Cascade Of Bifurcations Research Articles

Scenarios of hyperchaos occurrence in 4D Rössler system.

The generalized four-dimensional Rössler system is studied. Main bifurcation scenarios leading to a hyperchaos are described phenomenologically and their implementation in the model is demonstrated. In particular, we show that the formation of hyperchaotic invariant sets is related mainly to cascades (finite or infinite) of nondegenerate bifurcations of two types: period-doubling bifurcations of saddle cycles with a one-dimensional unstable invariant manifold and Neimark-Sacker bifurcations of stable cycles. The onset of the discrete hyperchaotic Shilnikov attractors containing a saddle-focus cycle with a two-dimensional unstable invariant manifold is confirmed numerically in a Poincaré map of the model. A new phenomenon, "jump of hyperchaoticity," when the attractor under consideration becomes hyperchaotic due to the boundary crisis of some other attractor, is discovered.

Discrete-Time Model of Seasonal Plankton Bloom

The most interesting results in modeling phytoplankton bloom were obtained based on a modification of the classical system of phytoplankton and zooplankton interaction. The modifications using delayed equations, as well as piecewise continuous functions with a delayed response to intoxication processes, made it possible to obtain adequate phytoplankton dynamics like in nature. This work develops a dynamic model of phytoplankton-zooplankton community consisting of two equations with discrete time. We use recurrent equations, which allows to describe delay in response naturally. The proposed model takes into account the phytoplankton toxicity and zooplankton response associated with phytoplankton toxicity. We use a discrete analogue of the Verhulst model to describe the dynamics of each of the species in the community under autoregulation processes. We use Holling-II type response function taking into account predator saturation to describe decrease in phytoplankton density due to its consumption by zooplankton. Growth and survival rates of zooplankton also depend on its feeding. Zooplankton mortality, caused by an increase in the toxic substances concentration with high density of zooplankton, is included in the limiting processes. An analytical and numerical study of the model proposed is made. The analysis shows that the stability loss of nontrivial fixed point corresponding to the coexistence of phytoplankton and zooplankton can occur through a cascade of period doubling bifurcations and according to the Neimark-Saker scenario leading to the appearance of quasiperiodic fluctuations as well. The proposed dynamic model of the phytoplankton and zooplankton community allows observing long-period oscillations, which is consistent with the results of field experiments. As well, the model have multistability areas, where a variation in initial conditions with the unchanged values of all model parameters can result in a shift of the current dynamic mode.

Preperiodicity and systematic extraction of periodic orbits of the quadratic map

Iteration of the quadratic map produces sequences of polynomials whose degrees explode as the orbital period grows more and more. The polynomial mixing all 335 period-12 orbits has degree [Formula: see text], while for the [Formula: see text] period-20 orbits the degree rises already to [Formula: see text]. Here, we show how to use preperiodic points to systematically extract exact equations of motion, one by one, without any need for iteration. Exact orbital equations provide valuable insight about the arithmetic structure and nesting properties of towers of algebraic numbers which define orbital points and bifurcation cascades of the map.

Simulation and Analysis of Complex Behaviors in a DC-DC Buck Converter

A new study of the Buck converter controlled by a smooth signal -changing the widely used non-smooth ramp signal by a smooth sine signal- operating in continuous conduction mode (CCM) is carried out in this paper, focusing on the analysis of bifurcation diagrams of relevant parameters that govern the system. Unexpectedly and contrary to what one would intuitively contemplate, new phenomena or complex dynamics of the non-smooth nature appear. Mainly border collisions and other non-smooth bifurcations are the main causes of the chaotic attractors and complex behaviors' emergence, even though the signal that controls the system is smooth. More importantly, a cascade of border collision bifurcations was discovered in the Buck converter controlled by a sine waveform, a kind of non-sooth bifurcation only reported in specific type of power inverters

Complexity and chaos control in a discrete-time Lotka–Volterra predator–prey system

