Form Of Limit Cycles Research Articles

Self-organized limit cycles in red-detuned atom-cavity systems

Recent experimental advances in the field of cold-atom cavity QED provide a powerful tool for exploring nonequilibrium correlated quantum phenomena beyond conventional condensed-matter scenarios. We present the dynamical phase diagram of a driven Bose-Einstein condensate coupled with the light field of a cavity, with a transverse driving field red-detuned from atomic resonance. We identify regions in parameter space showing dynamical instabilities in the form of limit cycles, which evolve into chaotic behavior in the strong driving limit. Such limit cycles originate from the interplay between cavity dissipation and atom-induced resonance frequency shift, which modifies the phase of cavity mode and gives excessive negative feedback on the atomic density modulation, leading to instabilities of the superradiant scattering. We find interesting merging of the limit cycles related by a ${Z}_{2}$ symmetry, and identify a limit cycle formed by purely atomic excitations. The effects of quantum fluctuations and atomic interactions are also investigated.

Digital Twin of the Drive System, Considering the Forces of Various Nature

Among the factors limiting the use of sliding seal actuators in robotic positioning systems, the main ones are two the compressibility of the working fluid (air or liquid) and friction. This article discusses the problem of choosing a friction model for solving problems of controlling positional systems, primarily with a pneumatic drive based on a digital twin – a detailed mathematical model. Friction is usually described as the process of mechanical interaction of touching bodies at their relative displacement in the plane of contact (external friction), or at the relative displacement of parallel layers of a liquid, gas, or deformable solid (internal friction, or viscosity). In most cases, friction is a useful phenomenon, making many common things possible, such as walking and braking on a car. On the other hand, friction can also cause undesirable consequences. For example, for high-precision mechanical systems, friction can degrade the overall performance of the system. The influence of friction is manifested by undesirable effects in the form of limit cycles, deviations in movement from a given trajectory, a decrease in positioning accuracy, etc. As one of the models a rather complex dynamic model of friction – the LuGre model was analyzed, which is widely used in many works. However, the authors stopped in this case with the simpler Karnopp model, which has the advantage of describing the interaction with the friction forces in the processes of transition from a state of rest to motion and vice versa. A general approach to the study of the dynamics and accuracy of the positional system is proposed, based on the use of a rationalized dimensionless mathematical model of the drive. Several numerical experiments were carried out on the obtained model, considering friction forces. Numerical experiments are shown in graphs.

Friction-induced vibration in a two-mass damped system

A two degree of freedom mass-spring-damper model on a moving lubricated belt is considered in this paper. The motions of the two masses are coupled through spring and damper and each mass interacts with a moving lubricated belt. The friction model includes the boundary, mixed and hydrodynamic regimes of lubrication contact regimes. Friction coefficient is presented as a function of sliding velocity at each mass interface that includes a combination of linear, exponential and cubic functions to account for relatively large coefficient at low sliding speed, followed by Stribeck effect at low to moderate speed ranges and finally a full hydrodynamic lubrication regime. A set of dimensionless parameters is introduced and system's dynamic response behavior as well as stability are studied for various combination of parameters. The results show that the unstable system response can be quasiperiodic, or periodic in the form of limit cycles away from the origin when the sliding velocity is relatively small. Triple and double periodic behavior is observed in addition to other responses when mass suspended to the ground is larger than the second mass. All responses become asymptotically stable when dimensionless sliding velocity threshold is exceeded. Finally, for a qualitative analysis, the method of averaging is used to solve the nonlinear coupled equations of motion in order to find the amplitude and frequency of steady-state vibration. A comparison between the numerical results from direct time integration and analytical findings shows that approximate solution is acceptable.

Dissipation-Induced Instabilities of a Spinor Bose-Einstein Condensate Inside an Optical Cavity.

We investigate the dynamics of a spinor Bose-Einstein condensate inside an optical cavity, driven transversely by a laser with a controllable polarization angle. We focus on a two-component Dicke model with complex light-matter couplings, in the presence of photon losses. We calculate the steady-state phase diagram and find dynamical instabilities in the form of limit cycles, heralded by the presence of exceptional points and level attraction. We show that the instabilities are induced by dissipative processes that generate nonreciprocal couplings between the two collective spins. Our predictions can be readily tested in state-of-the-art experiments and open up the study of nonreciprocal many-body dynamics out of equilibrium.

