scholarly journals Trapping Phenomenon Attenuates the Consequences of Tipping Points for Limit Cycles

2017 ◽  
Vol 7 (1) ◽  
Author(s):  
Everton S. Medeiros ◽  
Iberê L. Caldas ◽  
Murilo S. Baptista ◽  
Ulrike Feudel
Keyword(s):  
Limit Cycle ◽  
Limit Cycles ◽  
Nonlinear Dynamical Systems ◽  
Dynamical Behavior ◽  
Stable Limit Cycle ◽  
Tipping Points ◽  
Stable Limit ◽  
Nonlinear Dynamical ◽  
Natural Systems ◽  
Fold Bifurcation

Abstract Nonlinear dynamical systems may be exposed to tipping points, critical thresholds at which small changes in the external inputs or in the system’s parameters abruptly shift the system to an alternative state with a contrasting dynamical behavior. While tipping in a fold bifurcation of an equilibrium is well understood, much less is known about tipping of oscillations (limit cycles) though this dynamics are the typical response of many natural systems to a periodic external forcing, like e.g. seasonal forcing in ecology and climate sciences. We provide a detailed analysis of tipping phenomena in periodically forced systems and show that, when limit cycles are considered, a transient structure, so-called channel, plays a fundamental role in the transition. Specifically, we demonstrate that trajectories crossing such channel conserve, for a characteristic time, the twisting behavior of the stable limit cycle destroyed in the fold bifurcation of cycles. As a consequence, this channel acts like a “ghost” of the limit cycle destroyed in the critical transition and instead of the expected abrupt transition we find a smooth one. This smoothness is also the reason that it is difficult to precisely determine the transition point employing the usual indicators of tipping points, like critical slowing down and flickering.

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 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).

Sensitivity Enhancement of Cantilever-Based Sensors Using Feedback Delays

2010 ◽  
Vol 5 (4) ◽  
Author(s):  
Calvin Bradley ◽  
Mohammed F. Daqaq ◽  
Amin Bibo ◽  
Nader Jalili
Keyword(s):  
Limit Cycle ◽  
Limit Cycles ◽  
Stable Limit Cycle ◽  
Sensitivity Enhancement ◽  
Design Parameters ◽  
Feedback Signal ◽  
Stable Limit ◽  
Theoretical Understanding ◽  
Cycle Amplitude ◽  
Limit Cycle Amplitude

This paper entails a novel sensitivity-enhancement mechanism for cantilever-based sensors. The enhancement scheme is based on exciting the sensor at the clamped end using a delayed-feedback signal obtained by measuring the tip deflection of the sensor. The gain and delay of the feedback signal are chosen such that the base excitations set the beam into stable limit-cycle oscillations as a result of a supercritical Hopf bifurcation of the trivial fixed points. The amplitude of these limit-cycles is shown to be ultrasensitive to parameter variations and, hence, can be utilized for the detection of minute changes in the resonant frequency of the sensor. The first part of the manuscript delves into the theoretical understanding of the proposed mechanism and the operation concept. Using the method of multiple scales, an approximate analytical solution for the steady-state limit-cycle amplitude near the stability boundaries is obtained. This solution is then utilized to provide a comprehensive understanding of the effect of small frequency variations on the limit-cycle amplitude and the sensitivity of these limit-cycles to different design parameters. Once a deep theoretical understanding is established, the manuscript provides an experimental study to investigate the proposed concept. Experimental results demonstrate orders of magnitude sensitivity enhancement over the traditional frequency-shift method.

Robust Generation of Limit Cycles in Nonlinear Systems: Application on Two Mechanical Systems

2017 ◽  
Vol 12 (4) ◽  
Author(s):  
Ali Reza Hakimi ◽  
Tahereh Binazadeh
Keyword(s):  
Nonlinear Systems ◽  
Limit Cycle ◽  
Limit Cycles ◽  
Closed Loop ◽  
Stable Limit Cycle ◽  
Additional Term ◽  
Loop System ◽  
Stable Limit ◽  
Closed Loop System ◽  
Robust Performance

This paper studies inducing robust stable oscillations in nonlinear systems of any order. This goal is achieved through creating stable limit cycles in the closed-loop system. For this purpose, the Lyapunov stability theorem which is suitable for stability analysis of the limit cycles is used. In this approach, the Lyapunov function candidate should have zero value for all the points of the limit cycle and be positive in the other points in the vicinity of it. The proposed robust controller consists of a nominal control law with an additional term that guarantees the robust performance. It is proved that the designed controller results in creating the desirable stable limit cycle in the phase trajectories of the uncertain closed-loop system and leads to induce stable oscillations in the system's output. Additionally, in order to show the applicability of the proposed method, it is applied on two practical systems: a time-periodic microelectromechanical system (MEMS) with parametric errors and a single-link flexible joint robot in the presence of external disturbances. Computer simulations show the effective robust performance of the proposed controllers in generating the robust output oscillations.

