Regular Limit Cycles Research Articles

Heterogeneity induced control of chaotic systems to stable limit cycles

We explore the behavior of coupled chaotic oscillators where one unit has intrinsically dissimilar dynamics. We find that the presence of a single dissimilar chaotic system in the network manages to drive all the chaotic oscillators to regular limit cycles. Additionally the regular cycles that emerge are significantly smaller in size than the uncoupled chaotic attractors. Counterintuitively the more geometrically dissimilar the single distinct system is from the other chaotic oscillators, the stronger is the emergent control. Furthermore, the position of the dissimilar system in the network does not affect the control when the dissimilar element is markedly different. So surprisingly, enhanced heterogeneity in coupled systems leads to more pronounced and robust controllability. Our results can then potentially lead to the design of new control strategies in engineered systems, and also suggests mechanisms whereby naturally occurring complex systems can evolve to regular dynamics through coupling to heterogeneous sub-systems.

Self-induced stochastic resonance in an excitable potential well

The noise-induced dynamics of the oscillator, which is of an excitable potential and positive dissipation, is investigated in this paper. As a noise-induced coherence behavior, the self-induced stochastic resonance (SISR) caused by small random perturbations is explored in numerical simulations. It is found that when noise is small and finite, the location of the limit cycle induced by the SISR can be better determined when the prefactor of the WKB approximation of the Kramers rate is considered. Besides, unlike other excitable systems, only confined to a very small noise interval, the regular limit cycles induced by the SISR can be observed.

MHD limit cycles on FTU

The development of low-order tearing modes during density ramp-up in the high density regime on the Frascati Tokamak Upgrade is characterized by an initial ordinary stage, with a ‘one-to-one’ relation between mode amplitude and frequency, followed by the formation, on the amplitude/frequency plane, of ‘limit cycles’ with increasing area up to disruption for density limit if the density continues to grow. A critical mode amplitude for transition from smooth to cyclic behavior has been observed in experiments performed changing the line-averaged density, and the existence of such a threshold has been confirmed in experiments of real time control of tearing mode in the high density regime by means of electron cyclotron resonance heating.The amplitude and frequency modulations of the observed tearing mode (m and n are the poloidal and toroidal mode number, respectively) occur in few milliseconds, which is not in agreement with the diffusion resistive time of about two hundred milliseconds expected on the resonance from the non-linear theory. The origin of such modulations has been investigated, taking into account that in the high amplitude stages of the mode temporal evolution it is difficult to discriminate between non-linear effects and mode coupling mechanisms. Our analysis suggests that the formation of limit cycles could be due to a recursive island fragmentation, with a sort of self-healing phenomenon; in fact the island distortion increases before amplitude drops. Concerning the interaction with modes of different helicity, our experiments seem to indicate that the presence of the resonance in the plasma is necessary for the occurrence of deep and regular limit cycles for the 2/1 tearing mode.

Flexible dynamics of two quorum-sensing coupled repressilators.

Genetic oscillators play important roles in cell life regulation. The regulatory efficiency usually depends strongly on the emergence of stable collective dynamic modes, which requires designing the interactions between genetic networks. We investigate the dynamics of two identical synthetic genetic repressilators coupled by an additional plasmid which implements quorum sensing (QS) in each network thereby supporting global coupling. In a basic genetic ring oscillator network in which three genes inhibit each other in unidirectional manner, QS stimulates the transcriptional activity of chosen genes providing for competition between inhibitory and stimulatory activities localized in those genes. The "promoter strength", the Hill cooperativity coefficient of transcription repression, and the coupling strength, i.e., parameters controlling the basic rates of genetic reactions, were chosen for extensive bifurcation analysis. The results are presented as a map of dynamic regimes. We found that the remarkable multistability of the antiphase limit cycle and stable homogeneous and inhomogeneous steady states exists over broad ranges of control parameters. We studied the antiphase limit cycle stability and the evolution of irregular oscillatory regimes in the parameter areas where the antiphase cycle loses stability. In these regions we observed developing complex oscillations, collective chaos, and multistability between regular limit cycles and complex oscillations over uncommonly large intervals of coupling strength. QS coupling stimulates the appearance of intrachaotic periodic windows with spatially symmetric and asymmetric partial limit cycles which, in turn, change the type of chaos from a simple antiphase character into chaos composed of pieces of the trajectories having alternating polarity. The very rich dynamics discovered in the system of two identical simple ring oscillators may serve as a possible background for biological phenotypic diversification, as well as a stimulator to search for similar coupling in oscillator arrays in other areas of nature, e.g., in neurobiology, ecology, climatology, etc.

