Mixed-mode Oscillations Research Articles

Averaging method for the existence of singular canards in a two-cell inhibitory neural network

Abstract Studies of firing rate neuronal competition models, as observed in binocular rivalry, showed that the variation of the stimulus parameters can lead to rich and interesting dynamics. We use the two-cell inhibitory neural network model to address the canards accompanied by the L^s-typed mixed-mode oscillations (MMOs). Specifically, we prove the existence and asymptotic expression of canards for this two-cell model by the average method. Meanwhile, our simulation indicates that the results are in accordance with the theory. Our solution can provide ideas for determining and selecting weights in model fusion of machine learning.

Mixed-mode oscillations and chaos in a complex chemical reaction network involving heterogeneous catalysis.

The nonlinear dynamical behavior in a complex isothermal reaction network involving heterogeneous catalysis is studied. The method first determines the multiple steady states in the reaction network. This is followed by an analysis of bifurcation continuations to identify several kinds of bifurcations, including limit point, Bogdanov-Takens, generalized Hopf, period doubling, and generalized period doubling. Numerical simulations are performed around the period doubling and generalized period doubling bifurcations. Rich nonlinear behaviors are observed, including simple sustained oscillations, mixed-mode oscillations, non-mixed-mode chaotic oscillations, and mixed-mode chaotic oscillations. Concentration-time plots, 2D phase portraits, Poincaré maps, maximum Lyapunov exponents, frequency spectra, and cascade of bifurcations are reported. Period-doubling and period-adding routes leading to chaos are observed. Maximum Lyapunov exponents are positive for all the chaotic cases, but they are also positive for some non-chaotic orbits. This result diminishes the reliability of using maximum Lyapunov exponents as a tool for determining chaos in the network under study.

Dynamical effects of low-frequency and high-frequency current stimuli in a memristive Morris–Lecar neuron model

This paper deduces that the potassium and calcium ion currents in the two-dimensional (2D) Morris–Lecar neuron model can be respectively characterized by a passive memristor and a locally active memristor. Then a memristive Morris–Lecar neuron model with an equivalent circuit scheme is first constructed. The equilibrium trajectory and its stability are analyzed, which displays fold and Hopf bifurcations with proper model parameters. Bursting behavior and mixed-mode oscillations with after-depolarizing potential for low-frequency current stimulus are numerically disclosed. Besides, the bifurcation mechanisms for bursting behavior and mixed-mode oscillations are theoretically deduced. Furthermore, the firing patterns and antimonotonicity phenomenon for high-frequency current stimulus are numerically discovered. This study provides a new concept for building memristive bionic circuits from neuron models with well-defined ion channel currents.

Unveiling complex magnetic field configurations in red giant stars

The recent measurement of magnetic field strength inside the radiative interior of red giant stars has opened the way toward full 3D characterization of the geometry of stable large-scale magnetic fields. However, current measurements, which are limited to dipolar (ℓ = 1) mixed modes, do not properly constrain the topology of magnetic fields due to degeneracies on the observed magnetic field signature on such ℓ = 1 mode frequencies. Efforts focused toward unambiguous detections of magnetic field configurations are now key to better understand angular momentum transport in stars. We investigated the detectability of complex magnetic field topologies (such as the ones observed at the surface of stars with a radiative envelope with spectropolarimetry) inside the radiative interior of red giants. We focused on a field composed of a combination of a dipole and a quadrupole (quadrudipole) and on an offset field. We explored the potential of probing such magnetic field topologies from a combined measurement of magnetic signatures on ℓ = 1 and quadrupolar (ℓ = 2) mixed mode oscillation frequencies. We first derived the asymptotic theoretical formalism for computing the asymmetric signature in the frequency pattern for ℓ = 2 modes due to a quadrudipole magnetic field. To access asymmetry parameters for more complex magnetic field topologies, we numerically performed a grid search over the parameter space to map the degeneracy of the signatures of given topologies. We demonstrate the crucial role played by ℓ = 2 mixed modes in accessing internal magnetic fields with a quadrupolar component. The degeneracy of the quadrudipole compared to pure dipolar fields is lifted when considering magnetic asymmetries in both ℓ = 1 and ℓ = 2 mode frequencies. In addition to the analytical derivation for the quadrudipole, we present the prospect for complex magnetic field inversions using magnetic sensitivity kernels from standard perturbation analysis for forward modeling. Using this method, we explored the detectability of offset magnetic fields from ℓ = 1 and ℓ = 2 frequencies and demonstrate that offset fields may be mistaken for weak and centered magnetic fields, resulting in underestimating the magnetic field strength in stellar cores. We emphasize the need to characterize ℓ = 2 mixed-mode frequencies, (along with the currently characterized ℓ = 1 mixed modes), to unveil the higher-order components of the geometry of buried magnetic fields and to better constrain angular momentum transport inside stars.

