Boundary Crisis Research Articles

Finite-Time Analysis of Crises in a Chaotically Forced Ocean Model

We consider a coupling of the Stommel box model and the Lorenz model, with the goal of investigating the so-called crises that are known to occur given sufficient forcing. In this context, a crisis is characterized as the destruction of a chaotic attractor under a critical forcing strength. We document the variety of chaotic attractors and crises possible in our model, focusing on the parameter region where the Lorenz model is always chaotic and where bistability exists in the Stommel box model. The chaotic saddle collisions that occur in a boundary crisis are visualized, with the chaotic saddle computed using the Saddle-Straddle Algorithm. We identify a novel sub-type of boundary crisis, namely a vanishing basin crisis. For forcing strength beyond the crisis, we demonstrate the possibility of a merging between the persisting chaotic attractor and either a chaotic transient or a ghost attractor depending on the type of boundary crisis. An investigation of the finite-time Lyapunov exponents around crisis levels of forcing reveals a convergence between two near-neutral exponents, particularly at points of a trajectory most sensitive to divergence. This points to loss of hyperbolicity associated with crisis occurrence. Finally, we generalize our findings by coupling the Stommel box model to other strange attractors and thereby show that the behaviors are quite generic and robust.

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.

The fluctuational transition mechanism of non-hyperbolic chaotic invariant sets

In order to reveal the general escape mechanism of non-hyperbolic chaotic invariant sets under both weak noise limit and finite noise intensity, the simplest example of Hénon map, which represents the stretching and folding of the phase space, is taken to study the escape mechanism under two kinds of global bifurcation: the fractal boundary crisis and the attractor contact crisis. In this paper, we revealed the general exit mechanism by analyzing the escape paths derived from the shooting method and Monte Carlo simulation. Finally, to further demonstrate universality, an example of high-dimensional differential dynamical system, namely the transient chaos Duffing oscillator, is examined, which underscores the main idea of this paper analyzing specific deterministic structures not only enhances the comprehensibility of the escape process, but also allows for predictions of general escape behavior under weak noise intensity.

Identification of critical points in transient electroconvection dynamics using tensor bundles

We present and demonstrate a method to produce quantitative and qualitative descriptions of transient dynamics from empirical data, with the purpose of analyzing a novel transient discovered in liquid crystal electroconvection. By constructing a tensor bundle around an exemplar transient and creating a chart at every step aligned with the direction of propagation, we show that the Jacobian estimation problem can be reduced by a single dimension, relaxing data requirements and clarifying results. We apply this analysis to identify the onset of a boundary crisis in a predator–prey model. The resulting tensor bundle estimated from image data taken during a dynamical phase transition in a nematic liquid crystal details the behavior of the system along that trajectory, allowing topological analysis. Using this method, we quantify a saddle point in the phase space that drives the initial dynamics during a sudden increase in the driving voltage.

The bifurcation structure within robust chaos for two-dimensional piecewise-linear maps

We study two-dimensional, two-piece, piecewise-linear maps having two saddle fixed points. Such maps reduce to a four-parameter family and are well known to have a chaotic attractor throughout open regions of parameter space. The purpose of this paper is to determine where and how this attractor undergoes bifurcations. We explore the bifurcation structure numerically by using Eckstein’s greatest common divisor algorithm to estimate from sample orbits the number of connected components in the attractor. Where the map is orientation-preserving the numerical results agree with formal results obtained previously through renormalisation. Where the map is orientation-reversing or non-invertible the same renormalisation scheme appears to generate the bifurcation boundaries, but here we need to account for the possibility of some stable low-period solutions. Also the attractor can be destroyed in novel heteroclinic bifurcations (boundary crises) that do not correspond to simple algebraic constraints on the parameters. Overall the results reveal a broadly similar component-doubling bifurcation structure in the orientation-reversing and non-invertible settings, but with some additional complexities.

