scholarly journals Bifurcations of Finite-Time Stable Limit Cycles from Focus Boundary Equilibria in Impacting Systems, Filippov Systems, and Sweeping Processes

2018 ◽  
Vol 28 (10) ◽  
pp. 1850126 ◽  
Author(s):  
Oleg Makarenkov ◽  
Lakmi Niwanthi Wadippuli Achchige
Keyword(s):  
Limit Cycles ◽  
Finite Time ◽  
Stable Limit Cycle ◽  
Piecewise Smooth ◽  
Stable Limit ◽  
Bifurcation Of Limit Cycles ◽  
Filippov Systems ◽  
Boundary Equilibrium ◽  
Sweeping Processes ◽  
Linearized System

We establish a theorem on bifurcation of limit cycles from a focus boundary equilibrium of an impacting system, which is universally applicable to prove the bifurcation of limit cycles from focus boundary equilibria in other types of piecewise-smooth systems, such as Filippov systems and sweeping processes. Specifically, we assume that one of the subsystems of the piecewise-smooth system under consideration admits a focus equilibrium that lie on the switching manifold at the bifurcation value of the parameter. In each of the three cases, we derive a linearized system which is capable of concluding the occurrence of a finite-time stable limit cycle from the above-mentioned focus equilibrium when the parameter crosses the bifurcation value. Examples illustrate how conditions of our theorems lead to closed-form formulas for the coefficients of the linearized system.

Regularization of Planar Piecewise Smooth Systems with a Heteroclinic Loop

2021 ◽  
Vol 31 (15) ◽  
Author(s):  
Zhongjian Wang ◽  
Dingheng Pi
Keyword(s):  
Limit Cycles ◽  
Regular Point ◽  
Stable Limit Cycle ◽  
Piecewise Smooth ◽  
Stable Limit ◽  
Piecewise Smooth Systems ◽  
Heteroclinic Loop ◽  
Piecewise Smooth System ◽  
Stable Periodic ◽  
Smooth System

In this paper, we study bifurcations of the regularized systems of planar piecewise smooth systems, which have a visible fold-regular point and a sliding or grazing heteroclinic loop. Our results show that if the planar piecewise smooth system with a sliding heteroclinic loop undergoes sliding heteroclinic bifurcation, then the regularized system can bifurcate with a stable limit cycle passing through the regularized region and at most two limit cycles outside the regularized region. The regularized system can have at most three periodic orbits. When the upper subsystem is a Hamiltonian system, the regularized system can bifurcate with a semi-stable periodic orbit. Finally, we discuss two cases when the heteroclinic loop of a piecewise smooth system remains unbroken under a small perturbation. Our results show that the regularized system can bifurcate at most two limit cycles from an inner unstable grazing heteroclinic loop.

Analysis of chemical kinetic systems over the entire parameter space - I. The Sal’nikov thermokinetic oscillator

1988 ◽  
Vol 416 (1851) ◽  
pp. 391-402 ◽  
Keyword(s):  
Steady State ◽  
Parameter Space ◽  
Limit Cycles ◽  
Stable Limit Cycle ◽  
Chemical Kinetic ◽  
Phase Portraits ◽  
Stable Steady State ◽  
Stable Limit ◽  
Unstable Steady State ◽  
Undamped Oscillations

In this series of papers we re-examine, using recently developed techniques, some chemical kinetic models that have appeared in the literature with a view to obtaining a complete description of all the qualitatively distinct behaviour that the system can exhibit. Each of the schemes is describable by two coupled ordinary differential equations and contain at most three independent parameters. We find that even with these relatively simple chemical schemes there are regions of parameter space in which the systems display behaviour not previously found. Quite often these regions are small and it seems unlikely that they would be found via classical methods. In part I of the series we consider one of the thermally coupled kinetic oscillator models studied by Sal’nikov. He showed that there is a region in parameter space in which the system would be in a state of undamped oscillations because the relevant phase portrait consists of an unstable steady state surrounded by a stable limit cycle. Our analysis has revealed two further regions in which the phase portraits contain, respectively, two limit cycles of opposite stability enclosing a stable steady state and three limit cycles of alternating stability surrounding an unstable steady state. This latter region is extremely small, so much so that it could be reasonably neglected in any predictions made from the model.