Investigation of self-sustained beating oscillations in a Rijke burner

Thermoacoustic oscillations occur in a Rijke burner with a porous plug stabilized premixed flame. In addition to regular limit cycle oscillations, the thermoacoustic system also presents self-sustained periodic amplitude modulations, or beating oscillations, near the flame equivalence ratio limits of thermoacoustic instability. The beating oscillations exhibit approximately 1Hz or even lower frequency amplitude modulations for the acoustic pressure and the heat release rate. It is also found that low frequency flame pulsations arise with the amplitude modulations. The different characteristics of the beating oscillations near the upper and lower equivalence ratio limits are distinguished and explained by the coupling between the low frequency flame pulsations and the thermoacoustic instability. To simulate the beating oscillations, this paper proposes a flame model that incorporates the flame dynamics at the amplitude modulation time scale (∼s) and the acoustic time scale (∼ms). The simulation results of the thermoacoustic system suggest that this flame model can qualitatively predict the amplitude modulation caused by the slow pulsating flame temperature and heat release rate. Flame heat loss plays an important role in low frequency flame pulsations, and its effects on beating oscillations are discussed. Several simulation cases at different equivalence ratios are performed to retrieve the amplitude modulations near the upper and lower equivalence ratio limits. The results are in keeping with the experimental observations.

Persistent Chaos of Measles Epidemics in the Prevaccination United States Caused by a Small Change in Seasonal Transmission Patterns.

Epidemics of infectious diseases often occur in predictable limit cycles. Theory suggests these cycles can be disrupted by high amplitude seasonal fluctuations in transmission rates, resulting in deterministic chaos. However, persistent deterministic chaos has never been observed, in part because sufficiently large oscillations in transmission rates are uncommon. Where they do occur, the resulting deep epidemic troughs break the chain of transmission, leading to epidemic extinction, even in large cities. Here we demonstrate a new path to locally persistent chaotic epidemics via subtle shifts in seasonal patterns of transmission, rather than through high-amplitude fluctuations in transmission rates. We base our analysis on a comparison of measles incidence in 80 major cities in the prevaccination era United States and United Kingdom. Unlike the regular limit cycles seen in the UK, measles cycles in US cities consistently exhibit spontaneous shifts in epidemic periodicity resulting in chaotic patterns. We show that these patterns were driven by small systematic differences between countries in the duration of the summer period of low transmission. This example demonstrates empirically that small perturbations in disease transmission patterns can fundamentally alter the regularity and spatiotemporal coherence of epidemics.

Bifurcation analysis in a stage-structured predator–prey model with maturation delay

In this paper, we investigate the impact of maturation delay on the positive equilibrium solutions in a stage-structured predator–prey system. By analyzing the characteristic equation we derive the conditions for the emergence of Hopf bifurcation. By applying the normal form and the center manifold argument, the direction as well as the stability of periodic solutions bifurcating from Hopf bifurcation is explored. Results show that maturation delay can change the nature of the positive equilibrium solutions, and the loss of equilibrium stability occurs as a consequence of Hopf bifurcation. When Hopf bifurcation takes place, periodic solution arises and is further demonstrated to be asymptotically stable. In addition, the periodic solutions appear only for intermediate maturation delay, that is, there exists a delay window, outside of which the positive equilibrium is locally stable. Furthermore, numerical analysis shows that Hopf bifurcation is favored by a superior competition for adult predators to juveniles, a smaller mortality on juvenile and/or adult predators, and a higher resource carrying capacity. Interestingly, increasing food carrying capacity can lead to the emergence of irregular chaotic dynamics and regular limit cycles.

Caveats in modeling a common motif in genetic circuits

From a coarse-grained perspective, the motif of a self-activating species, activating a second species that acts as its own repressor, is widely found in biological systems, in particular in genetic systems with inherent oscillatory behavior. Here we consider a specific realization of this motif as a genetic circuit, termed the bistable frustrated unit, in which genes are described as directly producing proteins. Upon an improved resolution in time, we focus on the effect that inherent time scales on the underlying scale can have on the bifurcation patterns on a coarser scale. Time scales are set by the binding and unbinding rates of the transcription factors to the promoter regions of the genes. Depending on the ratio of these rates to the decay times of both proteins, the appropriate averaging procedure for obtaining a coarse-grained description changes and leads to sets of deterministic equations, which considerably differ in their bifurcation structure. In particular, the desired intermediate range of regular limit cycles fades away when the binding rates of genes are not fast as compared to the decay time of the proteins. Our analysis illustrates that the common topology of the widely found motif alone does not imply universal features in the dynamics.

