Boundary Crisis Research Articles

Paleoenvironmental Recovery After the Cretaceous/ Paleogene Boundary Crisis: Evidence From the Marine Bidart Section (SW France)

Abstract The study of Upper Cretaceous and Lower Paleogene benthic foraminifera from the Bidart section (SW France) provides detailed data on the paleobathymetry as well as paleoenvironmental conditions across the Cretaceous/Paleogene (K/Pg) boundary. A quantitative analysis of benthic foraminiferal assemblages from the Upper Maastrichtian Abathomphalus mayaroensis Biozone and the Danian Guembelitria cretacea, Parvularugoglobigerina eugubina, and Parasubbotina pseudobulloides Biozones was performed. Benthic foraminifera indicate that the upper Maastrichtian and lower Danian sediments at Bidart were deposited in the upper–middle part of the slope. Benthic foraminiferal assemblages indicate mesotrophic conditions during the late Maastrichtian and a strong decrease in the food supply to the sea floor coincident with the K/Pg boundary. This change in the trophic regime was related to the collapse of the food web triggered by the mass extinction of calcareous primary producers. Benthic assemblages in the lower...

Mechanism for boundary crises in quasiperiodically forced period-doubling systems

We investigate the mechanism for boundary crises in the quasiperiodically forced logistic map which is a representative model for quasiperiodically forced period-doubling systems. For small quasiperiodic forcing ɛ, a chaotic attractor disappears suddenly via a “standard” boundary crisis when it collides with the smooth unstable torus. However, when passing a threshold value of ɛ, a basin boundary metamorphosis occurs, and then the smooth unstable torus is no longer accessible from the interior of the basin of the attractor. For this case, using the rational approximations to the quasiperiodic forcing, it is shown that a nonchaotic attractor (smooth torus or strange nonchaotic attractor) as well as a chaotic attractor is destroyed abruptly through a new type of boundary crisis when it collides with an invariant “ring-shaped” unstable set which has no counterpart in the unforced case.

Dynamics of semiconductor lasers with bidirectional optoelectronic coupling: Stability, route to chaos, and entrainment

The dynamical behavior of two mutually coupled semiconductor lasers is studied. An optoelectronic coupling including a time delay in the propagation of the signals between the two lasers is considered. Starting from the appropriate rate equations for the photon and carrier densities, we investigate the stability of the fixed points and limit cycles of the system as a function of the coupling strength and the propagation time. From this analysis, a quasiperiodic route to chaos with boundary crisis events is identified as the responsible mechanism leading the system from regular to complex behavior. Several interesting phenomena are predicted for this system. Our analytical and numerical results are supported by experiments which are in good agreement with our predictions.

Bifurcation and chaos in a discrete genetic toggle switch system

Abstract A discrete genetic toggle switch system obtained by Euler method is first investigated. The conditions of existence for fold bifurcation and flip bifurcation are derived by using center manifold theorem and bifurcation theory. The numerical simulations, including bifurcation diagrams, phase portraits, and computation of Lyapunov exponents, not only show the consistence with the theoretical analysis but also exhibit the rich and complex dynamical behavior. We show the period 3 to 13 windows in different chaotic regions, period-doubling bifurcation or inverse period-doubling bifurcation from period-2 to 12 orbits leading to chaos, different kind of interior crisis and boundary crisis, intermittency behavior, chaotic set, chaotic non-attracting set, coexistence of period points with invariant cycles, and so on. The influence of the amplitude and frequency of excitable forcing on the system are also first considered by using numerical simulation. A different type of quasiperiodic orbits, jumping behaviors of quasiperiodic set from one set to another set, and the processes from quasiperiodic orbits to strange non-chaotic attractor are found.

Phase synchronization and crisis in coupled periodically driven chaotic oscillators

A nontrivial transition from the synchronous to the nonsynchronous state of two coupled nonautonomous oscillators is studied through the theory of two coupled circle maps (torus maps). For the case of nonidentical systems, the transition occurs without the connection to the change in the sign of Lyapunov exponent. We find that the scaling exponent for the intermittent slips near the critical point, which is measured by rotation sets (fluctuations in rotation vectors), depends on the system parameters unlike the case of phase synchronization for two coupled autonomous chaotic oscillators. We demonstrate that through the model of torus maps this nontrivial transition is induced by a boundary crisis on a torus geometry and intermittent slips could be described as the chaotic transient as a result of the crisis.