The Cyclicity of Period Annulus of Degenerate Quadratic Hamiltonian Systems with Polycycles S(2) or S(3) Under Perturbations of Piecewise Smooth Polynomials

2020 ◽  
Vol 30 (15) ◽  
pp. 2050230
Author(s):  
Jiaxin Wang ◽  
Liqin Zhao
Keyword(s):  
Hamiltonian Systems ◽  
Limit Cycles ◽  
Upper Bounds ◽  
Melnikov Function ◽  
Piecewise Smooth ◽  
Bifurcation Of Limit Cycles ◽  
First Order ◽  
Period Annulus ◽  
Quadratic Hamiltonian

In this paper, by using Picard–Fuchs equations and Chebyshev criterion, we study the bifurcation of limit cycles for degenerate quadratic Hamilton systems with polycycles [Formula: see text] or [Formula: see text] under the perturbations of piecewise smooth polynomials with degree [Formula: see text]. Roughly speaking, for [Formula: see text], a polycycle [Formula: see text] is cyclically ordered collection of [Formula: see text] saddles together with orbits connecting them in specified order. The discontinuity is on the line [Formula: see text]. If the first order Melnikov function is not equal to zero identically, it is proved that the upper bounds of the number of limit cycles bifurcating from each of the period annuli with the boundary [Formula: see text] and [Formula: see text] are respectively [Formula: see text] and [Formula: see text] (taking into account the multiplicity).

Stochastically perturbed limit cycles

1978 ◽  
Vol 15 (02) ◽  
pp. 311-320
Author(s):  
Charles J. Holland
Keyword(s):  
Dynamical Systems ◽  
Invariant Measure ◽  
Probability Measure ◽  
Limit Cycles ◽  
Noise Intensity ◽  
Stable Limit Cycle ◽  
Stable Limit ◽  
Additive White Noise ◽  
Small Intensity ◽  
Deterministic Dynamical Systems

In this paper we examine the effects of perturbing certain deterministic dynamical systems possessing a stable limit cycle by an additive white noise term with small intensity. We place assumptions on the system guaranteeing that when noise is present the corresponding random process generates an ergodic probability measure. We then determine the behavior of the invariant measure when the noise intensity is small.

scholarly journals An analysis of instabilities and limit cycles in glacier-dammed reservoirs

2019 ◽  
Author(s):  
Christian Schoof
Keyword(s):  
Limit Cycle ◽  
Limit Cycles ◽  
Water Storage ◽  
Drainage System ◽  
Stable Limit Cycle ◽  
Stable Limit ◽  
Outburst Floods ◽  
The World ◽  
The Stability ◽  
Glacial Hazards

Abstract. Glacier lake outburst floods are common glacial hazards around the world. How big such floods can become (either in terms of peak discharge or in terms of total volume released) depends on how they are initiated: what causes the runaway enlargement of a subglacial or other conduit to start, and how big can the lake get before that point is reached? Here we investigate how the spontaneous channelization of a linked-cavity drainage system controls the onset of floods. In agreement with previous work, we show that floods only occur in a band of water throughput rates, and identify stabilizing mechanisms that allow steady drainage of an ice-dammed reservoir. We also show how stable limit cycle solutions emerge from the instability, a show how and why the stability properties of a drainage system with spatially spread-out water storage differ from those where storage is localized in a single reservoir or lake.