Chaotic bursting patterns induced by transient chaos in a smooth three-dimensional dynamic model

This paper aims to report the transient-chaos-induced chaotic bursting oscillations in a smooth three-dimensional dynamic model with a low-frequency excitation when an order gap exists between the excitation frequency and the natural frequency. Based on the Nosé-Hoover system, a periodically driven nonlinear system is modelled, in which the transient chaos induced by boundary crisis route can be observed. Through theoretical analysis and numerical simulation, the stability conditions of the equilibrium point and six transient-chaos-induced chaotic bifurcation structures are obtained. Six typical cases in which different routes of the trajectories pass through the transient chaos areas, leading to different patterns of bursting oscillations. It can be found that not only the bifurcation structures but also the bifurcation delay behavior can influence the bursting patterns. In addition, in the transient chaos areas, the bifurcation delay behavior can result in the disappearance of the periodic signals in the small periodic windows.

Sharp changes in fractal basin of attraction in passive dynamic walking

A passive dynamic walker is a mechanical system that walks down a slope without any control, and gives useful insights into the dynamic mechanism of stable walking. This system shows specific attractor characteristics depending on the slope angle due to nonlinear dynamics, such as period-doubling to chaos and its disappearance by a boundary crisis. However, it remains unclear what happens to the basin of attraction. In our previous studies, we showed that a fractal basin of attraction is generated using a simple model over a critical slope angle by iteratively applying the inverse image of the Poincaré map, which has stretching and bending effects. In the present study, we show that the size and fractality of the basin of attraction sharply change many times by changing the slope angle. Furthermore, we improved our previous analysis to clarify the mechanisms for these changes and the disappearance of the basin of attraction based on the stretching and bending deformation in the basin formation process. These findings will improve our understanding of the governing dynamics to generate the basin of attraction in walking.

Complex dynamics of the passive biped robot with flat feet: Gait bifurcation, intermittency and crisis

During the past decades, the walking dynamics of different types of passive biped robots have garnered significant attention. This paper investigates the complex dynamics of a passive flat-footed robot, encompassing both the local bifurcation of limit cycles and global bifurcation, resulting in significant changes in the system’s walking pattern. To identify the robot’s dynamic behavior, the Lyapunov Exponents (LEs) are calculated using the small orthogonal perturbation vectors method. The obtained LEs diagram also unveils several subtle periodic windows scattered within the chaotic region, where interior crisis and intermittency occur. Moreover, two new kinds of gaits that coexist with the traditional period-1 ones are found. Of particular significance is the discovery of the double boundary crisis that caused by the critical period-3 unstable periodic orbit, and this crisis serves as the fundamental reason to the falling down on slopes of the flat-footed robots. By gaining a better understanding of the nature of passive dynamic walking, it becomes possible to design more efficient controllers that impose less stringent torque requirements on biped walking robots.

Stepwise deforestation during the Permian-Triassic boundary crisis linked to rising temperatures

Although the trajectory of the marine mass extinction at the ∼252-Ma Permian-Triassic (P-Tr) boundary has been well studied, details of the coeval collapse of terrestrial ecosystems remain murky. Here, we use hydrocarbon biomarker compositions and other geological records (i.e., organic carbon isotopes (δ13Corg), charcoal abundance, and Hg content) from a tropical peatland succession in southwestern China to reconstruct in detail the history of terrestrial ecosystem collapse during the P-Tr crisis. Our high-resolution hydrocarbon biomarker records reveal that this collapse proceeded in a stepwise manner with increasing intensity as the crisis unfolded. We recognize three discrete crisis stages: Stage I within the uppermost Xuanwei Formation and Stages II and III within the lowermost Kayitou Formation. Stage I, the early crisis stage, is marked by a significant decline in terrestrial biomass (continuing into the later stages), as recorded by reduced C29 steranes relative to total steranes and a concomitant reduction in the ratio of pristane to phytane (Pr/Ph). Stage II, the main crisis stage, records intensified soil erosion and sediment flux as revealed by rising dibenzofuran (DBF) content and high hopane/sterane ratios, the disappearance of coal seams, a sharp negative shift in δ13Corg, and peak concentrations of charcoal reflecting increased wildfire incidence. Stage III, the late crisis stage characterized by enhanced soil erosion, corresponds to peak values of Hg and Hg/TOC but no charcoal peak, suggesting intensified volcanism and a return to a humid climate. These stages closely follow temperature records, which show a stepwise rise during the crisis interval, implying that the deforestation process was strongly influenced by punctuated rises in temperature and/or its attendant effects (e.g., climate aridification). This P-Tr transition scenario suggests that global warming can trigger deforestation and reduce terrestrial carbon storage, thus serving as a positive climate feedback, with important implications for present-day climate change.

