Limit Cycle Attractor Research Articles

Circatidal rhythm of void volume in young healthy adults

Solar and lunar geophysical cycles are the major zeitgebers of biological oscillators, but most oscillating physiological systems are entrained to circadian rhythm. Although void volume oscillates, entrainment to geophysical cycle has not been shown. Study was done to determine if void volume oscillation of young healthy adults is entrained to geophysical cycle. Healthy undergraduates were placed on ad libitum regime of food and water intake, restricted exercise regime, but no intake of alcoholic beverages, caffeinated drinks, and tobacco. All voids were collected, timed, and measured for 72 h. Void volumes were corrected for body mass and binned to hourly time series. Hourly distribution of voids was tested for uniformity. Void volume time series was denoised, and the dominant oscillatory component was Hilbert transformed to analytical signal. The phase portrait was plotted in complex plane. The geometry of attractors in state space and recurrence were computed. There were 195 subjects, 128 males and 67 females. Geometrical mean age (range) was 26 years (18–37). Geometrical mean voids/day (50% inter-quantile range (IQR)) was 4 (4–5). Voids/hr were not uniformly distributed, . Geometrical mean void vol/kg/hr (50% IQR) was 0.8 ml (0.6–1.0). Void volume oscillation was rhythmic with peaks at 6.00 am and 6.00 pm, and troughs at 12.00 am and 12.00 pm, which coincided with Earth tides. The timescale of oscillation was 12 h. Phase portrait showed trajectories which converged on limit cycle attractor. Recurrence topology showed only periodic regime. Circatidal rhythm of void volume oscillation indicates entrainment to lunar geophysical cycle. The gravitation field of the moon is, therefore, the most likely zeitgeber.

Boundary-induced pattern formation from uniform temporal oscillation

Pattern dynamics triggered by fixing a boundary is investigated. By considering a reaction-diffusion equation that has a unique spatially uniform and limit cycle attractor under a periodic or Neumann boundary condition, and then by choosing a fixed boundary condition, we found three novel phases depending on the ratio of diffusion constants of activator to inhibitor: transformation of temporally periodic oscillation into a spatially periodic fixed pattern, travelling wave emitted from the boundary, and aperiodic spatiotemporal dynamics. The transformation into a fixed, periodic pattern is analyzed by crossing of local nullclines at each spatial point, shifted by diffusion terms, as is analyzed by using recursive equations, to obtain the spatial pattern as an attractor. The generality of the boundary-induced pattern formation as well as its relevance to biological morphogenesis is discussed.

Imperfect chimeras in a ring of four-dimensional simplified Lorenz systems

Analyzing the dynamics of networks of coupled oscillators possessing chimera states has attracted considerable attention in recent years. Here we study a ring of coupled 4D simplified Lorenz systems whose prototype was first studied by Li and Sprott [1]. An interesting feature of this system is the coexistence of a limit cycle and two symmetric strange attractors for some specific range of parameters. Investigating a network of this type under long range interactions, we find a multitude of very interesting chimera multi-chimera states, whose appearance we relate to initial conditions chosen within the basins of attraction of the limit cycle and strange attractor of the individual 4D system. Thus, for sufficiently low coupling, selecting more and more initial conditions from the strange attractor basins leads to one or more “rebel” particles departing from the synchronous group and producing fascinating “imperfect” chimera states. For larger coupling these “rebels” multiply and the system develops what one might call “perfect” chimera states with multiple “heads”.

Reviving Kalecki’s business cycle model in a growth context

In 1935 Kalecki formulated the first fully specified model of a macroeconomic dynamics in which he studied an endogenous and most elementary business cycle mechanism. To revive his insights, the present paper adapts his stationary economy to a growth context. Introducing a reasonable nonlinearity into the investment function, it then takes care that the model exhibits persistent cyclical behaviour that is characterized by a unique, globally attracting limit cycle. This feature is a robust property. A calibration of the numerical parameters achieves desired values for the cycle period as well as the amplitudes of the output-capital ratio and the capital growth rate. It is moreover demonstrated that a one-time shock to the dynamics can easily have long-lasting effects on the amplitudes. Despite common misgivings about delay differential equations (which here result from Kalecki’s implementation lag), the analysis can be conducted with a limited mathematical effort.