Bifurcations and mixed mode oscillations in a bi-stable plasma model with slow parametric excitation.

In this study, considering a bi-stable plasma model with slow parametric excitation, the bifurcation of periodic and chaotic responses as well as the resulting fast-slow motions is discussed analytically and numerically. For a nonautonomous fast sub-system, the generalized harmonic balancing method is utilized to obtain an averaged system. Bifurcation analysis about the averaged system shows that the critical manifolds form a S-shape structure. Meanwhile, supercritical and subcritical period doubling (PD) occurs on the upper branch simultaneously. As the frequency of the external excitation changes, bifurcation points on the limit cycle manifolds can present different relative locations. Moreover, an additional bi-stable structure induced by Cusp bifurcation emanates from the upper branch. On the other hand, the existence of a chaotic attractor and the corresponding boundary crisis phenomenon are verified using the Melnikov method and the basin of attraction. The structures of the numerical bifurcation diagram show good agreements with the analytical results. Considering two cases of low-frequency excitation, the corresponding fast-slow dynamics are discussed. It is found that, when the fast-slow flow passing the subcritical PD point, a low frequency with different magnitudes will lead to two patterns of bifurcation delay, i.e., the typical one and the excessive delay, which suppress the PD. As for the boundary crisis point, the slow passage effects show no distinct influence. Thus, three transition mechanisms based on two cases of the bifurcation structure are explained, including "fold of cycle-fold of cycle" type, "fold of cycle-delayed subcritical PD" type, and "fold of cycle-boundary crisis" type.

Transition to oscillatory instability of lean methane–air flames in microchannels

Micro-flow reactors and porous media burners are prone to instability in the form of flame with repetitive extinction and ignition (FREI) near the region of a temperature gradient. Although significant progress has been made in this field, there is still a lack of understanding regarding the nature of such stable-to-unstable transition. Some studies reported a smooth and continuous instability transition in microchannels, interpreting this phenomenon as a supercritical bifurcation. Meanwhile, others have observed that the transition is subcritical, with rapid, stepwise evolution of the stable flame to FREI. This study aims to rigorously determine the transition character theoretically and numerically using a two-dimensional model with a precise resolution of the bifurcation parameter (flow velocity) near the critical point and gradual wall temperature ramp. The results indicate that the transition is a subcritical bifurcation with strong hysteresis. However, the amplitude of the limit cycle born in the bifurcation point could be small compared to FREI-like oscillations. Moreover, further development of the instability depends significantly on the channel width. In relatively wide channels, the stable regime transforms immediately into the extinction/ignition one with rapid growth in amplitude. In moderate channels, complex transitional dynamics were observed with an evident pulsating regime, which evolved into FREI through mixed-mode oscillations. In narrow channels, the amplitude of transitional pulsating flame becomes comparable to the fully-developed extinction/ignition cycles, and the bifurcation diagram has a continuous character with no significant jumps or discontinuities. Such qualitative variation of transition character provides a possible explanation for contradictory interpretations presented in the literature.Novelty and Significance StatementCurrently, there is no consensus about the character of transition between the stable and extinction/ignition regimes in microchannels. Some studies have reported a continuous instability transition with an evident pulsating regime, interpreting it as a supercritical bifurcation, while others have observed subcritical bifurcation with a rapid transition. The novelty of this research lies in the rigorous determination of the bifurcation type and further instability development in microchannels of different widths based on an accurate simulation of flame dynamics near the critical point with very small steps of the flow velocity variation, along with the bifurcation theory. The study contributes to understanding the instability transition nature and may explain the different interpretations of the transitional phenomenon found in the literature. The results offer valuable insights for designing micro-scale and porous media combustion devices.