Chaos in driven Alfvén systems: boundary and interior crises

Chaotic transitions of nonlinear Alfvén waves via boundary and interior crises are studied. Alfvén crises are characterized using the unstable periodic orbits and their stable and unstable manifolds. In a period-3 periodic window of the bifurcation diagram, we identify a period-9 unstable periodic orbit that is responsible for both boundary and interior crises of two chaotic attractors. We demonstrate that these Alfvén crises arise from a homoclinic tangency.

Phase synchronization and crisis in coupled periodically driven chaotic oscillators

A nontrivial transition from the synchronous to the nonsynchronous state of two coupled nonautonomous oscillators is studied through the theory of two coupled circle maps (torus maps). For the case of nonidentical systems, the transition occurs without the connection to the change in the sign of Lyapunov exponent. We find that the scaling exponent for the intermittent slips near the critical point, which is measured by rotation sets (fluctuations in rotation vectors), depends on the system parameters unlike the case of phase synchronization for two coupled autonomous chaotic oscillators. We demonstrate that through the model of torus maps this nontrivial transition is induced by a boundary crisis on a torus geometry and intermittent slips could be described as the chaotic transient as a result of the crisis.

Chaos in driven Alfv�n systems: boundary and interior crises

Chaos in driven Alfv�n systems: boundary and interior crises

A new type of global bifurcation in H non map

A simple branch of solution on a bifurcation diagram, which begins at static bifurcation and ends at boundary crisis (or interior crisis in a periodic window), is generally a period-doubling cascade. A domain of solution in parameter space, enclosed by curves of static bifurcation and that of boundary crisis (or the interior of a periodic window), is the trace of branches of solution. A P-n branch of solution refers to the one starting from a period-n (n?1) solution and the corresponding domain in parameter space is named the P-n domain of solution. Because of the co-existence of attractors, there may be several branches within one interval on a bifurcation diagram and different domains of solution may overlap each other in some areas of the parameter space. A complex phenomenon, concerned both with the co-existence of attractors and the crises of chaotic attractors, was observed in the course of constructing domains of steady state solutions of the H?non map in parameter space by numerical methods. A narrow domain of period-m solutions firstly co-exists with (lies on) a big period-n (m?3n) domain. Then it enters the chaotic area of the big domain and becomes period-m windows. The co-existence of attractors disappears and is called the landing phenomenon. There is an interaction between the two domains in the course of landing: the chaotic area in the big domain is enlarged and there is a crisis step near the landing area.

MULTISTABILITY, BASIN BOUNDARY STRUCTURE, AND CHAOTIC BEHAVIOR IN A SUSPENSION BRIDGE MODEL

We consider the dynamics of the first vibrational mode of a suspension bridge, resulting from the coupling between its roadbed (elastic beam) and the hangers, supposed to be one-sided springs which respond only to stretching. The external forcing is due to time-periodic vortices produced by impinging wind on the bridge structure. We have studied some relevant dynamical phenomena in such a system, like periodic and quasiperiodic responses, chaotic motion, and boundary crises. In the weak dissipative limit the dynamics is mainly multistable, presenting a variety of coexisting attractors, both periodic and chaotic, with a highly involved basin of attraction structure.

Deterministic stochastic resonance in a Rössler oscillator.

We discuss the characteristics of stochastic resonancelike behavior observed in a deterministic system. If a periodically forced Rössler oscillator strays from the phase locking state, it exhibits the intermittent behavior known as phase slips. When the periodic force is modulated by a weak signal, the phase slips synchronize with the weak signal statistically. We numerically demonstrate, in terms of interslip intervals and signal to noise ratio, that the maximum synchronization can be achieved with the optimum intensity of chaotic fluctuations. It is shown that the stochastic resonancelike behavior can be observed regardless of the choice of parameters. The frequency dependence of the signal indicates that there is an optimum frequency for the maximum resonance. The phase slip rate is derived based on the fact that the phase slips are caused by a boundary crisis caused by an unstable-unstable pair bifurcation. The interslip distributions obtained from the derived slip rate and the approximation theory of the time-dependent Poisson process agree with those obtained by numerical simulations. In addition, the maximum enhancement of a weak signal is shown to be achieved by adjusting the chaotic fluctuations even if a signal becomes mixed with noise.

