Local stability implies global stability for the 2-dimensional Ricker map

Consider the difference equation $x_{k+1}=x_k e^{\alpha-x_{n-d}}$ where $\alpha$ is a positive parameter and d is a non-negative integer. The case d = 0 was introduced by W.E. Ricker in 1954. For the delayed version d >= 1 of the equation S. Levin and R. May conjectured in 1976 that local stability of the nontrivial equilibrium implies its global stability. Based on rigorous, computer aided calculations and analytical tools, we prove the conjecture for d = 1.Comment: for associated C++ program, mathematica worksheet and output, see http://www.math.u-szeged.hu/~krisztin/ricke

Global stability in a system using echo for position control

We consider a system of equations describing automatic position control by echo. The system can be reduced to a single differential equation with state-dependent delay. The delayed terms come from the control mechanism and the reaction time. H.-O. Walther [Differ. Integral Equ. 15(2002), No. 8, 923–944] proved that stable periodic motion is possible for large enough reaction time. We show that, for sufficiently small reaction lag, the control is perfect, i.e., the preferred position of the system is globally asymptotically stable

Global stability in a system using echo for position control

We consider a system of equations describing automatic position control by echo. The system can be reduced to a single differential equation with state-dependent delay. The delayed terms come from the control mechanism and the reaction time. H.-O. Walther [Differ. Integral Equ. 15(2002), No. 8, 923–944] proved that stable periodic motion is possible for large enough reaction time. We show that, for sufficiently small reaction lag, the control is perfect, i.e., the preferred position of the system is globally asymptotically stable

Enclosing the behavior of a hybrid automaton up to and beyond a Zeno point

Even simple hybrid automata like the classic bouncing ball can exhibit Zeno behavior. The existence of this type of behavior has so far forced a large class of simulators to either ignore some events or risk looping indefinitely. This in turn forces modelers to either insert ad-hoc restrictions to circumvent Zeno behavior or to abandon hybrid automata. To address this problem, we take a fresh look at event detection and localization. A key insight that emerges from this investigation is that an enclosure for a given time interval can be valid independent of the occurrence of a given event. Such an event can then even occur an unbounded number of times. This insight makes it possible to handle some types of Zeno behavior. If the post-Zeno state is defined explicitly in the given model of the hybrid automaton, the computed enclosure covers the corresponding trajectory that starts from the Zeno point through a restarted evolution

PARP-Inhibitor Treatment Prevents Hypertension Induced Cardiac Remodeling by Favorable Modulation of Heat Shock Proteins, Akt-1/GSK-3beta and Several PKC Isoforms.

Spontaneously hypertensive rat (SHR) is a suitable model for studies of the complications of hypertension. It is known that activation of poly(ADP-ribose) polymerase enzyme (PARP) plays an important role in the development of postinfarction as well as long-term hypertension induced heart failure. In this study, we examined whether PARP-inhibitor (L-2286) treatment could prevent the development of hypertensive cardiopathy in SHRs. 6-week-old SHR animals were treated with L-2286 (SHR-L group) or placebo (SHR-C group) for 24 weeks. Wistar-Kyoto rats were used as aged-matched, normotensive controls (WKY group). Echocardiography was performed, brain-derived natriuretic peptide (BNP) activity and blood pressure were determined at the end of the study. We detected the extent of fibrotic areas. The amount of heat-shock proteins (Hsps) and the phosphorylation state of Akt-1Ser473, glycogen synthase kinase (GSK)-3betaSer9, forkhead transcription factor (FKHR)Ser256, mitogen activated protein kinases (MAPKs), and protein kinase C (PKC) isoenzymes were monitored. The elevated blood pressure in SHRs was not influenced by PARP-inhibitor treatment. Systolic left ventricular function and BNP activity did not differ among the three groups. L-2286 treatment decreased the marked left ventricular (LV) hypertrophy which was developed in SHRs. Interstitial collagen deposition was also decreased by L-2286 treatment. The phosphorylation of extracellular signal-regulated kinase (ERK)1/2Thr183-Tyr185, Akt-1Ser473, GSK-3betaSer9, FKHRSer256, and PKC epsilonSer729 and the level of Hsp90 were increased, while the activity of PKC alpha/betaIIThr638/641, zeta/lambda410/403 were mitigated by L-2286 administration. We could detect signs of LV hypertrophy without congestive heart failure in SHR groups. This alteration was prevented by PARP inhibition. Our results suggest that PARP-inhibitor treatment has protective effect already in the early stage of hypertensive myocardial remodeling