“Quorum sensing” generated multistability and chaos in a synthetic genetic oscillator

We model the dynamics of the synthetic genetic oscillator Repressilator equipped with quorum sensing. In addition to a circuit of 3 genes repressing each other in a unidirectional manner, the model includes a phase-repulsive type of the coupling module implemented as the production of a small diffusive molecule-autoinducer (AI). We show that the autoinducer (which stimulates the transcription of a target gene) is responsible for the disappearance of the limit cycle (LC) through the infinite period bifurcation and the formation of a stable steady state (SSS) for sufficiently large values of the transcription rate. We found conditions for hysteresis between the limit cycle and the stable steady state. The parameters' region of the hysteresis is determined by the mRNA to protein lifetime ratio and by the level of transcription-stimulating activity of the AI. In addition to hysteresis, increasing AI-dependent stimulation of transcription may lead to the complex dynamic behavior which is characterized by the appearance of several branches on the bifurcation continuation, containing different regular limit cycles, as well as a chaotic regime. The multistability which is manifested as the coexistence between the stable steady state, limit cycles, and chaos seems to be a novel type of the dynamics for the ring oscillator with the added quorum sensing positive feedback.

Impulse pattern in bi-directionally coupled model neurons of different dynamics

The effects of bi-directional gap junction coupling of two model neurons with subthreshold oscillations have been examined when the individual neurons are operating at different dynamical states either in the tonic or bursting firing mode. Our simulations indicate that intermediate coupling strengths mostly lead to highly variable, often chaotic impulse patterns whereas transition to completely synchronized activity at high coupling strengths is generally going along with transitions to regular limit cycle activity. The synchronized activity pattern, however, can be completely different from the original pattern of the uncoupled neurons.

Limit Cycles for Non Smooth Differential Equations via Schwarzian Derivative

We consider a class of planar polynomial systems with discontinuous righthand sides and prove that, under certain hypotheses, it presents at most one singular limit cycle and two regular limit cycles. Furthermore the sum of the multiplicity of the regular limit cycles is less or equal than two. A key point in the proof is the study of the Schwarzian derivative of the return map. Finally, we give some examples reaching these bounds.

Resonance effects in the Bonhoeffer-van der Pol system

Resonance effects for the Bonhoeffer-van der Pol system are analysed in the range where a limit cycle and a stable focus coexist. As usual, resonance effects are obtained by external periodic forcing at initial points near the focus, which turns into a small limit cycle. For certain inputs, a transition to the regular limit cycle, assisted by resonance, is obtained.

Regular and irregular nonlinear pulsation in population II Cepheid models

view Abstract Citations (100) References (56) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Regular and Irregular Nonlinear Pulsations in Population II Cepheid Models Kovacs, Geza ; Buchler, J. Robert Abstract Linear and nonlinear pulsational properties of radiative stellar models relevant for W Vir, RV Tau, and yellow semiregular variables are surveyed. Masses and luminosities range from 0.4 - 0.8 M_sun; and 200 - 1000 L_sun; respectively. The chemical composition is fixed to X = 0.745, Z = 0.005. Three types of pulsational behavior occur (1) singly periodic pulsations (limit cycles) with a radius variation of one maximum (or minimum) per period which are called regular limit cycles, (2) limit cycles with 2, 4, 8, ... maxima (minima) per repetition period which are called period doubled limit cycles, (3) irregular pulsations which are chaotic in the sense of dynamical systems analysis. Publication: The Astrophysical Journal Pub Date: November 1988 DOI: 10.1086/166890 Bibcode: 1988ApJ...334..971K Keywords: Cepheid Variables; Semiregular Variable Stars; Stellar Mass; Stellar Models; Stellar Oscillations; Chemical Composition; Mass To Light Ratios; Stellar Luminosity; Stellar Structure; Astrophysics; STARS: CEPHEIDS; STARS: POPULATION II; STARS: PULSATION; STARS: RV TAURI; STARS: SEMIREGULAR VARIABLES full text sources ADS | data products SIMBAD (3)