Zirconium and neodymium isotopes record intensive felsic volcanism in South China region during the Permian-Triassic boundary crisis

The Permian-Triassic mass extinction (PTME), the largest biocrisis in Earth's history, is regarded as having been triggered by volcanism, although the source and nature of this volcanic activity in the South China region remains in debate. In this regard, determining the geochemical composition of the volcanic and terrestrial source fluxes to marine depositional systems is crucial for differentiating material derived from the Siberian Traps Large Igneous Province versus regional volcanic arcs. Here, we report zirconium (Zr) isotopic compositions (δ94/90Zr) of Permian-Triassic boundary strata from a deep carbonate-ramp setting at Xiakou, South China, providing new insights regarding the provenance of volcanic ash layers and their host silicic mudstones. The δ94/90Zr composition of magmatically sourced materials is generally controlled by degree of melt crystallization and has the potential to discriminate between primitive and evolved igneous sources. At Xiakou, the pre-PTME Lower Unit contains volcanic ash layers with δ94/90Zr (+0.05 ± 0.03‰, 2SD) similar to that of upper continental crustal (UCC), recording dominantly detrital zircon sources. About 10 kyr before the mass extinction, a single ash layer (Bed 249) records a transient excursion to mantle-like values (∼0‰), followed by a protracted shift across the Permian-Triassic boundary (PTB) toward signatures characteristic of more evolved felsic sources (+0.19 ± 0.12‰) accompanied by a positive shift in Nd isotopes (εNd(t)). The high δ94/90Zr of volcanic ash layers from the Middle Unit (corresponding approximately to the interval between the PTME and PTB) indicates dominance of regional felsic volcanic sources. In this interval, volcanic ash δ94/90Zr values at Xiakou are distinctly higher than those at Meishan, suggesting that primary volcanic signatures are better preserved in deep-water facies (Xiakou) than in shallow-water facies (Meishan). In the Middle (excluding the Bed 249) and Upper (post-PTME) units, positive covariation of δ94/90Zr with Al2O3/TiO2, Zr/TiO2 and Zr/Sc ratios was the product of mixing of detrital and volcanic materials, with the latter having a dominantly felsic source. This study provides new insight into the sources of zircons in PTB sections of South China, and it demonstrates the general utility of sedimentary Zr isotopic compositions in determining magmatic provenance.

Constraining oceanic carbonate chemistry evolution during the Cretaceous-Paleogene transition: Combined benthic and planktonic calcium isotope records from the equatorial Pacific Ocean