Acumen : an open-source testbed for cyber-physical systems research

Developing Cyber-Physical Systems requires methods and tools to support simulation and verification of hybrid (both continuous and discrete) models. The Acumen modeling and simulation language is an open source testbed for exploring the design space of what rigorousbut- practical next-generation tools can deliver to developers of Cyber- Physical Systems. Like verification tools, a design goal for Acumen is to provide rigorous results. Like simulation tools, it aims to be intuitive, practical, and scalable. However, it is far from evident whether these two goals can be achieved simultaneously. This paper explains the primary design goals for Acumen, the core challenges that must be addressed in order to achieve these goals, the “agile research method” taken by the project, the steps taken to realize these goals, the key lessons learned, and the emerging language design

Computer-aided proofs and algorithms in analysis

The computational power has increased dramatically since the appearance of the first computers, making them a vital tool in the analysis of dynamical systems. We present further applications of those two basic ideas, namely interval arithmetic and automatic differentiation, that address the question of the reliability of the results and the difficulty of calculating derivatives. In general, the result of a numerical calculation will be influenced by errors, since the set of the numbers represented by the machine is finite. This will inevitably lead to round-off and truncation errors. This should not be considered as a problem, but rather as the true nature of numerics. The notorious examples like evaluating 333.75y6+ x2(11x2y2−y6−121y4−2)+5.5y8+x/(2y) at (x, y) = (77617,33096) or plotting the polynomial t6 −6t5 +15t4 −20t3 +15t2 −6t +1 in a small neighborhood of 1, still result in unexpected outcomes, if one is unaware of the potential risks of the floating point computations. We mention the failure of a Patriot missile on February 25, 1991 or the explosion of the unmanned space rocket Ariane 5 on June 4, 1996 as practical examples of these potential risks becoming real. Therefore in mathematical proofs, where the beauty of the argument is its unquestionable truth itself, the usage of computers must be handled with extreme care. One technique, that is used to overcome these problems and make our computations rigorous, is called interval arithmetic. To calculate derivatives of a given function is often considered to be a hard problem, since in general with increasing the order or the dimension, the complexity of the formula of the derivative grows exponentially. The observation, that we do not need these formulae in general, but only certain values of the derivatives, is crucial to understand why automatic differentiation is so useful. The structure of the thesis is as follows. In Part I we give an introduction to the methods used in our papers. In Chapter 1 we get acquainted with the basic techniques, interval arithmetic, interval analysis, floating point computations and automatic differentiation. Chapter 2 gives an overview of the interaction between dynamical systems and different representations of the data. In Chapter 3 we take on the basic concept of automatic differentiation seen before, and present a method by Griewank et al. [17] to compute higher order derivatives of multivariate functions that will be used in Paper A. We go through the theory of graph representations in Chapter 4 by following the steps of Hohmann and Dellnitz [12] and Galias [15]. This theory may be used in qualitative analysis of maps. We give two applications in Paper B and Paper C. In addition, we give the proof of correctness of the algorithm for enclosing non-wandering points in Paper B. In Chapter 5 we introduce the reader to the method of self-consistent bounds by Zgliczy´nski and Mischaikow [44] and Zgliczy´nski [40, 42, 43] that may be used to analyze a certain class of dissipative partial differential equations. An application of this concept to a destabilized Kuramoto-Sivashinsky equation is given in Paper D. Chapter 6 gives a short overview of the results of the included papers. Part II is the main scientific contribution of this thesis, consisting of the formerly mentioned four papers

Global stability in a system using echo for position control

We consider a system of equations describing automatic position control by echo. The system can be reduced to a single differential equation with state-dependent delay. The delayed terms come from the control mechanism and the reaction time. H.-O. Walther [Differ. Integral Equ. 15(2002), No. 8, 923–944] proved that stable periodic motion is possible for large enough reaction time. We show that, for sufficiently small reaction lag, the control is perfect, i.e., the preferred position of the system is globally asymptotically stable