Infinite Period Bifurcation Research Articles

Noise and Parameter Mismatch in a Ring Network for Sensing Extremely Low Electric Fields

Over the past years, we have exploited the bistability features that are commonly found in many individual sensors to develop a network-based [ Acebrón et al. , 2003 ; Bulsara et al. , 2004 ; In et al. , 2003a ; In et al. , 2003b ; In et al. , 2005 ; In et al. , 2006 ; In et al. , 2012 ; Palacios et al. , 2005 ] approach to modeling, designing, and fabricating extremely sensitive magnetic- and electric-field sensors capable of resolving field changes as low as 150[Formula: see text]pT and 100[Formula: see text]fAmps, respectively. Higher sensitivity is achieved by exploiting the symmetry of the network to create infinite-period bifurcations that render the ensuing oscillations highly sensitive to symmetry-breaking effects from external signals. In this paper, we study the effects of noise on the response of a network-based electric-field sensor as well as the effects of parameter mismatch, which appear naturally due to material imperfections and noise. The results show that Signal-to-Noise Ratio (SNR) increases sharply near the onset of the infinite-period bifurcation, and they increase further as the coupling strength in the network increases while passing the threshold that leads to oscillatory behavior. Overall, the SNR indicates that the negative effects of highly contaminated signals are well-mitigated by the sensitivity response of the system. In addition, computer simulations show the network-based system to be robust enough to mismatches in system parameters, while the deviations from the nominal parameter values form regions where the oscillations persist. Noise has a smoothing effect over the boundaries of these regions.

Magnetically induced gluing bifurcations of three-tori in spherical Couette flows

The global dynamics of three-tori associated with homoclinic/heteroclinic global (infinite period) bifurcations are investigated for the magnetized spherical Couette problem, a paradigmatic model in geo- and astrophysical magnetohydrodynamics (MHD). A novel homoclinic bifurcation, involving collision between three-tori, is described. In addition, a heteroclinic bifurcation connecting two unstable two-tori with a stable three-torus is also analyzed. The role of the flow's spatial symmetries in this bifurcation scenario is also investigated. This bifurcation scenario gives rise to MHD flows that combine small with extremely large time scales.

Arnold tongue entrainment reveals dynamical principles of the embryonic segmentation clock.

Living systems exhibit an unmatched complexity, due to countless, entangled interactions across scales. Here, we aim to understand a complex system, that is, segmentation timing in mouse embryos, without a reference to these detailed interactions. To this end, we develop a coarse-grained approach, in which theory guides the experimental identification of the segmentation clock entrainment responses. We demonstrate period- and phase-locking of the segmentation clock across a wide range of entrainment parameters, including higher-order coupling. These quantifications allow to derive the phase response curve (PRC) and Arnold tongues of the segmentation clock, revealing its essential dynamical properties. Our results indicate that the somite segmentation clock has characteristics reminiscent of a highly non-linear oscillator close to an infinite period bifurcation and suggests the presence of long-term feedbacks. Combined, this coarse-grained theoretical-experimental approach reveals how we can derive simple, essential features of a highly complex dynamical system, providing precise experimental control over the pace and rhythm of the somite segmentation clock.

A Parametric Resonance for the Nonlocal Hirota–Maccari Equation

The nonlocal Hirota–Maccari equation is considered when a parametric excitation is acting over the frequency of a generic mode. Using the well-known asymptotic perturbation (AP) method, two coupled equations for the amplitude and phase can be obtained. We discovered the existence of an infinite-period bifurcation when the parametric force increases its value. Moreover, symmetry considerations suggest performing a global analysis of the two couples, in such a way that we find an energy-like function and corroborate and verify the existence of this infinite period bifurcation.

Collective Coherence Resonance in Networks of Optical Neurons

The dynamical properties of an optical neuron formed by a quantum dot semiconductor laser model subjected to optical injection and optical feedback are analyzed. The parameter space spanned by the injection strength and frequency detuning of the optical injection is systematically scanned and modulations of the bifurcation boundaries that induce complex scenarios are found, which enable new opportunities to introduce the optical setup as an optical neuron. The counterintuitive behavior of coherence resonance for different setups of a single‐driven optical neuron under optical feedback is also found. Following the results, the microscopically motivated quantum dot laser rate equation model is reduced to the normal form of a saddle‐node infinite period (SNIPER) bifurcation for low injection strengths and a network of four such SNIPER systems in a globally coupled setup is studied. A new phenomenon is observed, which is called collective coherence resonance. This novel dynamical behavior is connected to the coexistence of a network‐wide SNIPER bifurcation and a change in stability for the synchronized manifold, analyzed via the master stability function.