The Multiple Contributions of Jorgen Lund’s Ph.D. Dissertation, “Self-Excited, Stationary Whirl Orbits of a Journal in Sleeve Bearings,” RPI, 1966, Engineering Mechanics

2001 ◽  
Author(s):  
Dara W. Childs
Keyword(s):  
Limit Cycle ◽  
Limit Cycles ◽  
Stable Limit Cycle ◽  
Stable Limit ◽  
Rigid Rotor ◽  
Engineering Mechanics ◽  
Stable Limit Cycles

Abstract Lund set out to define the circumstances under which stable limit-cycle orbits could exist for the linearly unstable motion of a rigid rotor. He also undertook to examine the nature of these stable limit cycles when they are demonstrated to exist. He obviously succeeded in meeting both these objectives; however, Lund’s really remarkable and most useful contributions are covered “incidentally” in the course of developing his nonlinear analytical/computational solutions.

scholarly journals Flow-Excited Turbine Rotor Instability

1994 ◽  
Vol 1 (1) ◽  
pp. 37-51 ◽  
Author(s):  
Andrew D. Dimarogonas ◽  
Julio C. Gomez-Mancilla
Keyword(s):  
Limit Cycles ◽  
High Amplitude ◽  
Turbine Rotor ◽  
Design Stage ◽  
Stable Limit ◽  
Element Analysis ◽  
Tilting Pad ◽  
Linearized System ◽  
The Stability ◽  
Analytical Expressions

The problem of steam whirl is one of the technological limits that now prohibit the development of power-generating turbomachinery substantially above GW. Due to steam flow, self-excited vibrations develop at high loads in the form of stable limit cycles that, at even higher loads, deteriorate to chaotic vibration of high amplitude.A mathematical model is developed for stability analysis and for the development of a rational stability criterion to be used at the design stage. The bearing nonlinearity is introduced in the form of high-order coefficients of a Taylor expansion of the perturbation forces for fixed-arc slider bearings and employing nonlinear pad functions for the tilting pad bearings. The flow excitation is introduced in the form of follower force gradients related to the flow and the power generated.The study of the stable and unstable limit cycles, and the stability of the system in the large, beyond the linear analysis currently utilized, is done analytically for the De Laval rotor and numerically with finite element analysis of typical turbomachinery rotors.The range of loads for which limit cycles exist was found to be substantial. This is important for the operation of large machinery because such cycles permit the operation at loads much higher than the ones that correspond to the onset of instability of the linearized system. The conditions for the limit cycle deterioration into chaotic orbit are investigated. Analytical expressions have been obtained for the different stability thresholds for the De Laval rotor.

scholarly journals Modeling of Chaotic Processes by Means of Antisymmetric Neural ODEs

2021 ◽  
Author(s):  
Vasiliy Belozyorov ◽  
Danylo Dantsev
Keyword(s):  
Time Series ◽  
Continuous Function ◽  
Limit Cycles ◽  
Homoclinic Orbit ◽  
Chaotic Attractor ◽  
Stable Limit Cycle ◽  
Stable Limit ◽  
Activation Functions ◽  
Kolmogorov Theorem ◽  
Stable Limit Cycles

Abstract The main goal of this work is to construct an algorithm for modeling chaotic processes using special neural ODEs with antisymmetric matrices (antisymmetric neural ODEs) and power activation functions (PAFs). The central part of this algorithm is to design a neural ODEs architecture that would guarantee the generation of a stable limit cycle for a known time series. Then, one neuron is added to each equation of the created system until the approximating properties of this system satisfy the well-known Kolmogorov theorem on the approximation of a continuous function of many variables. In addition, as a result of such an addition of neurons, the cascade of bifurcations that allows generating a chaotic attractor from stable limit cycles is launched. We also consider the possibility of generating a homoclinic orbit whose bifurcations lead to the appearance of a chaotic attractor of another type. In conclusion, the conditions under which the found attractor adequately simulates the chaotic process are discussed. Examples are given.