ABSTRACT In this paper, we investigate a discrete-time Lotka–Volterra predator–prey model that can undergo flip bifurcations and Hopf bifurcations by using the centre manifold theorem and bifurcation theory. Bifurcation diagrams, maximum Lyapunov exponents, and phase portraits verify the theoretical analysis, which exhibits complex dynamical behaviours such as period-7, 13, 14, 21, 27, 32 orbits, a cascade of period-doubling bifurcation in period-2, 4, 8, quasi-periodic orbits and the chaotic sets. Furthermore, the feedback control method is used to stabilize the chaotic orbits at an unstable fixed point. This result further reveals richer dynamics of the discrete model compared with the continuous model. It is found out that the appropriately carrying capacity can be helpful to stabilize the model. Instead, the higher or lower carrying capacity may destabilize the system producing more complex dynamical behaviours.

Period multiplication cascade at the order-by-disorder transition in uniaxial random field XY magnets

Uniaxial random field disorder induces a spontaneous transverse magnetization in the XY model. Adding a rotating driving field, we find a critical point attached to the number of driving cycles needed to complete a limit cycle, the first discovery of this phenomenon in a magnetic system. Near the critical drive, time crystal behavior emerges, in which the period of the limit cycles becomes an integer n > 1 multiple of the driving period. The period n can be engineered via specific disorder patterns. Because n generically increases with system size, the resulting period multiplication cascade is reminiscent of that occurring in amorphous solids subject to oscillatory shear near the onset of plastic deformation, and of the period bifurcation cascade near the onset of chaos in nonlinear systems, suggesting it is part of a larger class of phenomena in transitions of dynamical systems. Applications include magnets, electron nematics, and quantum gases.

Model reduction for the collective dynamics of globally coupled oscillators: From finite networks to the thermodynamic limit.

Model reduction techniques have been widely used to study the collective behavior of globally coupled oscillators. However, most approaches assume that there are infinitely many oscillators. Here, we propose a new ansatz, based on the collective coordinate approach, that reproduces the collective dynamics of the Kuramoto model for finite networks to high accuracy, yields the same bifurcation structure in the thermodynamic limit of infinitely many oscillators as previous approaches, and additionally captures the dynamics of the order parameter in the thermodynamic limit, including critical slowing down that results from a cascade of saddle-node bifurcations.

Modeling and 1 : 1 Internal Resonance Analysis of Cable-Stayed Shallow Arches

In this paper, an analytical model of a cable-stayed shallow arch is developed in order to investigate the 1 : 1 internal resonance between modes of a cable and a shallow arch. Integrodifferential equations with quadratic and cubic nonlinearities are used to model the in-plane motion of a simple cable-stayed shallow arch. Nonlinear dynamic responses of a cable-stayed shallow arch subjected to external excitations with simultaneous 1 : 1 internal resonances are investigated. Firstly, the Galerkin method is used to discretize the governing nonlinear integral-partial-differential equations. Secondly, the multiple scales method (MSM) is used to derive the modulation equations of the system under external excitation of the shallow arch. Thirdly, the equilibrium, the periodic, and the chaotic solutions of the modulation equations are also analyzed in detail. The frequency- and force-response curves are obtained by using the Newton–Raphson method in conjunction with the pseudoarclength path-following algorithm. The cascades of period-doubling bifurcations leading to chaos are obtained by applying numerical simulations. Finally, the effects of key parameters on the responses are examined, such as initial tension, inclined angle of the cable, and rise and inclined angle of shallow arch. The comprehensive numerical results and research findings will provide essential information for the safety evaluation of cable-supported structures that have widely been used in civil engineering.