Chaos and bifurcation in the space-clamped FitzHugh–Nagumo system

The discrete space-clamped FitzHugh–Nagumo system obtained by the Euler method is investigated. It is proven that there are SNB and Naimark–Sacker bifurcations and chaotic phenomena in the sense of Marotto. The numerical simulations presented not only show the consistence of our results with the theoretical analysis but also exhibit the complex dynamical behaviors. This includes the period-9, 10, 16, 25, 27 and 42 orbits, quasi-period orbits, cascade of period-doubling bifurcations, pitchfork bifurcation, attractor merging crisis and boundary crisis, intermittency as well as supertransient and chaotic bands. The computation of Lyapunov exponents confirms the dynamical behaviors. We also consider the influence of periodic external signal on the original system via numerical simulations and find that γ and ω affect the dynamical behaviors significantly.

FRACTAL SPECTROSCOPY BY NOISE-FREE STOCHASTIC MULTIRESONANCE AT HIGHER HARMONICS

Noise-free stochastic resonance is investigated in two chaotic maps with periodically modulated control parameter close to a boundary crisis: the Hénon map and the kicked spin model. Response of the maps to the periodic signal at the fundamental frequency and its higher harmonics is examined. The systems show noise-free stochastic multiresonance, i.e. multiple maxima of the signal-to-noise ratio at the fundamental frequency as a function of the control parameter. The maxima are directly related to the fractal structure of the attractors and basins of attraction colliding at the crisis point. The signal-to-noise ratios at higher harmonics show more maxima, as well as dips where the signal-to-noise ratio is zero. This opens a way to use noise-free stochastic resonance to probe the fractal structure of colliding sets by a method which can be called "fractal spectroscopy". Using stochastic resonance at higher harmonics can reveal smaller details of the fractal structures, but the interpretation of results becomes more difficult. Quantitative theory based on a model of a colliding fractal attractor and a fractal basin of attraction is derived which agrees with numerical results for the signal-to-noise ratio at the fundamental frequency and at the first two harmonics, quantitatively for the Hénon map, and qualitatively for the kicked spin model. It is also argued that the maps under study belong to a more general class of threshold-crossing stochastic resonators with a modulated control parameter, and qualitative discussion of conditions under which stochastic multiresonance appears in such systems is given.

Basin Structure in the Two-Dimensional Dissipative Circle Map

Fractal basin structure in the two-dimensional dissipative circle map is examined in detail. Numerically obtained basin appears to be riddling in the parameter region where two periodic orbits co-exist near a boundary crisis, but it is shown to consist of layers of thin bands.

Chaos in a bienzymatic cyclic model with two autocatalytic loops

A general bienzymatic cyclic system including two autocatalytic loops is studied and used as a basic design principle for modelling extracellular matrix turnover. Using classical enzyme kinetic rates, the model is described by a set of four ordinary differential equations and numerically studied by bifurcation diagrams and Poincaré sections. We observe limit-cycle oscillations and chaotic behaviors arising from period-doubling cascades or intermittency. Chaotic oscillations originate from distinct strange attractors that undergo boundary and internal crisis. For some parameter values, the system presents several bistable areas, where a limit cycle coexists with another one or with a strange attractor. The dynamics are qualitatively modified when the weight of the autocatalytic loops on the system varies, resulting in the change in the number of attractors.

On the bifurcations of a rigid rotor response in squeeze-film dampers

The effectiveness of squeeze-film dampers in controlling vibrations in rotating machinery may be limited by the nonlinear interactions between large rotor imbalance forces with fluid-film forces induced by dampers operating in cavitated conditions. From a practical point of view, the occurrence of nonsynchronous and chaotic motion in rotating machinery is undesirable and should be avoided as they introduce cyclic stresses in the rotor, which in turn may rapidly induce fatigue failure. The bifurcations in the response of a rigid rotor supported by cavitated squeeze-film dampers resulting from such interactions are presented in this paper. The effects of design and operating parameters, namely the bearing parameter ( B), gravity parameter ( W), spring parameter ( S) and unbalance parameter ( U), on the bifurcations of the rotor response are investigated. Spring parameter ( S) values between 0 and 1 are considered. A spring parameter value of S=0 represents the special case of dampers without centering springs. With the exception of the case S=1, jump phenomena appeared to be a common bifurcation that occurred at certain combinations of B, W and U irrespective of the value of S. Period-doubling and secondary Hopf bifurcations which occurred for low values of S (⩽0.3) were not observed for the higher values S⩾0.5. For very low stiffness values ( S<0.1), a period-3 solution that formed a closed bifurcation curve consisting of a pair of saddle-nodes, was found. Period-doubling cascades of the period-1 and period-3 orbits, boundary crisis and type 3 intermittency are the routes to chaos for this system. This study also revealed that combination values of gravity and spring parameters although resulting in similar values of the static eccentricity ratio exhibited remarkably different bifurcation behavior. The possibility of bifurcation phenomena arising from the interactions between large rotor imbalance forces and fluid-film forces in cavitated dampers, occurring in industrial rotating machinery, cannot be de-emphasized.