Mixed-mode oscillations of an atomic force microscope in tapping mode.

In the phenomenon of mixed-mode oscillations, transitions between large-amplitude and small-amplitude oscillations may lead to anomalous jitter in the probe of a tapping mode atomic force microscope (TM-AFM) during the scanning process, thereby affecting the accuracy and clarity of the topographical images of the tested sample's surface. This work delves deeply into various mixed-mode oscillations and the corresponding formation mechanisms in TM-AFM under low-frequency resonant excitation. Through a detailed analysis of bifurcation sets of the fast subsystem, we found that the system's mixed-mode oscillations encompass the typical two coexisting branches and the novel three coexisting branches of equilibrium point attractors. In the stable case, a certain transition pattern in phase trajectory can be observed involving two jumps and four jumps, switching between quiescent and spiking states. In the bi-stable case, the trajectory undergoes distinct transitions decided by whether to pass through or crossover the middle branch of attractors when bifurcation occurs. By applying basin of attraction and fast-slow analysis methods, we unfold the dynamic mechanism of mixed-mode oscillations with distinct switching patterns. Our research contributes to a better understanding of complex oscillations of TM-AFM and provides valuable insights for improving image quality and measurement precision while mitigating detrimental oscillations.

Mixed-mode oscillations in a three-timescale coupled Morris-Lecar system.

Mixed-mode oscillations (MMOs) are complex oscillatory behaviors of multiple-timescale dynamical systems in which there is an alternation of large-amplitude and small-amplitude oscillations. It is well known that MMOs in two-timescale systems can arise either from a canard mechanism associated with folded node singularities or a delayed Andronov-Hopf bifurcation (DHB) of the fast subsystem. While MMOs in two-timescale systems have been extensively studied, less is known regarding MMOs emerging in three-timescale systems. In this work, we examine the mechanisms of MMOs in coupled Morris-Lecar neurons with three distinct timescales. We investigate two kinds of MMOs occurring in the presence of a singularity known as canard-delayed-Hopf (CDH) and in cases where CDH is absent. In both cases, we examine how features and mechanisms of MMOs vary with respect to variations in timescales. Our analysis reveals that MMOs supported by CDH demonstrate significantly stronger robustness than those in its absence. Moreover, we show that the mere presence of CDH does not guarantee the occurrence of MMOs. This work yields important insights into conditions under which the two separate mechanisms in two-timescale context, canard and DHB, can interact in a three-timescale setting and produce more robust MMOs, particularly against timescale variations.

Strong symmetry breaking rhythms created by folded nodes in a pair of symmetrically coupled, identical Koper oscillators.

The Koper model is a prototype system with two slow variables and one fast variable that possesses small-amplitude oscillations (SAOs), large-amplitude oscillations (LAOs), and mixed-mode oscillations (MMOs). In this article, we study a pair of identical Koper oscillators that are symmetrically coupled. Strong symmetry breaking rhythms are presented of the types SAO-LAO, SAO-MMO, LAO-MMO, and MMO-MMO, in which the oscillators simultaneously exhibit rhythms of different types. We identify the key folded nodes that serve as the primary mechanisms responsible for the strong nature of the symmetry breaking. The maximal canards of these folded nodes guide the orbits through the neighborhoods of these key points. For all of the strong symmetry breaking rhythms we present, the rhythms exhibited by the two oscillators are separated by maximal canards in the phase space of the oscillator.