Canard phenomenon in a slow-fast modified Leslie–Gower model

Geometrical Singular Perturbation Theory has been successful to investigate a broad range of biological problems with different time scales. The aim of this paper is to apply this theory to a predator-prey model of modified Leslie–Gower type for which we consider that prey reproduces mush faster than predator. This naturally leads to introduce a small parameter ϵ which gives rise to a slow-fast system. This system has a special folded singularity which has not been analyzed in the classical work [15]. We use the blow-up technique to visualize the behavior near this fold point P. Outside of this region the dynamics are given by classical regular and singular perturbation theory. This allows to quantify geometrically the attractive limit-cycle with an error of O(ϵ) and shows that it exhibits the canard phenomenon while crossing P.

Self-propulsion of a grain-filled dimer in a vertically vibrated channel

Steady dissipation of energy is a crucial property that distinguishes active particles from Brownian particles. However, it is not straightforward to explicitly model the dissipative property of existing active particles driven by a vibrating plate. We present a novel active particle that can be explicitly modeled by Newtonian dynamics of a conservative force field plus two asymmetrical dissipative terms. The particle is a dimer consisting of two ping-pong balls connected by a rigid rod, and its two balls are filled with granular particles of the same total mass but of different grain size. This dimer placed on a vibrating plate exhibits 3 types of motion – by tuning the frequency and the amplitude of the vibration, the dimer undergoes either a directed motion toward the small (or large) grain-filled side or an unbiased random motion. We investigate the various modes of motion both experimentally and numerically and show that the directed motion is a result of the asymmetric damping due to the size difference in the filling grains. Furthermore, the numerical simulation reveals that the dimer’s dynamics in either directed motion mode resembles a limit cycle attractor that is independent of its initial condition.

Predictive Coding for Dynamic Visual Processing: Development of Functional Hierarchy in a Multiple Spatiotemporal Scales RNN Model.

This letter proposes a novel predictive coding type neural network model, the predictive multiple spatiotemporal scales recurrent neural network (P-MSTRNN). The P-MSTRNN learns to predict visually perceived human whole-body cyclic movement patterns by exploiting multiscale spatiotemporal constraints imposed on network dynamics by using differently sized receptive fields as well as different time constant values for each layer. After learning, the network can imitate target movement patterns by inferring or recognizing corresponding intentions by means of the regression of prediction error. Results show that the network can develop a functional hierarchy by developing a different type of dynamic structure at each layer. The letter examines how model performance during pattern generation, as well as predictive imitation, varies depending on the stage of learning. The number of limit cycle attractors corresponding to target movement patterns increases as learning proceeds. Transient dynamics developing early in the learning process successfully perform pattern generation and predictive imitation tasks. The letter concludes that exploitation of transient dynamics facilitates successful task performance during early learning periods.

Limit cycles for the motion of finite-size particles in axisymmetric thermocapillary flows in liquid bridges

The motion of a small spherical particle of finite size in an axisymmetric thermocapillary liquid bridge is investigated numerically and experimentally. Due to the crowding of streamlines towards the free surface and the recirculating nature of the flow, advected particles visit the free surface repeatedly. The balance between centrifugal inertia and the strong short-range repulsive forces a particle experiences near the free surface leads to an attracting limit cycle for the particle motion. The existence of this limit cycle is established experimentally. It is shown that limit cycles obtained numerically by one-way-coupled simulations based on the Maxey–Riley equation and a particle–surface interaction model compare favorably with the experimental results if the thickness of the lubrication gap between the free surface and the surface of the particle is properly taken into account.

Demographic noise slows down cycles of dominance

We study the phenomenon of cyclic dominance in the paradigmatic Rock–Paper–Scissors model, as occurring in both stochastic individual-based models of finite populations and in the deterministic replicator equations. The mean-field replicator equations are valid in the limit of large populations and, in the presence of mutation and unbalanced payoffs, they exhibit an attracting limit cycle. The period of this cycle depends on the rate of mutation; specifically, the period grows logarithmically as the mutation rate tends to zero. We find that this behaviour is not reproduced in stochastic simulations with a fixed finite population size. Instead, demographic noise present in the individual-based model dramatically slows down the progress of the limit cycle, with the typical period growing as the reciprocal of the mutation rate. Here we develop a theory that explains these scaling regimes and delineates them in terms of population size and mutation rate. We identify a further intermediate regime in which we construct a stochastic differential equation model describing the transition between stochastically-dominated and mean-field behaviour.

