Order Of Nonlinear System Research Articles

Analysis of Noise-Induced Chaos-Order Transitions in Rulkov Model Near Crisis Bifurcations

Noise-induced transitions from chaos to order in nonlinear systems with crisis bifurcations are studied. In this study, a discrete-time Rulkov system is used as a conceptual model of the neuronal activity. We investigate probabilistic mechanisms of noise-induced transitions from chaotic spiking to quiescence in zones of crisis bifurcations. To analyze these transitions parametrically, we apply a mathematical technique based on the stochastic sensitivity functions and confidence domains. A stochastic phenomenon of the shifts of crisis bifurcation points and the expansion of the order window under increasing noise is discussed and analyzed. Using our analytical approach, we construct a parametric description of chaotic and regular regimes for the randomly forced Rulkov model.

Forwarding-based immersion and invariance control for $$\varvec{n}$$ n -dimensional strict-feedback nonlinear systems

Implementation of immersion and invariance (I&I) theorem becomes increasingly difficult as the degree of freedom increases. This inherent difficulty is mainly driven by the need to solve the partial differential equations in the immersion conditions. For a class of n-dimensional nonlinear systems in strict-feedback form, a novel method for designing asymptotically stabilizing control laws is developed in this work. At each step of the design, by defining a lower-order target system, the required mappings can be transformed into the virtual control inputs and the controlled system is able to exhibit the same responses as the target system. The order of nonlinear systems is reduced step by step by implementing I&I theorem repetitively. Unlike other control methodologies, the proposed algorithm does not require knowledge of Lyapunov functions in principle. In this paper, the design procedure and proof of closed-loop stability are described in detail. The TORA and magnetic levitation system are used to demonstrate the design procedures and controller performance via various simulations.

Regular low frequency cardiac output oscillations observed in critically ill surgical patients

A serendipitous finding during development of an automated “electronic flow chart” system to gather data on ICU patients [1] was the observation of low frequency oscillations in blood pressure that were not explained by systematic variability in the environment. [2] This finding has since been confirmed by others. [3,4] In the present report, hemodynamic data for critically ill surgical patients was continuously collected and visualized on a computer workstation to search for patterns not noted by standard monitoring. With this system, we observed low-frequency periodic oscillations in the cardiac output of ten patients, with regular periodicities of 4 to 280 minutes (average = 34 minutes). The mortality rate in these patients was 40%, while the mortality was only 10.8% in 83 similarly monitored intensive care unit (ICU) patients who did not develop regular oscillations in cardiac output. Interestingly, these oscillatory patterns appear to be associated with inadequate resuscitation of increased metabolic rates. The mathematical definition of “chaos” refers to irregular behavior that appears to be random but is actually deterministic. [5] A surprising finding concerning transitions between states of apparent randomness and order in nonlinear systems is that many systems become more organized after being disturbed. Chaotic behavior in biological systems may represent a normal physiologic state, while the loss of chaotic behavior may herald a pathophysiologic state. [6] The mechanism of the regular low frequency oscillations in cardiac output remains to be determined, but the high mortality rate suggests that it is a pathophysiologic marker, perhaps due to inadequate oxygen delivery in under-resuscitated shock. © 1997 John Wiley & Sons, Inc.

Subharmonics of any order in nonlinear systems of one degree of freedom: application to subharmonics of order 1/3

This paper consists of two parts. In the first one we discuss the subharmonics of any order in the case where the nonlinearity enters in the elastic forces. The basic differential equation is with coefficients not necessarily small. The “steady state” and “transient” solutions of the differential equation, and the conditions for the existence of the subharmonics and their stability in a steady state are examined. In the second part of the paper an application is given, namely, the investigation of the subharmonics of order 1/3 according to the theory of the first part. An illustrated example is given, the “inverse problem” is examined, and the conditions for the stability and instability of the “harmonic solution” are found.