scholarly journals STUDY POINTS OF REST HEREDITARITY DYNAMIC SYSTEMS VAN DER POL-DUFFING

2019 ◽  
pp. 71-77
Author(s):  
Е.Р. Новикова ◽  
Р.И. Паровик
Keyword(s):  
Limit Cycle ◽  
Dynamic Systems ◽  
Limit Cycles ◽  
Stable Limit Cycle ◽  
Viscous Friction ◽  
Oscillatory System ◽  
Stable Limit ◽  
Van Der Pol ◽  
Nonlinear Oscillatory System ◽  
Stable Limit Cycles

Using numerical modeling, oscillograms and phase trajectories were constructed to study the limit cycles of a van der Pol Duffing nonlinear oscillatory system with a power memory. The simulation results showed that in the absence of a power memory (α = 2, β = 1) or the classical van der Pol Duffing dynamical system, there is a single stable limit cycle, i.e. Lienar theorem holds. In the case of viscous friction (α = 2, 0 < β < 1), there is a family of stable limit cycles of various shapes. In other cases, the limit cycle is destroyed in two scenarios: a Hopf bifurcation (limit cycle-limit point) or (limit cycle-aperiodic process). Further continuation of the research may be related to the construction of the spectrum of Lyapunov maximal exponents in order to identify chaotic oscillatory regimes for the considered hereditary dynamic system (HDS). В работе с помощью численного моделирования построены осциллограммы и фазовые траектории с целью исследования предельных циклов нелинейной колебательной системы Ван-дер-Поля Дуффинга со степенной памятью. Результаты моделирования показали, что в случае отсутствия степенной памяти (α = 2, β = 1) или классической динамической системы Ван-дер-Поля Дуффинга, существует единственный устойчивый предельный цикл, т.е. выполняется теорема Льенара. В случае вязкого трения (α = 2, 0 < β < 1), существует семейство устойчивых предельных циклов различной формы. В остальных случаях происходит разрушение предельного цикла по двум сценариям: бифуркация Хопфа (предельный цикл-предельная точка) или (предельный циклапериодический процесс). Дальнейшее продолжение исследований может быть связано с построением спектра максимальных показателей Ляпунова с целью идентификации хаотических колебательных режимов для рассматриваемой эредитарной динамической системы (ЭДС).

On the Behaviour of Steamwhirl and Nonlinear Rotor-Bearing Systems

1993 ◽  
Author(s):  
Julio C. Gómez-Mancilla ◽  
Andrew D. Dimarogonas
Keyword(s):  
Limit Cycles ◽  
Tangential Force ◽  
Stable Limit ◽  
Slider Bearing ◽  
Chaotic Orbit ◽  
Element Analysis ◽  
Tilting Pad ◽  
Linearized System ◽  
Analytical Expressions ◽  
Stable Limit Cycles

Abstract The problem of steamwhirl is the technological limit which now prohibits the development of power generating turbomachinery substantially above 1 GW. Due to the steam flow, self excited vibrations develop at high loads, above the onset of instability of the linearized system, in the form of stable limit cycles which, at even higher loads, deteriorate to chaotic vibration. The bearing nonlinearity is introduced in the form of high order coefficients of a Taylor expansion of the perturbation forces for fixed-arc slider bearing and employing non-linear pad functions for the tilting pad bearings. The flow excitation is introduced in the form of radial and tangential force gradients related to the flow and power generated. The study of stable and unstable limit cycles and stability of the system in the large, beyond the linear analysis currently utilized, is done analytically for the DeLaval rotor and numerically with Finite Element analysis of typical turbomachinery rotors. The range of loads for which limit cycles exist was found to be substantial. This is important for the operation of large machinery because such limit cycles permit the operation at loads much higher than the ones which correspond to the onset of instability of the linearized system. The conditions for the limit cycle deterioration into chaotic orbit is studied. Analytical expressions have been obtained for the different thresholds for the DeLaval rotor.

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