Simple Limit Cycle Research Articles

Quantum asymptotic phases reveal signatures of quantum synchronization

Synchronization of quantum nonlinear oscillators has attracted much attention recently. To characterize the quantum oscillatory dynamics, we recently proposed a fully quantum-mechanical definition of the asymptotic phase, which is a key quantity in the synchronization analysis of classical nonlinear oscillators (Kato and Nakao 2022 Chaos 32 063133). In this work, we further extend this theory and introduce multiple asymptotic phases using the eigenoperators of the adjoint Liouville superoperator of the quantum nonlinear oscillator associated with different fundamental frequencies. We analyze a quantum van der Pol oscillator with Kerr effect in the strong quantum regime and show that the system has several different fundamental frequencies. By introducing order parameters and power spectra in terms of the associated quantum asymptotic phases, we reveal that phase locking of the system with a harmonic drive at several different frequencies, an explicit quantum signature observed only in the strong quantum regime, can be interpreted as synchronization on a torus rather than a simple limit cycle.

Dynamic behaviors of a Leslie-Gower model with strong Allee effect and fear effect in prey.

We incorporate the strong Allee effect and fear effect in prey into a Leslie-Gower model. The origin is an attractor, which implies that the ecological system collapses at low densities. Qualitative analysis reveals that both effects are crucial in determining the dynamical behaviors of the model. There can be different types of bifurcations such as saddle-node bifurcation, non-degenerate Hopf bifurcation with a simple limit cycle, degenerate Hopf bifurcation with multiple limit cycles, Bogdanov-Takens bifurcation, and homoclinic bifurcation.

Bistability and Chaos Emergence in Spontaneous Dynamics of Astrocytic Calcium Concentration

In this work, we consider a mathematical model describing spontaneous calcium signaling in astrocytes. Based on biologically relevant principles, this model simulates experimentally observed calcium oscillations and can predict the emergence of complicated dynamics. Using analytical and numerical analysis, various attracting sets were found and investigated. Employing bifurcation theory analysis, we examined steady state solutions, bistability, simple and complicated periodic limit cycles and also chaotic attractors. We found that astrocytes possess a variety of complex dynamical modes, including chaos and multistability, that can further provide different modulations of neuronal circuits, enhancing their plasticity and flexibility.

Stability switching curves in a Lotka–Volterra competition system with two delays

This study is devoted to consider the dynamic properties of a Lotka–Volterra competition model with two discrete delays. First, it reviews stability conditions of the no-delay model as a benchmark. Second, it examines the delay model and analytically establishes the stability switching curve on which stability is switched to instability and vice versa. Finally, numerical simulations are performed to confirm the theoretical results such as Hopf bifurcations and, further, demonstrate a wide variety of dynamics ranging from simple limit cycle to the existence of multi-stability and the birth of chaotic dynamics.

Устойчивость релейной динамической системы с нелинейным датчиком скорости и запаздыванием при действии постоянного возмущения

The paper considers the joint effect of the control delay and speed sensor output signal limiting on the stability of the relay dynamic system under the constant disturbance. It is shown that in this case a new property is detected in the system – the appearance of the unstable limit cycle. Phase trajectories are drawn to a stable limit cycle only from the area of initial conditions where their boundaries are determined by the trajectory of an unstable limit cycle. Using the method of Poincare mappings, the parameters of fixed points defining the unstable limit cycle as the boundary of the stability region are found. A simplified method for approximate determination of simple limit cycles and stability in the “large” is proposed based on the property of dynamic symmetry of the system. The method allows the study of the problem under consideration to be limited to applying shift and symmetry mappings to the switching lines.

Аналитическое построение точечных отображений релейной динамической системы с учетом запаздываний

The article considers solving the problem of analytical construction of Poincare maps for finding simple and complex limit cycles in a relay dynamic system involving constant perturbation and delay. Application of Neymark’s theory of multivariate point transformations allows reducing the problem under consideration to the search for a multifold fixed point thus overcoming the difficulty of finding complex periodic motions. The choice of switching lines as arcs without contact taking into account delays significantly simplified the task of analytical construction of point mapping. The results of analytical constructions are confirmed by numerical simulation of movements. The results obtained can find practical application in developing reactive systems for controlling the orientation and stabilization of the spacecraft. Compared with the previously known solution, a more complete result is obtained, which is of particular importance in the study of systems with high efficiency of executive bodies.

