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

Quantitative Characterization of α-Synuclein Aggregation in Living Cells through Automated Microfluidics Feedback Control

Aggregation of α-synuclein and formation of inclusions are hallmarks of Parkinson's disease (PD). Aggregate formation is affected by cellular environment, but it has been studied almost exclusively in cell-free systems. We quantitatively analyzed α-synuclein inclusion formation and clearance in a yeast cell model of PD expressing either wild-type (WT) α-synuclein or the disease-associated A53T mutant from the galactose (Gal)-inducible promoter. A computer-controlled microfluidics device regulated α-synuclein in cells by means of closed-loop feedback control. We demonstrated that inclusion formation is strictly concentration dependent and that the aggregation threshold of the A53T mutant is about half of the WT α-synuclein (56%). We chemically modulated the proteasomal and autophagic pathways and demonstrated that autophagy is the main determinant of A53T α-synuclein inclusions’ clearance. In addition to proposing a technology to overcome current limitations in dynamically regulating protein expression levels, our results contribute to the biology of PD and have relevance for therapeutic applications

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

Bifurcation Phenomena in Two-Dimensional Piecewise Smooth Discontinuous Maps

In recent years the theory of border collision bifurcations has been developed for piecewise smooth maps that are continuous across the border, and has been successfully applied to explain nonsmooth bifurcation phenomena in physical systems. However, many switching dynamical systems have been found to yield two-dimensional piecewise smooth maps that are discontinuous across the border. The theory for understanding the bifurcation phenomena in such systems is not available yet. In this paper we present the first approach to the problem of analysing and classifying the bifurcation phenomena in two-dimensional discontinuous maps, based on a piecewise linear approximation in the neighborhood of the border. We explain the bifurcations occurring in the static VAR compensator used in electrical power systems, using the theory developed in this paper. This theory may be applied similarly to other systems that yield two-dimensional discontinuous maps

Coexisting patterns of population oscillations: the degenerate Neimark Sacker bifurcation as a generic mechanism

We investigate a population dynamics model that exhibits a Neimark Sacker bifurcation with a period that is naturally close to 4. Beyond the bifurcation, the period becomes soon locked at 4 due to a strong resonance, and a second attractor of period 2 emerges, which coexists with the first attractor over a considerable parameter range. A linear stability analysis and a numerical investigation of the second attractor reveal that the bifurcations producing the second attractor occur naturally in this type of system.Comment: 8 pages, 3 figure

Design and validation of a virtual player for studying interpersonal coordination in the mirror game

This is the author accepted manuscript. The final version is available from the publisher via the DOI in this record.The mirror game has been recently proposed as a simple, yet powerful paradigm for studying interpersonal interactions. It has been suggested that a virtual partner able to play the game with human subjects can be an effective tool to affect the underlying neural processes needed to establish the necessary connections between the players, and also to provide new clinical interventions for rehabilitation of patients suffering from social disorders. Inspired by the motor processes of the central nervous system (CNS) and the musculoskeletal system in the human body, in this paper we develop a novel interactive cognitive architecture based on nonlinear control theory to drive a virtual player (VP) to play the mirror game with a human player (HP) in different configurations. Specifically, we consider two cases: the former where the VP acts as leader and the latter where it acts as follower. The crucial problem is to design a feedback control architecture capable of imitating and following or leading a human player in a joint action task. Movement of the end-effector of the VP is modeled by means of a feedback controlled Haken-Kelso-Bunz (HKB) oscillator, which is coupled with the observed motion of the HP measured in real time. To this aim, two types of control algorithms (adaptive control and optimal control) are used and implemented on the HKB model so that the VP can generate a human-like motion while satisfying certain kinematic constraints. A proof of convergence of the control algorithms is presented in the paper together with an extensive numerical and experimental validation of their effectiveness. A comparison with other existing designs is also discussed, showing the flexibility and the advantages of our control-based approach.This work was funded by the European Project AlterEgo FP7 ICT 2.9 - Cognitive Sciences and Robotics, Grant Number 600610

Simultaneous Border-Collision and Period-Doubling Bifurcations

We unfold the codimension-two simultaneous occurrence of a border-collision bifurcation and a period-doubling bifurcation for a general piecewise-smooth, continuous map. We find that, with sufficient non-degeneracy conditions, a locus of period-doubling bifurcations emanates non-tangentially from a locus of border-collision bifurcations. The corresponding period-doubled solution undergoes a border-collision bifurcation along a curve emanating from the codimension-two point and tangent to the period-doubling locus here. In the case that the map is one-dimensional local dynamics are completely classified; in particular, we give conditions that ensure chaos.Comment: 22 pages; 5 figure

Activity of bacterial seed endophytes of landrace durum wheat for control of Fusarium foot rot

Five bacterial endophytic isolates obtained from durum wheat seeds (Iandrace "Timilia reste nere") and identified as belonging to Pantoea (isolates A1, F7, F15 and GI) and Paenibacillus (isolate B) genera on the basis of 16S rDNA gene sequences, were assayed in vitro and in vivo for their ability to inhibit Fusarium culmorum growth and the disease (Fusarium foot rot) it causes in durum wheat. All isolates significantly reduced in vitro growth of F. culmorum in comparison with the control. After 120 hours of incubation, isolates B and GI showed the greatest mycelial growth inhibition, i.e., respectively, 76 and 74%. When durum wheat "Simeto" seeds were treated with bacterial isolates singly or in combinations and then inoculated with F. culmorum, all treatments with endophytes showed increased, but not statistically significant, seed germination. Except for isolate Al, all bacterial isolates stimulated vegetative parameters of durum wheat seedlings. Mixture of isolates F7 + F15 was the most effective in improving shoot height (+94%), root length (+47%) and vigour index (+81%). Mixture of isolates A1 + B reduced Fusarium foot rot incidence (-21%) and severity (-30%), and isolate A1 reduced incidence (-15%) and severity (-16%) of the disease. These results indicate potential of bacterial seed endophytes, identified in this study, for control of Fusarium foot rot and suggest that bacterial seed endophytes may provide a new biocontrol agent for an environmentally sustainable durum wheat disease management programme