Strong Symmetry Breaking in Coupled, Identical Lengyel–Epstein Oscillators via Folded Singularities

We study pairs of symmetrically coupled, identical Lengyel-Epstein oscillators, where the coupling can be through both the fast and slow variables. We find a plethora of strong symmetry breaking rhythms, in which the two oscillators exhibit qualitatively different oscillations, and their amplitudes differ by as much as an order of magnitude. Analysis of the folded singularities in the coupled system shows that a key folded node, located off the symmetry axis, is the primary mechanism responsible for the strong symmetry breaking. Passage through the neighborhood of this folded node can result in splitting between the amplitudes of the oscillators, in which one is constrained to remain of small amplitude, while the other makes a large-amplitude oscillation or a mixed-mode oscillation. The analysis also reveals an organizing center in parameter space, where the system undergoes an asymmetric canard explosion, in which one oscillator exhibits a sequence of limit cycle canards, over an interval of parameter values centered at the explosion point, while the other oscillator executes small amplitude oscillations. Other folded singularities can also impact properties of the strong symmetry breaking rhythms. We contrast these strong symmetry breaking rhythms with asymmetric rhythms that are close to symmetric states, such as in-phase or anti-phase oscillations. In addition to the symmetry breaking rhythms, we also find an explosion of anti-phase limit cycle canards, which mediates the transition from small-amplitude, anti-phase oscillations to large-amplitude, anti-phase oscillations.

Modelling and analysis of cAMP-induced mixed-mode oscillations in cortical neurons: Critical roles of HCN and M-type potassium channels.

Cyclic AMP controls neuronal ion channel activity. For example hyperpolarization-activated cyclic nucleotide-gated (HCN) and M-type K+ channels are activated by cAMP. These effects have been suggested to be involved in astrocyte control of neuronal activity, for example, by controlling the action potential firing frequency. In cortical neurons, cAMP can induce mixed-mode oscillations (MMOs) consisting of small-amplitude, subthreshold oscillations separating complete action potentials, which lowers the firing frequency greatly. We extend a model of neuronal activity by including HCN and M channels, and show that it can reproduce a series of experimental results under various conditions involving and inferring with cAMP-induced activation of HCN and M channels. In particular, we find that the model can exhibit MMOs as found experimentally, and argue that both HCN and M channels are crucial for reproducing these patterns. To understand how M and HCN channels contribute to produce MMOs, we exploit the fact that the model is a three-time scale dynamical system with one fast, two slow, and two super-slow variables. We show that the MMO mechanism does not rely on the super-slow dynamics of HCN and M channel gating variables, since the model is able to produce MMOs even when HCN and M channel activity is kept constant. In other words, the cAMP-induced increase in the average activity of HCN and M channels allows MMOs to be produced by the slow-fast subsystem alone. We show that the slow-fast subsystem MMOs are due to a folded node singularity, a geometrical structure well known to be involved in the generation of MMOs in slow-fast systems. Besides raising new mathematical questions for multiple-timescale systems, our work is a starting point for future research on how cAMP signalling, for example resulting from interactions between neurons and glial cells, affects neuronal activity via HCN and M channels.