Action potential initiation in a two-compartment model of pyramidal neuron mediated by dendritic Ca2+ spike

Dendritic Ca2+ spike endows cortical pyramidal cell with powerful ability of synaptic integration, which is critical for neuronal computation. Here we propose a two-compartment conductance-based model to investigate how the Ca2+ activity of apical dendrite participates in the action potential (AP) initiation to affect the firing properties of pyramidal neurons. We have shown that the apical input with sufficient intensity triggers a dendritic Ca2+ spike, which significantly boosts dendritic inputs as it propagates to soma. Such event instantaneously shifts the limit cycle attractor of the neuron and results in a burst of APs, which makes its firing rate reach a plateau steady-state level. Delivering current to two chambers simultaneously increases the level of neuronal excitability and decreases the threshold of input-output relation. Here the back-propagating APs facilitate the initiation of dendritic Ca2+ spike and evoke BAC firing. These findings indicate that the proposed model is capable of reproducing in vitro experimental observations. By determining spike initiating dynamics, we have provided a fundamental link between dendritic Ca2+ spike and output APs, which could contribute to mechanically interpreting how dendritic Ca2+ activity participates in the simple computations of pyramidal neuron.

Self-Organized Behavior Generation for Musculoskeletal Robots.

With the accelerated development of robot technologies, control becomes one of the central themes of research. In traditional approaches, the controller, by its internal functionality, finds appropriate actions on the basis of specific objectives for the task at hand. While very successful in many applications, self-organized control schemes seem to be favored in large complex systems with unknown dynamics or which are difficult to model. Reasons are the expected scalability, robustness, and resilience of self-organizing systems. The paper presents a self-learning neurocontroller based on extrinsic differential plasticity introduced recently, applying it to an anthropomorphic musculoskeletal robot arm with attached objects of unknown physical dynamics. The central finding of the paper is the following effect: by the mere feedback through the internal dynamics of the object, the robot is learning to relate each of the objects with a very specific sensorimotor pattern. Specifically, an attached pendulum pilots the arm into a circular motion, a half-filled bottle produces axis oriented shaking behavior, a wheel is getting rotated, and wiping patterns emerge automatically in a table-plus-brush setting. By these object-specific dynamical patterns, the robot may be said to recognize the object's identity, or in other words, it discovers dynamical affordances of objects. Furthermore, when including hand coordinates obtained from a camera, a dedicated hand-eye coordination self-organizes spontaneously. These phenomena are discussed from a specific dynamical system perspective. Central is the dedicated working regime at the border to instability with its potentially infinite reservoir of (limit cycle) attractors “waiting” to be excited. Besides converging toward one of these attractors, variate behavior is also arising from a self-induced attractor morphing driven by the learning rule. We claim that experimental investigations with this anthropomorphic, self-learning robot not only generate interesting and potentially useful behaviors, but may also help to better understand what subjective human muscle feelings are, how they can be rooted in sensorimotor patterns, and how these concepts may feed back on robotics.

The detection and stabilisation of limit cycle for deterministic finite automata

ABSTRACTIn this paper, the topological structure properties of deterministic finite automata (DFA), under the framework of the semi-tensor product of matrices, are investigated. First, the dynamics of DFA are converted into a new algebraic form as a discrete-time linear system by means of Boolean algebra. Using this algebraic description, the approach of calculating the limit cycles of different lengths is given. Second, we present two fundamental concepts, namely, domain of attraction of limit cycle and prereachability set. Based on the prereachability set, an explicit solution of calculating domain of attraction of a limit cycle is completely characterised. Third, we define the globally attractive limit cycle, and then the necessary and sufficient condition for verifying whether all state trajectories of a DFA enter a given limit cycle in a finite number of transitions is given. Fourth, the problem of whether a DFA can be stabilised to a limit cycle by the state feedback controller is discussed. Criteria for limit cycle-stabilisation are established. All state feedback controllers which implement the minimal length trajectories from each state to the limit cycle are obtained by using the proposed algorithm. Finally, an illustrative example is presented to show the theoretical results.