Dynamical systems approach to study thermoacoustic transitions in a liquid rocket combustor.

Liquid rockets are prone to large amplitude oscillations, commonly referred to as thermoacoustic instability. This phenomenon causes unavoidable developmental setbacks and poses a stern challenge to accomplish the mission objectives. Thermoacoustic instability arises due to the nonlinear interaction between the acoustic and the reactive flow subsystems in the combustion chamber. In this paper, we adopt tools from dynamical systems and complex systems theory to understand the dynamical transitions from a state of stable operation to thermoacoustic instability in a self-excited model multielement liquid rocket combustor based on an oxidizer rich staged combustion cycle. We observe that this transition to thermoacoustic instability occurs through a sequence of bursts of large amplitude periodic oscillations. Furthermore, we show that the acoustic pressure oscillations in the combustor pertain to different dynamical states. In contrast to a simple limit cycle oscillation, we show that the system dynamics switches between period-3 and period-4 oscillations during the state of thermoacoustic instability. We show several measures based on recurrence quantification analysis and multifractal theory, which can diagnose the dynamical transitions occurring in the system. We find that these measures are more robust than the existing measures in distinguishing the dynamical state of a rocket engine. Furthermore, these measures can be used to validate models and computational fluid dynamics simulations, aiming to characterize the performance and stability of rockets.

Computation Model of Cyber-Physical Immunosensor System

This paper initiates a study toward developing and applying computation models of cyber-physical immunosensor systems (CPISS). The focus is on the mathematical description of continuous population dynamics combined with dynamic logic used for discrete events. First, we introduce a class of lattice differential equations with time delay simulating antigen-antibody interactions within immunopixels. The spatial operator is modeling diffusion-like interaction between immunopixels. We then use the syntax of dynamic logic to describe discrete states of the immunopixel as a result of fluorescencing. An electrical signal which is simulated as a number of immunopixels' fluorescencing is important from the viewpoint of CPISS design. Stability research is focused on the notion of practical stability. For this purpose, we constructed a specific randomized multivariate algorithm, which provides a probabilistic estimate of the practical stability of the immunosensor system. In particular, we use the Monte Carlo technique. It analyzes both initial conditions and time delay and rate parameters. The experimental results obtained provide a complete analysis of immunosensor model stability with respect to changes of time delay, namely, as the time delay was increased, the stable endemic solution changed at a critical value to a stable limit cycle. Furthermore, when increasing the time delay, the behavior changed from convergence to simple limit cycle to convergence to complicated limit cycles with an increasing number of local maxima and minima per cycle until, at sufficiently high time delay, the behavior became chaotic. Such behavior can be seen using both phase portraits, tile plots, and simulated electrical signal. Mathematical models and algorithms, which are developed in this paper, may be considered as additional skills of CPISS. There is shown their software implementation as methods in language R.

Nonlinear flutter response of pre-heated functionally graded panels

The thermo-mechanical behavior of functionally graded (FG) panels is investigated in supersonic air flow. The three-node triangular element based on the Mindlin plate theory is employed to account for the transverse shear strains, and the von-Karman nonlinear strain–displacement relation is utilized considering the geometric nonlinearity. The effective material properties of the FG material are assumed to vary through the thickness according to simple power law distribution. The aeroelastic equation is established using the first-order piston theory, the linear rule of mixture and the principle of virtual work. A multi-mode approach is utilized to form the reduced-order model. Nonlinear flutter response is obtained by solving the reduced-order aeroelastic equation in time using the Runge–Kutta fourth-order method. Numerical simulation reveals that the multi-mode reduced-order model has a good convergence property. By using the 24-mode model the variation of flutter amplitude with dimensionless dynamic pressure, and the route of nonlinear flutter response from simple harmonic limit cycle oscillation (LCO) to non-harmonic periodic oscillation are examined.

Effects of Moderate Noise on a Limit Cycle Oscillator: Counterrotation and Bistability

The effects of noise on the dynamics of nonlinear systems is known to lead to many counterintuitive behaviors. Using simple planar limit cycle oscillators, we show that the addition of moderate noise leads to qualitatively different dynamics. In particular, the system can appear bistable, rotate in the opposite direction of the deterministic limit cycle, or cease oscillating altogether. Utilizing standard techniques from stochastic calculus and recently developed stochastic phase reduction methods, we elucidate the mechanisms underlying the different dynamics and verify our analysis with the use of numerical simulations. Last, we show that similar bistable behavior is found when moderate noise is applied to the FitzHugh-Nagumo model, which is more commonly used in biological applications.