Oscillations and collective excitability in a model of stochastic neurons under excitatory and inhibitory coupling.

We study a model with excitable neurons modeled as stochastic units with three states, representing quiescence, firing, and refractoriness. The transition rates between quiescence and firing depend exponentially on the number of firing neighbors, whereas all other rates are kept constant. This model class was shown to exhibit collective oscillations (synchronization) if neurons are spiking autonomously, but not if neurons are in the excitable regime. In both cases, neurons were restricted to interact through excitatory coupling. Here we show that a plethora of collective phenomena appear if inhibitory coupling is added. Besides the usual transition between an absorbing and an active phase, the model with excitatory and inhibitory neurons can also undergo reentrant transitions to an oscillatory phase. In the mean-field description, oscillations can emerge through supercritical or subcritical Hopf bifurcations, as well as through infinite period bifurcations. The model has bistability between active and oscillating behavior, as well as collective excitability, a regime where the system can display a peak of global activity when subject to a sufficiently strong perturbation. We employ a variant of the Shinomoto-Kuramoto order parameter to characterize the phase transitions and their system-size dependence.

Oscillation death in coupled counter-rotating identical nonlinear oscillators.

We study oscillatory and oscillation suppressed phases in coupled counter-rotating nonlinear oscillators. We demonstrate the existence of limit cycle, amplitude death, and oscillation death, and also clarify the Hopf, pitchfork, and infinite period bifurcations between them. Especially, the oscillation death is a new type of oscillation suppressions of which the inhomogeneous steady states are neutrally stable. We discuss the robust neutral stability of the oscillation death in non-conservative systems via the anti-parity-time-symmetric phase transitions at exceptional points in terms of non-Hermitian systems.

PC -symmetry-protected edge states in interacting driven-dissipative bosonic systems

A main objective of topological photonics is the design of disorder-resilient optical devices. Many prospective applications would benefit from nonlinear effects, which not only are naturally present in real systems but also are needed for switching in computational processes, while the underlying particle interactions are a key ingredient for the manifestation of genuine quantum effects. A particularly attractive switching mechanism of dynamical systems are infinite-period bifurcations into limit cycles, as these set on with a finite amplitude. Here we describe how to realize this switching mechanism by combining attractive and repulsive particle interactions in a driven-dissipative Su-Schrieffer-Heeger model, such as realized in excitonic lasers and condensates so that the system displays a nonhermitian combination of parity and charge-conjugation (PC) symmetry. We show that this symmetry survives in the nonlinear case and induces infinite-period and limit-cycle bifurcations (distinct from a Hopf bifurcation) where the system switches from a symmetry-breaking stationary state into a symmetry-protected power-oscillating state of finite amplitude. These protected dynamical solutions display a number of characteristic features, among which are their finite amplitude at onset, their arbitrary long oscillation period close to threshold, and the symmetry of their frequency spectrum which provides a tuneable frequency comb. Phases with different transition scenarios are separated by exceptional points in the stability spectrum, involving nonhermitian degeneracies of symmetry-protected excitations.

Phase slips in superconducting weak links

Superconducting vortices and phase slips are primary mechanisms of dissipation in superconducting, superfluid, and cold atom systems. While the dynamics of vortices is fairly well described, phase slips occurring in quasi-one dimensional superconducting wires still elude understanding. The main reason is that phase slips are strongly non-linear time-dependent phenomena that cannot be cast in terms of small perturbations of the superconducting state. Here we study phase slips occurring in superconducting weak links. Thanks to partial suppression of superconductivity in weak links, we employ a weakly nonlinear approximation for dynamic phase slips. This approximation is not valid for homogeneous superconducting wires and slabs. Using the numerical solution of the time-dependent Ginzburg-Landau equation and bifurcation analysis of stationary solutions, we show that the onset of phase slips occurs via an infinite period bifurcation, which is manifested in a specific voltage-current dependence. Our analytical results are in good agreement with simulations.

Travelling-wave solutions bifurcating from relative periodic orbits in plane Poiseuille flow

Travelling-wave solutions are shown to bifurcate from relative periodic orbits in plane Poiseuille flow at Re = 2000 in a saddle-node infinite period bifurcation. These solutions consist in self-sustaining sinuous quasi-streamwise streaks and quasi- streamwise vortices located in the bulk of the flow. The lower branch travelling-wave solutions evolve into spanwise localized states when the spanwise size Lz of the domain in which they are computed is increased. On the contrary, upper branch of travelling-wave solutions develop multiple streaks when Lz is increased. Upper branch travelling-wave solutions can be continued into coherent solutions of the filtered equations used in large-eddy simulations where they represent turbulent coherent large-scale motions.