A novel route to a Hopf bifurcation scenario in switched systems with dead-zone

Planar switched systems with dead-zone are analyzed. In particular, we consider the effects of a perturbation which is applied to a linear control law and, due to the perturbation, the control changes from purely positional to position–velocity control. This type of a perturbation leads to a novel Hopf-like discontinuity induced bifurcation. We show that this bifurcation leads to the creation of a small scale limit cycle attractor, which scales as the square root of the bifurcation parameter. We then investigate numerically a planar switched system with a positional feedback law, dead-zone and time delay in the switching function. Using the same parameter values as for the switched system without time delay in the switching function, we show a Hopf-like bifurcation scenario which exhibits a qualitative and a quantitative agreement with the scenario analyzed for the non-delayed system.

Phase-amplitude reduction of transient dynamics far from attractors for limit-cycling systems.

Phase reduction framework for limit-cycling systems based on isochrons has been used as a powerful tool for analyzing the rhythmic phenomena. Recently, the notion of isostables, which complements the isochrons by characterizing amplitudes of the system state, i.e., deviations from the limit-cycle attractor, has been introduced to describe the transient dynamics around the limit cycle [Wilson and Moehlis, Phys. Rev. E 94, 052213 (2016)]. In this study, we introduce a framework for a reduced phase-amplitude description of transient dynamics of stable limit-cycling systems. In contrast to the preceding study, the isostables are treated in a fully consistent way with the Koopman operator analysis, which enables us to avoid discontinuities of the isostables and to apply the framework to system states far from the limit cycle. We also propose a new, convenient bi-orthogonalization method to obtain the response functions of the amplitudes, which can be interpreted as an extension of the adjoint covariant Lyapunov vector to transient dynamics in limit-cycling systems. We illustrate the utility of the proposed reduction framework by estimating the optimal injection timing of external input that efficiently suppresses deviations of the system state from the limit cycle in a model of a biochemical oscillator.

Coexisting Oscillation and Extreme Multistability for a Memcapacitor‐Based Circuit

The coexisting oscillations are observed with a memcapacitor‐based circuit that consists of two linear inductors, two linear resistors, and an active nonlinear charge‐controlled memcapacitor. We analyze the dynamics of this circuit and find that it owns an infinite number of equilibrium points and coexisting attractors, which means extreme multistability arises. Furthermore, we also show the stability of the infinite many equilibria and analyze the coexistence of fix point, limit cycle, and chaotic attractor in detail. Finally, an experimental result of the proposed oscillator via an analog electronic circuit is given.

Research on a six-order chaotic circuit with three memristors

A memristor is a nonlinear nanoscale-sized element with memory function, and it has an italic type 8 voltage-current relation curve that looks like a pinched hysteresis loop characteristic. The memristor is utilized to construct chaotic circuit, which has attracted the attention of the researchers. At present, most of studies focus on applying one or two memristors to the chaotic circuit. In order to study the multi memristor chaotic circuit, in this work we propose a six-order chaotic circuit with two flux-controlled memristors and a charge-controlled memristor. A corresponding six-order nonlinear dynamic differential equation of the circuit state variables is established. The dynamic properties of the circuit are demonstrated in detail. The analyses of equilibria and equilibrium stability show that the circuit has an equilibrium located in the three-dimensional space which is constituted by memristor internal state variables, and it is found that the equilibrium stability is determined by the circuit parameters and the initial states of three memristors. The Lyapunov exponent spectra and bifurcation diagrams of the circuit imply that it can produce two bifurcation behaviors by adjusting its parameters, which are Hopf bifurcation and anti-period doubling bifurcation. The hyperchaos, transient chaos and intermittency cycle phenomena are found in the same system. The dynamical behavior of this circuit is dependent on the initial state of memristor, showing different orbits such as chaotic oscillation, periodic oscillation and stable sink under different initial states. Finally, the simulation results indicate that some strange attractors like lotus type and superposition type are observed when voltage and electricity signal in observing chaotic attractors are generalized to power and energy signal, respectively. And the attractor production between the energy signals of the memristors are studied. Specially, when different initial conditions of three memristors are used to simulate the circuit, we can find the coexistence phenomenon of chaotic attractors with different topological structures or quasi-periodic limit cycle and chaotic attractor. The six-order chaotic oscillating circuit is mainly composed of three parts:the parallel connection between a flux-controlled memristor and capacitor, the serial connection between a charge-controlled memristor and inductor, and another flux-controlled memristor that is alone and floating, which enriches the application of memristor in high-order chaotic circuit. Compared with most of other chaotic systems, it has many circuit parameters and very complex topological structure, which enhances the complexity of chaotic system and the randomness of the generated signal. It is more difficult to decipher the encrypted information in chaotic secure communication, and thus it has better security performance and safety performance.

