On the Regulators with Random Noises in Dynamic Block

The problem of controlling stochastic linear systems with quadratic criterion is considered. A class of optimal controllers which are equivalent to the separation theorem regulator is determined. For all of such controllers the quadratic functional has the same value. The effects of disregarded disturbances which are modeled by random noises in the dynamic block of the regulator are investigated. It is shown that the equivalent (in the classic propounding) controllers respond to these noises in different ways. Sometimes an "equivalent optimal" regulator may be less receptive towards additional disturbances than the standard one (which comes from the separation theorem). The optimal regulator is found which takes into account the presence of such noises

Noise-induced transitions and shifts in a climate-vegetation feedback model

Motivated by the extremely important role of the Earth’s vegetation dynamics in climate changes, we study the stochastic variability of a simple climate–vegetation system. In the case of deterministic dynamics, the system has one stable equilibrium and limit cycle or two stable equilibria corresponding to two opposite (cold and warm) climate–vegetation states. These states are divided by a separatrix going across a point of unstable equilibrium. Some possible stochastic scenarios caused by different externally induced natural and anthropogenic processes inherit properties of deterministic behaviour and drastically change the system dynamics. We demonstrate that the system transitions across its separatrix occur with increasing noise intensity. The climate–vegetation system therewith fluctuates, transits and localizes in the vicinity of its attractor. We show that this phenomenon occurs within some critical range of noise intensities. A noise-induced shift into the range of smaller global average temperatures corresponding to substantial oscillations of the Earth’s vegetation cover is revealed. Our analysis demonstrates that the climate–vegetation interactions essentially contribute to climate dynamics and should be taken into account in more precise and complex models of climate variability.</jats:p

Стохастическая генерация пачечных колебаний в трехмерной модели Хиндмарш–Роуз

We study the effect of random disturbances on the dynamics of the three-dimensional Hindmarsh–Rose model of neuronal activity. Due to the strong nonlinearity, even the original deterministic system ex- hibits diverse and complex dynamic regimes (various types of periodic oscillations, oscillations zones with period doubling and adding, coexistence of several attractors, chaos). In this paper, we consider a para- metric zone where a stable equilibrium is the only attractor. We show that even in this zone with simple deterministic dynamics, under the random disturbances, such complex effect as the stochastic generation of bursting oscillations can occur. For a small noise, random states concentrate near the equilibrium. With the increase of the noise intensity, random trajectories can go far from the stable equilibrium, and along with small-amplitude oscillations around the equilibrium, bursts are observed. This phenomenon is analysed using the mathematical methods based on the stochastic sensitivity function technique. An algorithm of estimation of critical values for noise intensity is proposedВ работе изучается воздействие случайных возмущений на динамику трехмерной модели Хи- ндмарш–Роуз нейронной активности. Благодаря сильной нелинейности даже исходная детер- минированная система демонстрирует весьма разнообразные и сложные динамические режимы (периодические колебания разных типов, удвоение и добавление периода колебаний, сосуществова- ние нескольких аттракторов, хаос). В данной статье рассматривается параметрическая зона, в которой единственным аттрактором является устойчивое равновесие. Показывается, что да- же в этой зоне с простой детерминированной динамикой под влиянием случайных возмущений в системе может наблюдаться такое сложное явление, как стохастическая генерация пачеч- ных колебаний. При малых шумах случайные состояния концентрируются вблизи устойчивого равновесия. При увеличении интенсивности шума траектории могут проходить далеко от рав- новесия и наряду с малоамплитудными осцилляциями вокруг равновесия наблюдаются пачечные колебания. Проводится анализ этого явления с помощью математических методов, основанных на технике функций стохастической чувствительности, и предлагается алгоритм оценки критических значений интенсивности шума, вызывающего пачечные колебани

Coexisting Attractors and Multistate Noise-Induced Intermittency in a Cycle Ring of Rulkov Neurons

We study dynamics of a unidirectional ring of three Rulkov neurons coupled by chemical synapses. We consider both deterministic and stochastic models. In the deterministic case, the neural dynamics transforms from a stable equilibrium into complex oscillatory regimes (periodic or chaotic) when the coupling strength is increased. The coexistence of complete synchronization, phase synchronization, and partial synchronization is observed. In the partial synchronization state either two neurons are synchronized and the third is in antiphase, or more complex combinations of synchronous and asynchronous interaction occur. In the stochastic model, we observe noise-induced destruction of complete synchronization leading to multistate intermittency between synchronous and asynchronous modes. We show that even small noise can transform the system from the regime of regular complete synchronization into the regime of asynchronous chaotic oscillations