Analysis of Successive Doubly Nested Mixed-Mode Oscillations

In previous works [Inaba & Kousaka, 2020; Inaba & Tsubone, 2020; Inaba et al., 2023], significant bifurcation structures referred to as nested Mixed-Mode Oscillations (MMOs) were found to be present in forced Bonhoeffer–van der Pol (BVP) oscillators. It is well known that unnested Mixed-Mode Oscillation-Incrementing Bifurcations (MMOIBs) can generate [Formula: see text] oscillations (i.e. [Formula: see text] followed by [Formula: see text] repeated [Formula: see text] times) for successive values of [Formula: see text], where [Formula: see text] and [Formula: see text] are adjacent fundamental simple MMOs, e.g. [Formula: see text] and [Formula: see text], where [Formula: see text] is an integer. Furthermore, it has been confirmed that MMOIBs can generate nested MMOs. Let two adjacent unnested MMOIB-generated MMOs be denoted [Formula: see text] and [Formula: see text]. Then, singly nested MMOIBs can generate [Formula: see text] for successive values of [Formula: see text], i.e. [Formula: see text] followed by [Formula: see text] repeated [Formula: see text] times, between the [Formula: see text]- and [Formula: see text]-generating regions. The sequential generation of singly nested MMOs has been investigated in detail in previous work [Ito et al., 2021]. Nested MMOs can, however, be at least doubly nested. In this study, we investigate doubly nested MMOs considering a constrained nonautonomous BVP oscillator containing an idealized diode. Based on the observed dynamics of this system, Poincaré return maps are rigorously constructed in one dimension. Therefore, we can solve the successive saddle-node bifurcations using a nested (i.e. double-loop) bisection method. We track 60 successive doubly nested MMOIBs and we do not rule out the possibility that the 58 scaling constants corresponding to the MMOIB intervals converge to unity. We note that because we solve the bifurcation equation avoiding the use of a method that requires the careful selection of the initial conditions (e.g. the Newton–Raphson), we can accurately track the saddle-node bifurcations without missing any doubly nested MMO sequences.

Emergent dynamics in fractional-order Wilson–Cowan neural network systems

The firing dynamics of excitable systems are critical to understand organized responses in cortical networks. In this paper, we examine a fractional-order Wilson–Cowan (W–C) neural network model composed of excitatory and inhibitory neuron populations, utilizing Caputo’s fractional-order derivative formalism to explore the influence of fractional-order dynamics on firing behavior. The significance of extending to the fractional-order domain lies in the model’s theoretical framework, which inherently retains memory and hereditary characteristics. We investigate memory-dependent response functions and average neuronal characteristics, enabling us to formulate a fractional-order model that incorporates past dynamics into the neuronal populations’ features. This generalized model is capable of producing alternations between spiking and bursting phenomena, including mixed-mode oscillations (MMOs). Using stability and bifurcation analyses, we delineate the parameter space within which variations in firing patterns emerge. One notable impact of the model’s memory property is the potential stabilization of neuronal activity. We demonstrate that our numerical findings are in alignment with the analytical predictions and that the memory trace depends on the fractional-order dynamics.

Unraveling the impact of temperature on the reaction kinetics of the electro-oxidation of methanol on Pt(1 0 0)

Methanol is one of the key molecules in the challenge towards a sustainable future, particularly as a renewable hydrogen carrier fuel and as a low-carbon and net carbon-neutral liquid chemical. For most applications, it is imperative to understand the impact of temperature on the methanol electro-oxidation reaction (MEOR). In this study, the influence of the temperature on the kinetics of the MEOR and the parallel reaction pathways is assessed by investigating responses in both conventional and oscillatory regimes using a single-crystal Pt(100) electrode. Our findings demonstrate that chronoamperometric measurements under steady-state conditions provide more reliable values for apparent activation energies compared to transient conditions. Furthermore, a temperature-dependent shift in the dominance of specific oxidation pathways is observed, analogous to a kinetic and thermodynamic control mechanism, preventing the complete poisoning of the electrode surface. Specifically, oxidation pathways leading to the formation of reaction byproducts predominate at lower temperatures, while the oxidation pathway via COad becomes dominant at temperatures exceeding 30 °C. Moreover, our research shows that, at shorter times, temperature changes minimally affect the mean potential required to sustain the applied current during the oscillations in a galvanostatic experiment, which is closely linked with the voltaic efficiency. However, over longer periods, when mass transport phenomena become significant and mixed-mode oscillations occur, elevated temperatures increase the mean potential, resulting in reduced voltaic efficiency. Therefore, to facilitate the complete conversion of methanol to CO2 without increasing the mean potential for current maintenance, it is essential not only to increase the temperature but also to improve the mass transport conditions to mitigate the mixed-mode oscillations, despite their lower minima reached during oscillation. This idea challenges the conventional assumption that a lower minimum potential implies a lower mean potential during oscillations. This advancement propels our understanding to a more sophisticated level, providing valuable insights to guide the materials design to increase the conversion efficiency and optimize the operating temperature of devices crucial to energy conversion.