Invariant curves for the discretised van der Pol equation

It is well known that the stiff van der Pol equation has a strongly attractive limit cycle. In this paper it is shown that the Euler method applied to the van der Pol equation with small step size, small compared to the perturbation parameter, admits an attractive invariant closed curve close to the limit cycle. To describe closed curves in the vicinity of the limit cycle, 14 charts are introduced. A general graph transform result is derived and applied in these charts. The proof of the main result relies on the contraction principle in a suitable function space. Estimates are given for the distance of the invariant curve to the limit cycle.

Entrainment Maps.

Circadian oscillators found across a variety of species are subject to periodic external light-dark forcing. Entrainment to light-dark cycles enables the circadian system to align biological functions with appropriate times of day or night. Phase response curves (PRCs) have been used for decades to gain valuable insights into entrainment; however, PRCs may not accurately describe entrainment to photoperiods with substantial amounts of both light and dark due to their reliance on a single limit cycle attractor. We have developed a new tool, called an entrainment map, that overcomes this limitation of PRCs and can assess whether, and at what phase, a circadian oscillator entrains to external forcing with any photoperiod. This is a 1-dimensional map that we construct for 3 different mathematical models of circadian clocks. Using the map, we are able to determine conditions for existence and stability of phase-locked solutions. In addition, we consider the dependence on various parameters such as the photoperiod and intensity of the external light as well as the mismatch in intrinsic oscillator frequency with the light-dark cycle. We show that the entrainment map yields more accurate predictions for phase locking than methods based on the PRC. The map is also ideally suited to calculate the amount of time required to achieve entrainment as a function of initial conditions and the bifurcations of stable and unstable periodic solutions that lead to loss of entrainment.

Mixed-Mode Oscillations in a piecewise linear system with multiple time scale coupling

In this work, we analyze a four dimensional slow–fast piecewise linear system with three time scales presenting Mixed-Mode Oscillations. The system possesses an attractive limit cycle along which oscillations of three different amplitudes and frequencies can appear, namely, small oscillations, pulses (medium amplitude) and one surge (largest amplitude). In addition to proving the existence and attractiveness of the limit cycle, we focus our attention on the canard phenomena underlying the changes in the number of small oscillations and pulses. We analyze locally the existence of secondary canards leading to the addition or subtraction of one small oscillation and describe how this change is globally compensated for or not with the addition or subtraction of one pulse.

Bounds for Deterministic and Stochastic Dynamical Systems using Sum-of-Squares Optimization

We describe methods for proving upper and lower bounds on infinite-time averages in deterministic dynamical systems and on stationary expectations in stochastic systems. The dynamics and the quantities to be bounded are assumed to be polynomial functions of the state variables. The methods are computer-assisted, using sum-of-squares polynomials to formulate sufficient conditions that can be checked by semidefinite programming. In the deterministic case, we seek tight bounds that apply to particular local attractors. An obstacle to proving such bounds is that they do not hold globally; they are generally violated by trajectories starting outside the local basin of attraction. We describe two closely related ways past this obstacle: one that requires knowing a subset of the basin of attraction, and another that considers the zero-noise limit of the corresponding stochastic system. The bounding methods are illustrated using the van der Pol oscillator. We bound deterministic averages on the attracting limit cycle above and below to within 1%, which requires a lower bound that does not hold for the unstable fixed point at the origin. We obtain similarly tight upper and lower bounds on stochastic expectations for a range of noise amplitudes. Limitations of our methods for certain types of deterministic systems are discussed, along with prospects for improvement.