How a small noise generates large-amplitude oscillations of volcanic plug and provides high seismicity

A non-linear behavior of dynamic model of the magma-plug system under the action of N-shaped friction force and stochastic disturbances is studied. It is shown that the deterministic dynamics essentially depends on the mutual arrangement of an equilibrium point and the friction force branches. Variations of this arrangement imply bifurcations, birth and disappearance of stable limit cycles, changes of the stability of equilibria, system transformations between mono- and bistable regimes. A slope of the right increasing branch of the friction function is responsible for the formation of such regimes. In a bistable zone, the noise generates transitions between small and large amplitude stochastic oscillations. In a monostable zone with single stable equilibrium, a new dynamic phenomenon of noise-induced generation of large amplitude stochastic oscillations in the plug rate and pressure is revealed. A beat-type dynamics of the plug displacement under the influence of stochastic forcing is studied as well

Noise-induced generation of saw-tooth type transitions between climate attractors and stochastic excitability of paleoclimate

Motivated by important paleoclimate applications we study a three dimensional model of the Quaternary climatic variations in the presence of stochastic forcing. It is shown that the deterministic system exhibits a limit cycle and two stable system equilibria. We demonstrate that the closer paleoclimate system to its bifurcation points (lying either in its monostable or bistable zone) the smaller noise generates small or large amplitude stochastic oscillations, respectively. In the bistable zone with two stable equilibria, noise induces a complex multimodal stochastic regime with intermittency of small and large amplitude stochastic fluctuations. In the monostable zone, the small amplitude stochastic oscillations localized in the vicinity of unstable equilibrium appear along with the large amplitude oscillations near the stable limit cycle. For the analysis of these noise-induced effects, we develop the stochastic sensitivity technique and use the Mahalanobis metric in the three-dimensional case. To approximate the distribution of random trajectories in Poincare sections, we use a method of confidence ellipses. A spatial configuration of these ellipses is defined by the stochastic sensitivity and noise intensity. The glaciation/deglaciation transitions going between two polar Earth’s states with the warm and cold climate become easier and quicker with increasing the noise intensity. Our stochastic analysis demonstrates a near 100 ky saw-tooth type climate self fluctuations known from paleoclimate records. In addition, the enhancement of noise intensity blurs the sharp climate cycles and reduces the glaciation-deglaciation periods of the Earth’s paleoclimate

Analysis of noise-induced eruptions in a geyser model

Motivated by important geophysical applications we study a non-linear model of geyser dynamics under the influence of external stochastic forcing. It is shown that the deterministic dynamics is substantially dependent on system parameters leading to the following evolutionary scenaria: (i) oscillations near a stable equilibrium and a transient tendency of the phase trajectories to a spiral sink or a stable node (pre-eruption regime), and (ii) fast escape from equilibrium (eruption regime). Even a small noise changes the system dynamics drastically. Namely, a low-intensity noise generates the small amplitude stochastic oscillations in the regions adjoining to the stable equilibrium point. A small buildup of noise intensity throws the system over its separatrix and leads to eruption. The role of the friction coefficient and relative pressure in the deterministic and stochastic dynamics is studied by direct numerical simulations and stochastic sensitivity functions technique

Coexisting Attractors and Multistate Noise-Induced Intermittency in a Cycle Ring of Rulkov Neurons

We study dynamics of a unidirectional ring of three Rulkov neurons coupled by chemical synapses. We consider both deterministic and stochastic models. In the deterministic case, the neural dynamics transforms from a stable equilibrium into complex oscillatory regimes (periodic or chaotic) when the coupling strength is increased. The coexistence of complete synchronization, phase synchronization, and partial synchronization is observed. In the partial synchronization state either two neurons are synchronized and the third is in antiphase, or more complex combinations of synchronous and asynchronous interaction occur. In the stochastic model, we observe noise-induced destruction of complete synchronization leading to multistate intermittency between synchronous and asynchronous modes. We show that even small noise can transform the system from the regime of regular complete synchronization into the regime of asynchronous chaotic oscillations