Nested mixed-mode oscillations in the forced van der Pol oscillator

The forced van der Pol oscillator has played a fundamental role in the development of nonlinear science. It is notable that the van der Pol oscillator in the absence of an AC forcing term explains the underlying mechanism that induces limit cycles and relaxation oscillations; the forced van der Pol oscillator was the first electric oscillator that, via measurements of the emitted sound, was inferred to exhibit chaotic behavior (van der Pol and van der Mark, 1927). This oscillator thus had a significant influence on the development of chaos theory. It was subsequently demonstrated, via a nonstandard analysis undertaken in the 1980s, that a canard explosion was present in the dynamics exhibited by this oscillator. In previous works (Inaba and Kousaka, (2020); Inaba et al., (2023)), we established the existence of bifurcation structures that are referred to as nested mixed-mode oscillations (MMOs); these structures are generated by a forced Bonhoeffer–van der Pol (BVP) oscillator. Nested MMOs, despite their complexity, are robust, repeatable, and composed of higher dimensional oscillations. In this study, we show that nested MMOs can also be produced by the forced van der Pol oscillator. This paper is of extreme importance as it demonstrates that nested MMOs are not a phenomenon that can only be observed in specific BVP systems. Based on the fact that nested MMO bifurcations also occur in the forced van der Pol oscillator, this study indicates that nested MMOs could represent a phenomenon that is more widely observed than previously believed.

Bifurcation mechanisms underlying the nested mixed-mode oscillations in a canard-generating driven Bonhoeffer–van der Pol oscillator

In previous works (Inaba and Kousaka, 2020; Inaba and Tsubone, 2020; Inaba et al, 2022) the bifurcation structures referred to as nested mixed-mode oscillations (MMOs) were discovered; these structures can be generated by a driven slow–fast Bonhoeffer–van der Pol (BVP) oscillator. It is known that this oscillator can generate canards in the absence of perturbations. In this work, we investigate nested MMOs in a canard-generating driven BVP oscillator near a supercritical Hopf bifurcation point subjected to weak periodic perturbations. The analysis of this system is undertaken using the techniques presented by Kawakami, (1984). These techniques include the use of a shooting algorithm to derive the parameters of the forced oscillators at which bifurcations occur by solving the simultaneous equations that identify the conditions in which the solution of the forced oscillators is periodic and the characteristic multiplier(s) with the largest absolute value of the periodic solution on the Poincaré section crosses the unit circle. We also obtain several two-parameter bifurcation diagrams that describe the system investigated here. One motivation for this study is to demonstrate that the bifurcation structures present in nested MMOs are likely to be a widespread phenomenon. We hypothesize here that nested MMOs represent a stable phenomenon; this is likely to be the case because they appear and disappear consistently and are bound by sequential saddle–node and period-doubling bifurcation curves; this hypothesis is supported by the observation that codimension-two bifurcation points or cusps do not appear in the region close to these bifurcation boundaries.

Complex adaptive learning cortical neural network systems for solving time-fractional difference equations with bursting and mixed-mode oscillation behaviours