The Mesozoic-Cenozoic transition is a period of biogeochemical cycle perturbations. The strongest of them is the Cretaceous-Paleogene boundary (K-Pg) crisis, characterized by one of the most important extinctions of planktonic marine calcifiers in Earth's history. One of the primary drivers of this biocalcification crisis is thought to be the increase in atmospheric CO2 concentration and ocean acidification triggered by the Chicxulub Impact, and/or Deccan volcanism. Because it reflects changes of the calcium cycle and/or depends on parameters of the carbonate system, the Ca isotope composition of carbonate minerals precipitated from seawater (δ44/40Ca) offers the potential to reconstruct some of the environmental changes that occurred. Here we present new high-resolution planktonic and benthic foraminiferal δ44/40Ca, δ18O, δ13C, and Sr/Ca records across the K-Pg transition from Shatsky rise (Leg 198; ODP Site 1209, Hole C). The δ44/40Ca record displays a succession of rapid shifts of ca. −0.4‰ across the K-Pg transition. They are similar though not identical between the planktonic and benthic records. These shifts took place on a timescale significantly shorter than the residence time of Ca in the oceans and are therefore unlikely to result from global disequilibrium in the oceanic Ca budget. Instead, changes in the fractionation factor between carbonate minerals and seawater in response to changes in precipitation rates may explain the observed δ44/40Ca and Sr/Ca record. The benthic and planktonic δ44/40Ca records show a late Maastrichtian and an early Danian negative excursions best explained by a succession of episodes of ocean alkalinity increase related to increased continental weathering caused by CO2 emissions from Deccan volcanism and the aftermath of the K-Pg biocalcification crisis. Carbonate compensation via carbonate sediment dissolution, biological carbonate compensation via reduction of biocalcification, and/or an increase in continental weathering must have occurred to offset the excess CO2, ultimately resulting in rapid changes in ocean carbonate chemistry, in combination with reduced surface alkalinity export in response to the early Paleogene planktonic biomineralization crisis.

Multistability of gaits, the basin of attraction and its external topology in the simplest passive walking model on stairs

For the passive walking robot, the basin of attraction is very small and thin, and thus makes it a great challenge to produce stable passive dynamic walking. This paper reports systematically the effect of initial conditions on gait dynamics of the simplest passive walking model on stairs. We find two new gaits with period-3 and period-4, and they both lead to higher periodic gaits and chaos by period-doubling bifurcations. The critical period-3 unstable orbit generated via the cyclic-fold bifurcation takes part in the annihilation of chaotic attractors twice, and this double boundary crisis becomes the main cause to the falling down on stairs of the passive walker. In particular, we compute the Lyapunov exponents to verify the effectiveness of the bifurcation process of these new gaits. More importantly, through analyzing the topology of phase space, the external domain of basin of attraction is classified in detail and marked precisely with different falling patterns. We believe such findings to add significantly to the knowledge of the selection of initial values and to be potentially helpful in the design and optimization of passive walking robots.

Nonlinear Dynamic Bifurcation and Chaos Characteristics of Piezoelectric Composite Lattice Sandwich Plates

This paper investigates the nonlinear bifurcation and chaos characteristics of piezoelectric composite lattice sandwich plates at 1:3 internal resonance. The nonlinear vibration partial differential equation is first discretized to become an ordinary differential equation by applying the Galerkin method. The main resonance modulation equations and the parametric resonance for the external excitation frequency that is close to the system’s second-order modal frequency are then obtained by using the multiscale method. The Newton–Raphson method is subsequently applied to obtain the bifurcation diagram of the steady-state equilibrium of modulation equation with varying system parameters. The equilibrium stability is finally analyzed. We confirm the existence of static bifurcation such as saddle-node bifurcation, pitchfork bifurcation, and Hopf dynamic bifurcation. A detailed analysis is also conducted on the complex nonlinear jump phenomenon caused by the presence of multiple nonlinear steady-state solution regions. In the dynamic Hopf bifurcation interval, the fourth-order Runge–Kutta method is used to continue to track the dynamic periodic solution of the modulation equation in Cartesian coordinates. It is found that there are multiple boundary crises and attractor merging crises in the vibration system for piezoelectric composite lattice sandwich plates.

Bifurcations, relaxation time, and critical exponents in a dissipative or conservative Fermi model.

We investigated the time evolution for the stationary state at different bifurcations of a dissipative version of the Fermi-Ulam accelerator model. For local bifurcations, as period-doubling bifurcations, the convergence to the inactive state is made using a homogeneous and generalized function at the bifurcation parameter. It leads to a set of three critical exponents that are universal for such bifurcation. Near bifurcation, an exponential decay describes convergence whose relaxation time is characterized by a power law. For global bifurcation, as noticed for a boundary crisis, where a chaotic transient suddenly replaces a chaotic attractor after a tiny change of control parameters, the survival probability is described by an exponential decay whose transient time is given by a power law.