Evaluation of reaction fluxes in stationary and oscillating far-from-equilibrium biological systems

The complex spatio-temporal structures that appear in chemical and biological systems require far-from-equilibrium conditions which may lead to the circulation of reaction fluxes. We investigate how time asymmetry of cross-correlation functions of concentration fluctuations may be exploited to determine reaction fluxes at the cellular level. Using simulations of the master equation as a reference, we show that, far from a bifurcation, the Langevin approach provides a reliable tool to compute analytical expressions for time correlation functions. Biochemical mechanisms associated with bistability and oscillations issued from a Hopf bifurcation or a saddle–node infinite period bifurcation are considered. We show that the blind use of the simple relation obtained when assuming a linear deterministic dynamics often leads to a poor estimation of the value of the reaction flux and even of its sign.

Symmetry-Breaking as a Paradigm to Design Highly-Sensitive Sensor Systems

A large class of dynamic sensors have nonlinear input-output characteristics, often corresponding to a bistable potential energy function that controls the evolution of the sensor dynamics. These sensors include magnetic field sensors, e.g., the simple fluxgate magnetometer and the superconducting quantum interference device (SQUID), ferroelectric sensors and mechanical sensors, e.g., acoustic transducers, made with piezoelectric materials. Recently, the possibilities offered by new technologies and materials in realizing miniaturized devices with improved performance have led to renewed interest in a new generation of inexpensive, compact and low-power fluxgate magnetometers and electric-field sensors. In this article, we review the analysis of an alternative approach: a symmetry-based design for highly-sensitive sensor systems. The design incorporates a network architecture that produces collective oscillations induced by the coupling topology, i.e., which sensors are coupled to each other. Under certain symmetry groups, the oscillations in the network emerge via an infinite-period bifurcation, so that at birth, they exhibit a very large period of oscillation. This characteristic renders the oscillatory wave highly sensitive to symmetry-breaking effects, thus leading to a new detection mechanism. Model equations and bifurcation analysis are discussed in great detail. Results from experimental works on networks of fluxgate magnetometers are also included.

Optical injection enables coherence resonance in quantum-dot lasers

We demonstrate that optically injected semiconductor quantum-dot lasers operated in the frequency-locked regime exhibit the counterintuitive effect of coherence resonance, i.e., the regularity of noise-induced spiking is a non-monotonic function of the spontaneous emission noise, and it is optimally correlated at a non-zero value of the noise intensity. We uncover the mechanism of coherence resonance from a microscopically based model of the quantum-dot laser structure, and show that it is related to excitability under optical injection and to a saddle-node infinite period (SNIPER) bifurcation occurring for small injection strength at the border of the frequency locking regime. By a model reduction we argue that the phenomenon of coherence resonance is generic for a wide class of optically injected lasers.

Coherent solutions for the fundamental resonance of the Boussinesq equation

We consider the Boussinesq equation in an infinite wall under an external resonant weak excitation. Two slow flow equations are obtained and amplitude and phase modulation equations as well as external force-response and frequency–response curves are determined. Energy considerations are used in order to study the global behavior of the two slow flow equations and to demonstrate the existence of closed orbits around the equilibrium points with a second frequency in addition to the forcing frequency. Two period quasi-periodic motions are present with amplitudes depending on the initial conditions. Moreover, in certain cases librations can occur with a frequency depending on the amplitude of the external excitation. If the external excitation increases, the modulation period for the slow flow equations becomes infinite and an infinite-period bifurcation occurs, while the modulation amplitude remains finite and the libration transforms into a closed orbit. Finally, we demonstrate that the separatrix of the two slow flow equations corresponds to an asymptotically steady-state and phase-locked periodic solution.

Edge states for the turbulence transition in the asymptotic suction boundary layer

AbstractWe demonstrate the existence of an exact invariant solution to the Navier–Stokes equations for the asymptotic suction boundary layer. The identified periodic orbit with a very long period of several thousand advective time units is found as a local dynamical attractor embedded in the stability boundary between laminar and turbulent dynamics. Its dynamics captures both the interplay of downstream-oriented vortex pairs and streaks observed in numerous shear flows as well as the energetic bursting that is characteristic for boundary layers. By embedding the flow into a family of flows that interpolates between plane Couette flow and the boundary layer, we demonstrate that the periodic orbit emerges in a saddle–node infinite-period (SNIPER) bifurcation of two symmetry-related travelling-wave solutions of plane Couette flow. Physically, the long period is due to a slow streak instability, which leads to a violent breakup of a streak associated with the bursting and the reformation of the streak at a different spanwise location. We show that the orbit is structurally stable when varying both the Reynolds number and the domain size.