대향류 확산화염에서 맥동 불안정성의 비선형 거동에 대한 수치해석

대향류 확산화염에서 확산-전도 불안정성에 의한 맥동불안정성의 비선형 거동을 수치 해석적으로 연구하였다. Lewis 수를 1보다 충분히 크게 두고 일차원 준정상상태의 화염의 해로부터 Damkohler 수를 섭동시켜 시간에 따른 화염의 전개를 계산하였다. 맥동 불안정성에 의한 비선형 화염전개는 세 가지 다른 형태, 즉 교란이 점점 감소되는 경우, 교란이 증폭되어 안정된 주기적 진동이 일어나는 경우, 그리고 교란이 계속 증폭되어 화염이 소염되는 경우 등으로 나타났다. 스트레치를 받지 않는 화염의 결과와 달리 대향류 유동장의 화염에서는 안정된 한계순환 맥동 불안정이 존재하였다. 세 가지 다른 형태의 화염 전개를 보이는 임계 Damkohler 수를 계산하여 동적 소염이 일어나는 영역을 표시하였고, 이는 층류소화염의 국소소염 계산에 이용될 수 있다. 불안정성이 나타나는 갈래질의 구조는 초임계 및 임계이하 Hopf 갈래질로 나타났다. 특정한 . Damkohler 수의 영역에서 안정된 한계 순환 갈래질이 나타났으며, 화염온도가 증가함에 따라 영역이 축소되어 안정된 한계순환이 일어나는 영역은 사라지고 불안정한 한계순환 갈래질이 나타났다. 안정된 한계순환 영역이 확장되는 영역이 존재하며, 이는 단순한 한계순환 불안정성이 주기배증에 의한 Rossler 갈래질이 나타나면서 한계 영역이 확장되었다.

S1502-1-2 シンプルな非線形最適制御を利用して二足歩行から始める実践ロボット教育カリキュラム(工学基礎としてのメカトロ導入教育,社会変革を技術で廻す機械工学)

There is no robot around us in our society at the current stage if we define a robot as an autonomous machine working in the arena of offices, homes, and disaster sites, etc. outside the factories. Mechatronics, dynamics and robotics involving humans are the world of strong nonlinearity. The realized movements of humanoid biped robot using simple nonlinear optimal control use only small active power with simple chaotic limit cycles and showed robustness during walking, running everywhere even on the ice and snow. This paper showed the application of the simple nonlinear optimal control to various movements making full use of instability as a source of driving force and developed into practical education curriculum as an introduction with humanoid biped robot to Robotics having an intellectual and emotional appeal.

New Canards Bursting and Canards Periodic-Chaotic Sequence

A trajectory following the repelling branch of an equilibrium or a periodic orbit is called a canards solution. Using a continuation method, we find a new type of canards bursting which manifests itself in an alternation between the oscillation phase following attracting the limit cycle branch and resting phase following a repelling fixed point branch in a reduced leech neuron model. Via periodic-chaotic alternating of infinite times, the number of windings within a canards bursting can approach infinity at a Gavrilov–Shilnikov homoclinic tangency bifurcation of a simple saddle limit cycle.

A-26 障害予防とパワーアップのためのロバストな「ナンバ・テニス」の提唱 : 平衡点不安定を利用する地面を蹴らない動きの生成(テニス・バドミントン2)

We realized the simple self-sustained humanlike robust walking & running NANBA of humanoid biped robot GENBE based on distributed control of physical body in a martial art, which uses only small active power with simple chaotic limit cycle utilizing instability. Instability makes the natural movement and can be applied to walking of a physically handicapped person, rehabilitation, sports, and so on. This paper showed the development of robust NANBA walking of humanoid Biped Robot GENBE to NANBA-Tennis making full use of instability as a source of driving force.

Chaotic motions and limit cycle flutter of two-dimensional wing in supersonic flow

Based on the piston theory of supersonic flow and the energy method, the flutter motion equations of a two-dimensional wing with cubic stiffness in the pitching direction are established. The aeroelastic system contains both structural and aerodynamic nonlinearities. Hopf bifurcation theory is used to analyze the flutter speed of the system. The effects of system parameters on the flutter speed are studied. The 4th order Runge-Kutta method is used to calculate the stable limit cycle responses and chaotic motions of the aeroelastic system. Results show that the number and the stability of equilibrium points of the system vary with the increase of flow speed. Besides the simple limit cycle response of period 1, there are also period-doubling responses and chaotic motions in the flutter system. The route leading to chaos in the aeroelastic model used here is the period-doubling bifurcation. The chaotic motions in the system occur only when the flow speed is higher than the linear divergent speed and the initial condition is very small. Moreover, the flow speed regions in which the system behaves chaos are very narrow.