Reversing the irreversible: From limit cycles to emergent time symmetry

In 1979 Penrose hypothesized that the arrows of time are explained by the hypothesis that the fundamental laws are time irreversible. That is, our reversible laws, such as the standard model and general relativity are effective, and emerge from an underlying fundamental theory which is time irreversible. In Cort\^{e}s and Smolin (2014a, 2014b, 2016) we put forward a research program aiming at realizing just this. The aim is to find a fundamental description of physics above the planck scale, based on irreversible laws, from which will emerge the apparently reversible dynamics we observe on intermediate scales. Here we continue that program and note that a class of discrete dynamical systems are known to exhibit this very property: they have an underlying discrete irreversible evolution, but in the long term exhibit the properties of a time reversible system, in the form of limit cycles. We connect this to our original model proposal in Cort\^{e}s and Smolin (2014a), and show that the behaviours obtained there can be explained in terms of the same phenomenon: the attraction of the system to a basin of limit cycles, where the dynamics appears to be time reversible. Further than that, we show that our original models exhibit the very same feature: the emergence of quasi-particle excitations obtained in the earlier work in the space-time description is an expression of the system's convergence to limit cycles when seen in the causal set description.

Control of chaos: Lie algebraic exact linearization approach for the Lü system

In this article, nonlinear control of the Lu chaos is presented by transforming a Lu system in a linear controllable system by using the Lie algebraic exact linearization method. Numerically, control of chaos does not have much place in current literature unless one actually finds and displays the controller in the chaotic system. The controller for the Lu system is designed for an arbitrary output. Based on this controller, both linear and nonlinear outputs are discussed in detail. Numerical simulations are carried out for all necessary aspects of the Lu problem that justify our analytical findings. Moreover, the stabilization of the Lu system based on the linear controllable system is discussed in the form of limit cycles and Hopf bifurcation. In particular, the numerical Hopf bifurcation diagram is presented using the numerical continuation technique.

Attractive manifolds in end milling

The attractive manifolds formed in end milling are analyzed. Such states are formed if the trajectories of steady periodic deformational tool displacements are unstable overall or in individual sections. In the dynamic milling system, attractive manifolds in the form of limit cycles, invariant tori, and strange (chaotic) attractors may appear. Bifurcations of attractive manifolds in parameter space are analyzed at length, with examples. Interest centers on how attractive manifolds affect manufacturing quality.

Bifurcation and chaotic in a model for activated sludge reactors

A dynamical model of an activated sludge process system is considered and analyzed. Numerical techniques are used to show when the system exhibits chaos. Three choices of bifurcation parameters produce different pictures of solution behavior in the form of limit cycles, two-torus and chaotic behavior. For some range of the reactor residence time the model exhibits chaotic behavior as well. Practical criteria are also derived for the effects of feed conditions and purge fraction on the dynamic characteristics of the bioreactor model.

Improved stability results for uncertain discrete-time state-delayed systems in the presence of nonlinearities

In this paper, delay-dependent linear matrix inequality (LMI)-based stability conditions have been developed for uncertain discrete-time state-delayed systems, in which nonlinear effects in the form of limit cycles may arise due to the finite word length implementation of such systems. Two problems have been addressed in this paper. The first problem considers the discrete-time system to be under the combined influence of quantization and overflow nonlinearities and in the second problem, the system is under the influence of saturation nonlinearities. The criteria developed are based on utilizing both the delay partitioning method and reciprocally convex approach. It is demonstrated with the help of numerical examples that the presented criteria are able to yield less conservative results along with lower computational complexity than previously reported criteria.

Sequential dynamics of master–slave systems

Following on previous work, we discuss a scenario in which two neural ensembles have a master–slave arrangement. We will consider a more particular case in which the master system exhibits transient dynamics due to a stable heteroclinic sequence (SHS) and the slave system has attractors in the form of limit cycles. We will give sufficient conditions that guarantee the existence of a SHS in the form of a tube in the full phase space, together with an open stable heteroclinic channel that surrounds it. We present a numerical observation of chaotic-like transient behaviour different from well-known transient chaos.

MULTIPLE ATTRACTORS AND BIFURCATIONS IN HARD OSCILLATORS DRIVEN BY CONSTANT INPUTS

Hard oscillators are dynamical systems that show the coexistence of qualitatively different attractors, in the form of limit cycles and equilibrium points. In the presence of external inputs their dynamic behavior is significantly different from those of oscillators, called soft, with a limit cycle as unique attractor. This paper studies the dynamics of a simple hard oscillator under the influence of a constant external input. It is shown that, despite the apparent simplicity, when the input strength and the oscillator's natural frequency are varied the system exhibits many different bifurcation phenomena, including global bifurcations as saddle-node on limit cycle and homoclinic bifurcations. The model under investigation can play a role in neuroscience, as it exhibits two different mechanisms of class I neural excitability and one mechanism for class II. It also highlights a mechanism of transition between the two classes.