Transition characteristics of the dynamic behavior of a vehicle wheel-rail vibro-impact system

<abstract> <p>A two-degree-of-freedom vehicle wheel-rail impact vibration system model is developed, and the equivalent impact stiffness and damping of the rail are fitted applying ABAQUS, taking into account the high and low irregularity generated by the welded joints of the rail. A wheel-rail periodic interface with fixed impact was selected as the Poincaré map, and the fourth-order Runge-Kutta numerical method with variable step size was used to solve the system response. The dynamic characteristics of the system are investigated using a combination of the Bifurcation diagram, Phase plane diagram, the Poincaré map, the Time-domain diagram and the Frequency-domain diagram. It is verified that the vehicle wheel-rail impact vibration system has Hopf bifurcation, Neimark-Sacker bifurcation, Period-doubling bifurcation and Boundary crisis, and there are rich and complex nonlinear dynamic behavior changes. The research on the bifurcation and chaos characteristics of vehicle wheel-rail impact vibration systems can provide a reference for improving the stability of vehicle operation in engineering practice as well as the prediction and control of chaos in vehicle vibration reduction design.</p> </abstract>

A Unitary Association-based conodont biozonation of the Smithian–Spathian boundary (Early Triassic) and associated biotic crisis from South China

The Smithian–Spathian boundary (SSB) crisis played a prominent role in resetting the evolution and diversity of the nekton (ammonoids and conodonts) during the Early Triassic recovery. The late Smithian nektonic crisis culminated at the SSB, ca. 2.7 Myr after the Permian–Triassic boundary mass extinction. An accurate and high-resolution biochronological frame is needed for establishing patterns of extinction and re-diversification of this crisis. Here, we propose a new biochronological frame for conodonts that is based on the Unitary Associations Method (UAM). In this new time frame, the SSB can thus be placed between the climax of the extinction and the onset of the re-diversification. Based on the study of new and rich conodont collections obtained from five sections (of which four are newly described here) in the Nanpanjiang Basin, South China, we have performed a thorough taxonomical revision and described one new genus and 21 new species. Additionally, we have critically reassessed the published conodont data from 16 other sections from South China, and we have used this new, standardized dataset to construct the most accurate, highly resolved, and laterally reproducible biozonation of the Smithian to early Spathian interval for South China. The resulting 11 Unitary Association Zones (UAZ) are intercalibrated with lithological and chemostratigraphical (δ13Ccarb) markers, as well as with ammonoid zones, thus providing a firm basis for an evolutionary meaningful and laterally consistent definition of the SSB. Our UAZ8, which is characterized by the occurrence of Icriospathodus ex gr. crassatus, Triassospathodus symmetricus and Novispathodus brevissimus, is marked by a new evolutionary radiation of both conodonts and ammonoids and is within a positive peak in the carbon isotope record. Consequently, we propose to place the SSB within the separation interval intercalated between UAZ7 and UAZ8 thus leaving some flexibility for future refinement and updating.

Wuchiapingian (Lopingian, late Permian) brachiopod fauna from Guangdong Province, southeastern China: systematics and contribution to the Lopingian recovery