A chaotic crisis between chaotic saddle and attractor in forced Duffing oscillators

We investigate crises in forced Duffing oscillators by the generalized cell mapping digraph method to efficiently complete the global analysis of non-linear systems, which includes global transient analysis through digraphic algorithms based on a strictly theoretical correspondence between generalized cell mappings and digraphs. A process of generalized cell mapping method is developed to refine persistent and transient self-cycling sets. The refining procedures of persistent and transient self-cycling sets are respectively given on the basis of their definitions in the cell state space. A chaotic boundary crisis and a chaotic interior crisis are discovered. A chaotic boundary crisis owing to a collision between chaotic attractor and saddle occurs in its basin boundary possessing a fractal structure. In such a case the chaotic attractor together with its basin of attraction is suddenly destroyed as the parameter passes through the critical value, and the chaotic saddle also undergoes an abrupt enlargement in its size. Namely, the chaotic attractor is converted into an incremental portion of the chaotic saddle after the collision. For a chaotic interior crisis, there is a sudden increase in the size of a chaotic attractor as the parameter passes through the critical value. For the chaotic interior crisis, it is demonstrated that the chaotic attractor collides with a chaotic saddle in its basin interior when the crisis occurs. This chaotic saddle is an invariant and non-attracting set. The origin and evolution of the chaotic saddle are investigated as well.

Dynamical Systems Approach to Space and Astrophysical Turbulence

A new transition mechanism to Alfven chaos via boundary crisis in space and astrophysical plasmas is reported. This crisis appears in a complex plasma region in the presence of a large number of coexisting attractors. We characterize an example of double boundary crises using the unstable periodic orbit determined from the numerical solution of the driven-dissipative derivative nonlinear Schrodinger equation, and demonstrate how the same unstable periodic orbit causes the appearance/disappearance of two chaotic attractors due to two successive homoclinic tangencies.

Multiple attractors and crisis route to chaos in a model food-chain

An attempt has been made to identify the mechanism, which is responsible for the existence of chaos in narrow parameter range in a realistic ecological model food-chain. Analytical and numerical studies of a three species food-chain model similar to a situation likely to be seen in terrestrial ecosystems has been carried out. The study of the model food chain suggests that the existence of chaos in narrow parameter ranges is caused by the crisis-induced sudden death of chaotic attractors. Varying one of the critical parameters in its range while keeping all the others constant, one can monitor the changes in the dynamical behaviour of the system, thereby fixing the regimes in which the system exhibits chaotic dynamics. The computed bifurcation diagrams and basin boundary calculations indicate that crisis is the underlying factor which generates chaotic dynamics in this model food-chain. We investigate sudden qualitative changes in chaotic dynamical behaviour, which occur at a parameter value a 1=1.7804 at which the chaotic attractor destroyed by boundary crisis with an unstable periodic orbit created by the saddle-node bifurcation. Multiple attractors with riddled basins and fractal boundaries are also observed. If ecological systems of interacting species do indeed exhibit multiple attractors etc., the long term dynamics of such systems may undergo vast qualitative changes following epidemics or environmental catastrophes due to the system being pushed into the basin of a new attractor by the perturbation. Coupled with stochasticity, such complex behaviours may render such systems practically unpredictable.

Entropy and bifurcations in a chaotic laser.

We compute bounds on the topological entropy associated with a chaotic attractor of a semiconductor laser with optical injection. We consider the Poincaré return map to a fixed plane, and are able to compute the stable and unstable manifolds of periodic points globally, even though it is impossible to find a plane on which the Poincaré map is globally smoothly defined. In this way, we obtain the information that forms the input of the entropy calculations, and characterize the boundary crisis in which the chaotic attractor is destroyed. This boundary crisis involves a periodic point with negative eigenvalues, and the entropy associated with the chaotic attractor persists in a chaotic saddle after the bifurcation.