Multiple Limit Cycles in Laser Interference Transduced Resonators

2011 ◽  
Author(s):  
David B. Blocher ◽  
Richard H. Rand ◽  
Alan T. Zehnder
Keyword(s):  
Limit Cycle ◽  
Limit Cycles ◽  
Thermal Stresses ◽  
Stable Limit Cycle ◽  
Laser Interferometry ◽  
Numerical Continuation ◽  
Stable Limit ◽  
Laser Interference ◽  
Direct Numerical Integration ◽  
Stable Limit Cycles

Nanoscale resonators whose motion is measured through laser interferometry are known to exhibit stable limit cycle motion. Motion of the resonator through the interference field modulates the amount of light absorbed by the resonator and hence the temperature field within it. The resulting coupling of motion and thermal stresses can lead to self oscillation, i.e. a limit cycle. In this work the coexistence of multiple stable limit cycles is demonstrated in an analytic model. Numerical continuation and direct numerical integration are used to study the structure of the solutions to the model. The effect of damping is discussed as well as the properties that would be necessary for physical devices to exhibit this behavior.

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

2020 ◽  
Vol 14 (9) ◽  
pp. 3175-3194
Author(s):  
Christian Schoof
Keyword(s):  
Limit Cycle ◽  
Limit Cycles ◽  
Drainage System ◽  
Stable Limit Cycle ◽  
Peak Discharge ◽  
Stable Limit ◽  
Outburst Floods ◽  
The World ◽  
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 the flood, 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 can control the onset of floods. In agreement with previous work, we show that floods only occur in a band of water throughput rates in which steady reservoir drainage is unstable, and we identify stabilizing mechanisms that allow steady drainage of an ice-dammed reservoir. We also show how stable limit cycle solutions emerge from the instability and identify parameter regimes in which the resulting floods cause flotation of the ice dam. These floods are likely to be initiated by flotation rather than the unstable enlargement of a distributed drainage system.

Robust Stable Limit Cycle Generation in Multi-Input Mechanical Systems

Robotica ◽  
2021 ◽  
pp. 1-12
Author(s):  
Tahereh Binazadeh ◽  
Mahsa Karimi
Keyword(s):  
Limit Cycle ◽  
Limit Cycles ◽  
Sliding Mode ◽  
Mechanical Systems ◽  
Stable Limit Cycle ◽  
Hamiltonian Function ◽  
Practical Case ◽  
Stable Limit ◽  
Mode Method ◽  
Stable Limit Cycles

SUMMARY This paper proposes a robust controller for the generation of stable limit cycles in multi-input mechanical systems subjected to model uncertainties. The proposed idea is based on Port-Controlled Hamiltonian (PCH) model and energy-based control by considering the Hamiltonian function as the Lyapunov function. For this purpose, first, a nominal controller is designed by shaping the energy function of the system according to the structure of the desired limit cycle. Then, an additional robustifying control term is designed based on the integral sliding mode method with the selection of an appropriate sliding surface. Finally, computer simulations for two practical case studies are provided to confirm the effectiveness of the proposed controller in the generation of stable limit cycles in the presence of uncertainties.

Bimolecular kinetic scheme with a stable and an unstable limiting cycle of two oscillating components

1980 ◽  
Vol 45 (8) ◽  
pp. 2135-2142 ◽  
Author(s):  
Antonín Tockstein ◽  
Karel Komers
Keyword(s):  
Limit Cycle ◽  
Limit Cycles ◽  
Rate Constants ◽  
Reaction Kinetic ◽  
Stable Limit Cycle ◽  
Kinetic Scheme ◽  
Stable Limit ◽  
Basic Scheme ◽  
Limiting Cycle ◽  
Consecutive Reactions

A simple reaction kinetic scheme of two oscillating components was found, whose analog solution shows a stable limit cycle separated from a stable focus by an unstable limit cycle. The scheme leads to trajectories which alternate between the types of limit cycles and stable spirals on changing the rate constants. Modifications of the basic scheme, which consists of five consecutive reactions with two accelerating parallel steps, were studied and its chemical realization was proposed.

scholarly journals STUDYING THE STABILITY BY USING LOCAL LINEARIZATION METHOD

2020 ◽  
Vol 2 (1) ◽  
pp. 27-31
Author(s):  
Abdulghafoor Jasim Salim ◽  
Kais Ismail Ebrahem ◽  
Suhirman
Keyword(s):  
Singular Point ◽  
Limit Cycle ◽  
Autoregressive Model ◽  
Stable Limit Cycle ◽  
Linearization Method ◽  
Stable Limit ◽  
Local Linearization ◽  
Non Linear ◽  
The Stability ◽  
Local Linearization Method

Abstract: In this paper we study the stability of one of a non linear autoregressive model with trigonometric term  by using local linearization method proposed by Tuhro Ozaki .We find the singular point ,the stability of the singular point and the limit cycle. We conclude  that the proposed model under certain conditions have a non-zero singular point which is  a asymptotically salable ( when  0 ) and have an  orbitaly stable limit cycle . Also we give some examples in order to explain the method. Key Words : Non-linear Autoregressive model; Limit cycle; singular point; Stability.

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