AbstractA diverse Wuchiapingian brachiopod fauna, which contains 57 species in 28 genera, is described from the Shuizhutang Formation at the Liannan section, Guangdong province, southeastern China. Four new species Tyloplecta liannanensis n. sp., Linoproductus huananensis n. sp., Araxathyris minor n. sp., and Permophricodothyris flata n. sp. are proposed. From well-preserved Liannan specimens, characteristics of the shell microstructures in Permianella are revised, and different morphologies of muscle scars in Permophricodothyris are distinctly shown. Until now, only several Wuchiapingian brachiopod faunas have been found in South China. Compared with these faunas, the Liannan fauna shows much higher α diversity and is more like faunas from southeastern China than those from the Yangtze area in faunal composition. The Liannan fauna is dominated with Neochonetes, Transennatia, Orthothetina, Permophricodothyris, and Cathaysia, which are normally larger and more strongly ornamented than their Changhsingian counterparts. The Wuchiapingian brachiopods in South China are represented mainly by the Douling fauna and Shuizhutang fauna. The Douling fauna has relatively low diversity and presents the survival stage after the Guadalupian–Lopingian boundary crisis. The Shuizhutang fauna has a much higher diversity and more key Changhsingian taxa and shows a rapid radiation stage. Faunal compositions of the two faunas indicate that the initial recovery of brachiopods occurred mainly at the genus level followed by a more rapid radiation at both genus and species levels.UUID: http://zoobank.org/474cd988-6ec3-4d16-b045-f84d5fe3fb66

Dynamics, multistability, and crisis analysis of a sine-circle nontwist map.

We propose a one-dimensional dynamical system, the sine-circle nontwist map, that can be considered a local approximation of the standard nontwist map and an extension of the paradigmatic sine-circle map. The map depends on three parameters, exhibiting a simple mathematical form but with a rich dynamical behavior. We identify periodic, quasiperiodic, and chaotic solutions for different parameter sets with the Lyapunov exponent and Slater's theorem. From the bifurcation analysis, we determine two bifurcation lines, those that depend on just two of the control parameters, for which the bifurcation that occurs is of the saddle-node type. In order to investigate multistability, we analyze the bifurcation diagrams in the two directions of parameter variation and we observe some regions of hysteresis, representing the coexistence of different attractors. We also analyze different multistable scenarios, as single attractor, coexistence of periodic attractors, coexistence of chaotic and periodic attractors, chaotic behavior, and coexistence of different chaotic bands, by the Lyapunov exponent and the analysis of the domain occupied by the solutions. From the parameter spaces constructed, we observe the prevalence of single attractor and only chaotic behavior scenarios. The multistable scenario is, mostly, formed by different periodic attractors. Lastly, we analyze the crisis in chaotic attractors and we identify the interior and the boundary crisis. From our results, the boundary crisis plays a key role for the extinction of multistability.

Transient behaviors and equilibria-analysis-based boundary crisis analysis in a smooth 4D dynamical system

This paper addresses the transient behavior mechanism analysis of a new chaotic system. Interesting transient behaviors of the new system are analyzed under the attractor merging crisis. The transient chaos and transient quasi-periodic behaviors are observed, under the boundary crisis, when the system has two stable equilibria. Furthermore, the reason of the two transient behaviors is revealed, with the conditions and relevant threshold for the boundary crisis are found, by introducing a simplified method of Lorenz time series. The lifetime of transient quasi-periodicity and parameters are found to meet the power-law distribution and have long memory, which indicates that the lifetime of transient quasi-periodicity is predictable. Therefore, the transient quasi-periodicity has potential values in the field of weak signal amplitude prediction.

The Birth of a Hidden Attractor Through Boundary Crisis

Hidden attractors have been reported in a significant number of dynamical systems, according to their definition related to the existence of saddle points. Since it has now become common knowledge that there are different types of saddle invariant sets besides a fixed saddle point, it is of great interest to investigate the subject. In this paper, a sudden appearance of a hidden chaotic attractor is observed with the variation of a control parameter in some two-dimensional mapping systems. In order to reveal the mechanism behind the sudden appearance of hidden attractors, the generalized cell mapping method with GPU parallel computing is employed in order to obtain very accurate invariant sets. It is found that hidden attractors are generated from an existing chaotic saddle through a boundary crisis, where a hidden chaotic attractor is touching a periodic or chaotic saddle on its basin boundary. Three examples of two-dimensional mapping systems are given to demonstrate the birth of hidden attractors through a boundary crisis.