significant areas where a detailed understanding is presently lacking. The clearest results to date concern equilibria undergoing bifurcations at switching boundaries, and limit cycles undergoing grazing and sliding bifurcations. After discussing fundamental concepts, such as topological equivalence of two piecewise smooth systems, discontinuity-induced bifurcations are defined for equilibria and limit cycles. Conditions for equilibria to exist in n-dimensions are given, followed by the conditions under which they generically undergo codimension-one bifurcations. The extent of knowledge of their unfoldings is also summarized. Codimension-one bifurcations of limit cycles and boundary-intersection crossing are described together with techniques for their classification. Codimension-two bifurcations are discussed with suggestions for further study

Shrinking Point Bifurcations of Resonance Tongues for Piecewise-Smooth, Continuous Maps

Resonance tongues are mode-locking regions of parameter space in which stable periodic solutions occur; they commonly occur, for example, near Neimark-Sacker bifurcations. For piecewise-smooth, continuous maps these tongues typically have a distinctive lens-chain (or sausage) shape in two-parameter bifurcation diagrams. We give a symbolic description of a class of "rotational" periodic solutions that display lens-chain structures for a general $N$-dimensional map. We then unfold the codimension-two, shrinking point bifurcation, where the tongues have zero width. A number of codimension-one bifurcation curves emanate from shrinking points and we determine those that form tongue boundaries.Comment: 27 pages, 6 figure

Bifurcations in the Lozi map

We study the presence in the Lozi map of a type of abrupt order-to-order and order-to-chaos transitions which are mediated by an attractor made of a continuum of neutrally stable limit cycles, all with the same period.Comment: 17 pages, 12 figure

Monolingual and bilingual spanish-catalan speech recognizers developed from SpeechDat databases

Under the SpeechDat specifications, the Spanish member of SpeechDat consortium has recorded a Catalan database that includes one thousand speakers. This communication describes some experimental work that has been carried out using both the Spanish and the Catalan speech material. A speech recognition system has been trained for the Spanish language using a selection of the phonetically balanced utterances from the 4500 SpeechDat training sessions. Utterances with mispronounced or incomplete words and with intermittent noise were discarded. A set of 26 allophones was selected to account for the Spanish sounds and clustered demiphones have been used as context dependent sub-lexical units. Following the same methodology, a recognition system was trained from the Catalan SpeechDat database. Catalan sounds were described with 32 allophones. Additionally, a bilingual recognition system was built for both the Spanish and Catalan languages. By means of clustering techniques, the suitable set of allophones to cover simultaneously both languages was determined. Thus, 33 allophones were selected. The training material was built by the whole Catalan training material and the Spanish material coming from the Eastern region of Spain (the region where Catalan is spoken). The performance of the Spanish, Catalan and bilingual systems were assessed under the same framework. The Spanish system exhibits a significantly better performance than the rest of systems due to its better training. The bilingual system provides an equivalent performance to that afforded by both language specific systems trained with the Eastern Spanish material or the Catalan SpeechDat corpus.Peer ReviewedPostprint (published version

A Stochastic Broadcast Pi-Calculus

In this paper we propose a stochastic broadcast PI-calculus which can be used to model server-client based systems where synchronization is always governed by only one participant. Therefore, there is no need to determine the joint synchronization rates. We also take immediate transitions into account which is useful to model behaviors with no impact on the temporal properties of a system. Since immediate transitions may introduce non-determinism, we will show how these non-determinism can be resolved, and as result a valid CTMC will be obtained finally. Also some practical examples are given to show the application of this calculus.Comment: In Proceedings QAPL 2011, arXiv:1107.074

Smoothing tautologies, hidden dynamics, and sigmoid asymptotics for piecewise smooth systems

Switches in real systems take many forms, such as impacts, electronic relays, mitosis, and the implementation of decisions or control strategies. To understand what is lost, and what can be retained, when we model a switch as an instantaneous event, requires a consideration of so-called hidden terms. These are asymptotically vanishing outside the switch, but can be encoded in the form of nonlinear switching terms. A general expression for the switch can be developed in the form of a series of sigmoid functions. We review the key steps in extending the Filippov's method of sliding modes to such systems. We show how even slight nonlinear effects can hugely alter the behaviour of an electronic control circuit, and lead to `hidden' attractors inside the switching surface.Comment: 12 page

ENVIRONMENTAL AND ECONOMIC IMPACTS OF SOIL EROSION AND FERTILITY MINING IN NORTHERN TANZANIA

This paper develops a soil conservation model that is relevant to smallholder farmers who apply little or no fertilizer. Empirical results drawn from northern Tanzania imply that, ignoring fertility mining problem in model specification leads to overestimation of profits for farms that apply little or no fertilizer. The model also shows that, the impact of output price on soil conservation efforts depends on the curvature of the soil erosion function.Soil erosion, Fertility mining, Soil conservation, Price policy, Soil erosion function, Sub-Saharan Africa, Tanzania., Environmental Economics and Policy, Land Economics/Use,

Comparison between measured and predicted turbulence frequency spectra in ITG and TEM regimes

The observation of distinct peaks in tokamak core reflectometry measurements - named quasi-coherent-modes (QCMs) - are identified as a signature of Trapped-Electron-Mode (TEM) turbulence [H. Arnichand et al. 2016 Plasma Phys. Control. Fusion 58 014037]. This phenomenon is investigated with detailed linear and nonlinear gyrokinetic simulations using the \gene code. A Tore-Supra density scan is studied, which traverses through a Linear (LOC) to Saturated (SOC) Ohmic Confinement transition. The LOC and SOC phases are both simulated separately. In the LOC phase, where QCMs are observed, TEMs are robustly predicted unstable in linear studies. In the later SOC phase, where QCMs are no longer observed, ITG modes are identified. In nonlinear simulations, in the ITG (SOC) phase, a broadband spectrum is seen. In the TEM (LOC) phase, a clear emergence of a peak at the TEM frequencies is seen. This is due to reduced nonlinear frequency broadening of the underlying linear modes in the TEM regime compared with the ITG regime. A synthetic diagnostic of the nonlinearly simulated frequency spectra reproduces the features observed in the reflectometry measurements. These results support the identification of core QCMs as an experimental marker for TEM turbulenc