A study of bifurcation behaviour in oblique turning operation

This paper presents results of linear stability analysis in turning using nonlinear force–feed dynamic model by considering three-dimensional cutting tool geometry. The modified analytical equations for cutting insert with three-dimensional tool geometry are derived by including relative displacements of the tool with respect to a two-degree of freedom work-piece model. The critical stability points obtained as a function of feed are validated against time-domain solutions. Simulation results are shown in the form of limit cycles and bifurcation diagram. Influence of static-feed term on cutting dynamics over a cantilever work-piece is illustrated along with the experimental results.

Characteristics of Transcriptional Activity in Nonlinear Dynamics of Genetic Regulatory Networks

Microarray measurements of mRNA abundances is a standard tool for evaluation of transcriptional activity in functional genomics. The methodology underlying these measurements assumes existence of a direct link between transcription levels, that is, gene-specific mRNA copy numbers present in the cell, and transcription rates, that is, the numbers of gene-specific mRNA molecules synthesized per unit of time. In this paper, the question of whether or not such a tight interdependence may exist is examined in the context of nonlinear dynamics of genetic regulatory networks. Using the equations of chemical kinetics, a model has been constructed that is capable of explicitly taking into consideration nonlinear interactions between the genes through the teamwork of transcription factors. Jacobian analysis of stability has shown that steady state equilibrium is impossible in such systems. However, phase space compressibility is found to be negative, thus suggesting that asymptotic stability may exist and assume either the form of limit cycle or of a chaotic attractor. It is argued that in rapidly fluctuating or chaotic systems, direct evaluation of transcription rates through transcription levels is highly problematic. It is also noted that even if a hypothetical steady state did exist, the knowledge of transcription levels alone would not be sufficient for the evaluation of transcription rates; an additional set of parameters, namely the mRNA decay rates, would be required. An overall conclusion of the work is that the measurements of mRNA abundances are not truly representative of the functionality of genes and structural fidelity of the genetic codes.

Two distinct mechanisms of coherence in randomly perturbed dynamical systems

We carefully examine two mechanisms--coherence resonance and self-induced stochastic resonance--by which small random perturbations of excitable systems with large time scale separation may lead to the emergence of new coherent behaviors in the form of limit cycles. We analyze what controls the degree of coherence in these two mechanisms and classify their very different properties. In particular we show that coherence resonance arises only at the onset of bifurcation and is rather insensitive against variations in the noise amplitude and the time scale separation ratio. In contrast, self-induced stochastic resonance may arise away from bifurcations and the properties of the limit cycle it induces are controlled by both the noise amplitude and the time scale separation ratio.

Design of a Class of Stabilizing Nonlinear State Feedback Controllers with Bounded Inputs

This paper addresses the problem of designing stabilizing global linearization controllers subject to bounded inputs. It is well-known that saturation of the input variables usually leads to a serious performance degradation of even instability of processes if undesired attractors are present. Instability occurs in open-loop unstable plants once the controller hits the constraints. If the plant is open-loop stable, saturation may also lead to instability in the form of limit cycles. This paper provides conditions under which the existence of stable global linearization controllers for SISO and MIMO plants is ensured. These conditions will set up the basis to derive a tuning technique that, preserving stability, renders a quite acceptable performance in terms of closed-loop response. Two problem categories, and solution strategies, will be considered: (1) If the set of constrained inputs Ub is given, a feasible region in the state space is calculated and a low-gain static controller computed to preserve stability. Performance is then improved by adding and saturating and external controller. (2) If the feasible set in the state space and the desired dynamic behavior are given, a feasible set, Ub, and thus a global linearization controller, is computed. Performance is improved through an external and possibly saturated controller. The methodology we propose represents a simple way of designing nonlinear state feedback controllers for systems affected by severe nonlinearities and undesired attractors such as chemical reactors and it is particulary attractive for open-loop unstable plants. It applies to SISO as well as to MIMO systems and is illustrated through two simulation examples that involve continuous-stirred tank reactors. The first of them consists of the control of a non-isothermal reactor at the unstable state, while the second consists of the control of level and concentration.

Predictions of the existence, frequency, and amplitude of physiological tremor in normal man based on measured frequency-response characteristics.

Mathematical models for 1) musculoskeletal dynamics, and 2) reflex feedback, based on the results of the authors' frequency-response measurements on normal adult male human subjects, are combined to produce a model for physiological tremor in such subjects. An analysis of this model shows that the system will be unstable to small disturbances (that is, tremor will occur) under certain conditions of external loading. Further, when the system is unstable, nonlinearities in the model produce responses in the form of limit cycles, and both the frequency and amplitude of the resulting tremor can be calculated. For constant loads applied through a constant compliance, the model predicts the onset of tremor at low loads, a maximum intensity of tremor at loads corresponding to 30-50 percent of maximum voluntary effort, and a decrease in the tremor amplitude at still higher loads.