Electronic Implementation of a Repressilator with Quorum Sensing Feedback

We investigate the dynamics of a synthetic genetic repressilator with quorum sensing feedback. In a basic genetic ring oscillator network in which three genes inhibit each other in unidirectional manner, an additional quorum sensing feedback loop stimulates the activity of a chosen gene providing competition between inhibitory and stimulatory activities localized in that gene. Numerical simulations show several interesting dynamics, multi-stability of limit cycle with stable steady-state, multi-stability of different stable steady-states, limit cycle with period-doubling and reverse period-doubling, and infinite period bifurcation transitions for both increasing and decreasing strength of quorum sensing feedback. We design an electronic analog of the repressilator with quorum sensing feedback and reproduce, in experiment, the numerically predicted dynamical features of the system. Noise amplification near infinite period bifurcation is also observed. An important feature of the electronic design is the accessibility and control of the important system parameters.

Bifurcations with imperfect SO(2) symmetry and pinning of rotating waves

Rotating waves are periodic solutions in SO(2) equivariant dynamical systems. Their precession frequency changes with parameters and it may change sign, passing through zero. When this happens, the dynamical system is very sensitive to imperfections that break the SO(2) symmetry and the waves may become trapped by the imperfections, resulting in steady solutions that exist in a finite region in parameter space. This is the so-called pinning phenomenon. In this study, we analyse the breaking of the SO(2) symmetry in a dynamical system close to a Hopf bifurcation whose frequency changes sign along a curve in parameter space. The problem is very complex, as it involves the complete unfolding of high codimension. A detailed analysis of different types of imperfections indicates that a pinning region surrounded by infinite-period bifurcation curves appears in all cases. Complex bifurcational processes, strongly dependent on the specifics of the symmetry breaking, appear very close to the intersection of the Hopf bifurcation and the pinning region. Scaling laws of the pinning region width and partial breaking of SO(2) to Z m are also considered. Previous as well as new experimental and numerical studies of pinned rotating waves are reviewed in the light of the new theoretical results.

BIFURCATIONS IN A FREQUENCY-WEIGHTED KURAMOTO OSCILLATORS NETWORK

This paper focuses on the origin of rich dynamics in a frequency-weighted Kuramoto-oscillator network, which embeds the periodical activity patterns of human beings to understand the collective behaviorial dynamics in human population. We present analytical results of the dynamics of the model by reducing the N-dimensional system to solvable low-dimensional equations. The bifurcation analysis reveals that there exist supercritical Hopf bifurcation and infinite-period saddle-node bifurcation, which explain the rich dynamics including the incoherent, oscillatory, and synchronous states observed in this model.

“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.

An Analysis of Overall Network Architecture Reveals an Infinite-period Bifurcation Underlying Oscillation Arrest in the Segmentation Clock

Unveiling the mechanisms through which the somitogenesis regulatory network exerts spatiotemporal control of the somitic patterning has required a combination of experimental and mathematical modeling strategies. Significant progress has been made for the zebrafish clockwork. However, due to its complexity, the clockwork of the amniote segmentation regulatory network has not been fully elucidated. Here, we address the question of how oscillations are arrested in the amniote segmentation clock. We do this by constructing a minimal model of the regulatory network, which privileges architectural information over molecular details. With a suitable choice of parameters, our model is able to reproduce the oscillatory behavior of the Wnt, Notch and FGF signaling pathways in presomitic mesoderm (PSM) cells. By introducing positional information via a single Wnt3a gradient, we show that oscillations are arrested following an infinite-period bifurcation. Notably: the oscillations increase their amplitude as cells approach the anterior PSM and remain in an upregulated state when arrested; the transition from the oscillatory regime to the upregulated state exhibits hysteresis; and opposing Fgf8 and RA gradients along the PSM naturally arise in our simulations. We hypothesize that the interaction between a limit cycle (originated by the Notch delayed-negative feedback loop) and a bistable switch (originated by the Wnt-Notch positive cross-regulation) is responsible for the observed segmentation patterning. Our results agree with previously unexplained experimental observations and suggest a simple plausible mechanism for spatiotemporal control of somitogenesis in amniotes.