Complex networks have been programmed to mimic the input and output functions in multiple biophysical algorithms of cortical neurons at spiking resolution. Prior research has demonstrated that the ineffectual features of membranes can be taken into account by discrete fractional commensurate, non-commensurate and variable-order patterns, which may generate multiple kinds of memory-dependent behaviour. Firing structures involving regular resonator chattering, fast, chaotic spiking and chaotic bursts play important roles in cortical nerve cell insights and execution. Yet, it is unclear how extensively the behaviour of discrete fractional-order excited mechanisms can modify firing cell attributes. It is illustrated that the discrete fractional behaviour of the Izhikevich neuron framework can generate an assortment of resonances for cortical activity via the aforesaid scheme. We analyze the bifurcation using fragmenting periodic solutions to demonstrate the evolution of periods in the framework’s behaviour. We investigate various bursting trends both conceptually and computationally with the fractional difference equation. Additionally, the consequences of an excitable and inhibited Izhikevich neuron network (INN) utilizing a regulated factor set exhibit distinctive dynamic actions depending on fractional exponents regulating over extended exchanges. Ultimately, dynamic controllers for stabilizing and synchronizing the suggested framework are shown. This special spiking activity and other properties of the fractional-order model are caused by the memory trace that emerges from the fractional-order dynamics and integrates all the past activities of the neuron. Our results suggest that the complex dynamics of spiking and bursting can be the result of the long-term dependence and interaction of intracellular and extracellular ionic currents.

Reduced-dimensional skip-inception feature-aggregated classified proportional-integral-derivative for suppression of mixed-mode oscillations in hydropower units

Reduced-dimensional skip-inception feature-aggregated classified proportional-integral-derivative for suppression of mixed-mode oscillations in hydropower units

Experimental investigation of the period-adding bifurcation route to chaos in plasma.

Since the characteristic timescales of the various transport processes inside the discharge plasma span several orders of magnitude, it can be regarded as a typical fast-slow system. Interestingly, in this work, a special kind of complex oscillatory dynamics composed of a series of large-amplitude relaxation oscillations and small-amplitude near-harmonic oscillations, namely, mixed-mode oscillations (MMOs), was observed. By using the ballast resistance as the control parameter, a period-adding bifurcation sequence of the MMOs, i.e., from L^{s} to L^{s+1}, was obtained in a low-pressure DC glow discharge system. Meanwhile, a series of intermittently chaotic regions caused by inverse saddle-node bifurcation was embedded between the two adjacent periodic windows. The formation mechanism of MMOs was analyzed, and the results indicated that the competition between electron production and electron loss plays an important role. Meanwhile, the nonlinear time series analysis technique was used to study the dynamic behavior quantitatively. The attractor in the reconstructed phase space indicated the existence of the homoclinic orbits of type Γ^{-}. In addition, by calculating the largest Lyapunov exponent (LLE), the chaotic nature of these states was confirmed and quantitatively characterized. With the decrease in the ballast resistance, the return map of the chaotic state gradually changed from the nearly one-dimensional single-peak structure to the multibranch structure, which indicates that the dissipation of the system decreased. By further calculating the correlation dimension, it was shown that the complexity of the strange attractors increased for higher-order chaotic states.

Inferring Coupling Strengths of Mixed-mode Oscillations in Red Giant Stars Using Deep Learning

Asteroseismology is a powerful tool that may be applied to shed light on stellar interiors and stellar evolution. Mixed modes, behaving like acoustic waves in the envelope and buoyancy modes in the core, are remarkable because they allow for probing the radiative cores and evanescent zones of red giant stars. Here, we have developed a neural network that can accurately infer the coupling strength, a parameter related to the size of the evanescent zone, of solar-like stars in ∼5 ms. In comparison with existing methods, we found that only ∼43% of inferences were in agreement with a difference less than 0.03 in a sample of ∼1700 Kepler red giants. To understand the origin of these differences, we analyzed a few of these stars using independent techniques such as the Monte Carlo Markov Chain method and echelle diagrams. Through our analysis, we discovered that these alternate techniques are supportive of the neural-net inferences. We also demonstrate that the network can be used to yield estimates of coupling strength and period spacing in stars with structural discontinuities. Our findings suggest that the rate of decline in the coupling strength in the red giant branch is greater than previously believed. These results are in closer agreement with calculations of stellar-evolution models than prior estimates, further underscoring the remarkable success of stellar evolution theory and computation. Additionally, we show that the uncertainty in measuring period spacing increases rapidly with diminishing coupling strength.