1G2-3 人間型二足ロボット「源兵衛」による身体に負担の少ない歩行・走行・起き上がり(医療福祉ロボットI,一般講演)

In the previous paper, we realized the simple self-sustained humanlike robust walking & running NANBA of humanoid biped robot GENBE based on distributed control of physical body in a martial art utilizing instability without ZMP (Zero Moment Point) control, which uses only small active power with simple chaotic limit cycle utilizing instability, further developing into autonomous walking, runnung, instantaneous turn and the simple autonomous shock avoidance during falling down and instantaneous rising up. Instability makes the natural movement. This paper introduced experimentally the robustness of humanoid biped robot GENBE who walks and runs everywhere even on the ice and snow, developing to the rising up with light load.

A33 人間型二足ロボット「源兵衛」の「ナンバ歩き・ナンバ走り」のロバスト性のメカニズム(ロボット・スキル)

In the previous paper, we realized the simple self-sustained humanlike robust walking & running NANBA of humanoid biped robot GENBE based on distributed control of physical body in a martial art utilizing instability without ZMP (Zero Moment Point) control, which uses only small active power with simple chaotic limit cycle utilizing instability, further developing into autonomous walking, running, instantaneous turn and the simple autonomous shock avoidance during falling down and instantaneous rising up. Instability makes the natural movement. This paper made clear experimentally by using high-speed video camera the mechanism of robustness of humanoid biped robot GENBE who walks and runs everywhere even on the ice and snow.

Limit cycles in the presence of convection: A traveling wave analysis

We consider a diffusion model with limit cycle reaction functions. In an unbounded domain, diffusion spreads the pattern outwards from the source. Convection adds instability to the reaction-diffusion system. The result of this instability is a readiness to create a pattern. We choose the Lambda-Omega reaction functions for their simple limit cycle. We carry out a transformation of the dependent variables into polar form. From this we consider the initiation of the pattern to approximate a traveling wave. We carry out numerical experiments to test our analysis. These confirm the premise of the analysis, that the initiation can be modeled by a traveling wave. Furthermore, the analysis produces a good estimate of the numerical results. Most significantly, we confirm that the pattern consists of two different types.

Vascular coupling induces synchronization, quasiperiodicity, and chaos in a nephron tree

The paper presents a study of synchronization phenomena in a system of 22 nephrons supplied with blood from a common cortical radial artery. The nephrons are assumed to interact via hemodynamic and vascularly propagated coupling, both mediated by vascular connections. Using anatomic and physiological criteria, the nephrons are divided into groups: cortical nephrons and medullary nephrons with short, intermediate and long Henle loops. Within each of these groups the delay parameters of the internal feedback regulation are given a random component to represent the internephron variability. For parameters that generate simple limit cycle dynamics in the pressure and flow regulation of single nephrons, the ensemble of coupled nephrons showed steady state, quasiperiodic or chaotic dynamics, depending on the interaction strengths and the arterial blood pressure. When the solutions were either quasiperiodic or chaotic, cortical nephrons synchronized to a single frequency, but the longer medullary nephrons formed two clusters with different frequencies. Under no physiologically realistic combination of parameters did all nephrons assume a common frequency. Our results suggest a greater variability in the nephron dynamics than is apparent from measurements performed on cortical nephrons only. This variability may explain the development of chaotic dynamics in tubular pressure records from hypertensive rats.

Analysis and synthesis of perturbed Duffing oscillators

AbstractAnalysis and synthesis of perturbed Duffing oscillators have been presented. The oscillations in such systems are regarded as limit cycles in perturbed Hamiltonian systems under polynomial perturbations of sixth degree and are analysed by using the Melnikov function. It has been proved that there exists a polynomial perturbation depending on the zeros of the Melnikov function so that the system considered can have either two simple limit cycles, or one limit cycle of multiplicity 2, or one simple limit cycle. A synthesis of such oscillators based on the Melnikov's theory has been proposed. Copyright © 2006 John Wiley & Sons, Ltd.