Sensitivity and uncertainty analysis of a reduced-order model of nonlinear BWR dynamics: I. Forward sensitivity analysis

This work applies the forward sensitivity analysis method to the reduced-order Boiling Water Reactor (BWR) model originally developed by March-Leuba, Perez and Cacuci [“Universality and Aperiodic Behavior of Nuclear Reactors,” Nucl. Sci. Eng. 86, 401, 1984.] to compute first-order sensitivities of the state functions (power, precursors concentration, reactivity, void fraction and fuel temperature) underlying this model. This reduced-order BWR model comprises point neutron kinetics equations coupled to thermal-hydraulics equations that describe the time evolution of the fuel temperature and coolant density in the recirculation loop. When the heat transfer from the reactor to the coolant is increased under high-power/low-flow operating conditions, this model’s state functions can bifurcate, via a period-doubling cascade of bifurcations, from their steady-state values into a chaotic dynamic evolution. Numerical results for the sensitivities of the state functions with respect to a model parameter that controls the injected neutron population are presented for the following regions in phase-space: (i) the stable region; (ii) the period-one region between the 1st-order bifurcation and the 2nd-order bifurcations; (iii) the period-two region between the 2nd-order bifurcations and the 3rd-order bifurcations; (iv) the period-three region between the 3rd-order bifurcations and the 4th-order bifurcations; (v) the aperiodic/chaotic region, which arises from an infinite cascade of period-doubling bifurcations produced when the heat transfer from reactor to coolant is increased past a critical value. In particular, it is shown that the amplitudes of the oscillations of the sensitivities in the chaotic region reach massive values ±O1023 already within 150 s after the initial perturbation. These results confirm the intuitive expectation that the BWR model is extremely unstable in the chaotic region.Ongoing work aims at performing a comprehensive sensitivity analysis of this reduced-order BWR model using the adjoint sensitivity analysis methodology, by investigating the time evolutions of the sensitivities of all state functions with respect to all model parameters, in the five regions in phase-space that are identified in this work. The sensitivities thus obtained will be used in a subsequent work to perform a comprehensive computation of the standard deviations that would be induced in the BWR model’s state functions by uncertainties that affect the BWR model’s imprecisely known parameter.

The boundary of chaos for interval mappings

A goal in the study of dynamics on the interval is to understand the transition to positive topological entropy. There is a conjecture from the 1980's that the only route to positive topological entropy is through a cascade of period doubling bifurcations. We prove this conjecture in natural families of smooth interval maps, and use it to study the structure of the boundary of mappings with positive entropy. In particular, we show that in families of mappings with a fixed number of critical points the boundary is locally connected, and for analytic mappings that it is a cellular set.

Large oscillation of electrostatically actuated curved beams

This paper investigates the responses of initially curved micro-beams subjected to an electrostatic excitation. A Euler–Bernoulli beam theory is utilized to derive the governing equation of motion. A reduced-order model was developed by discretizing the equation of motion using straight beam mode shapes as basis functions in a Galerkin expansion. The results show evidence of the superharmonic resonances of order-three and two in addition to primary resonance. The co-existence of multiple stable orbits observed around only one stable equilibrium and under excitation waveforms with RMS voltage less than the snap-back voltage. These branches are a branch of small orbits within a narrow potential well and two branches of medium-sized and large orbits within a wider potential well. The transition between them results in the characteristic of the double peaks appearing in the frequency-response curve. We also report a bubble structure along the medium-sized branch consists of a cascade of period-doubling bifurcations and a cascade of reverse period-doubling bifurcations. A chaotic attractor develops within that structure at larger excitation levels. It demonstrates evidence of chaos with a wide-based spectrum and an elevated noise-floor. Odd-periodic windows appear also within the attractor including period-three (P-3), period-five (P-5) and period-six (P-6) windows. The chaotic attractor terminates in a cascade of reverse period-doubling bifurcations of period-four (P-4) orbits and period-two (P-2) orbits.

Understanding the origin of extreme events in El Niño southern oscillation.

We investigate a low-dimensional slow-fast model to understand the dynamical origin of El Niño southern oscillation. A close inspection of the system dynamics using several bifurcation plots reveals that a sudden large expansion of the attractor occurs at a critical system parameter via a type of interior crisis. This interior crisis evolves through merging of a cascade of period-doubling and period-adding bifurcations that leads to the origin of occasional amplitude-modulated extremely large events. More categorically, a situation similar to homoclinic chaos arises near the critical point; however, atypical global instability evolves as a channellike structure in phase space of the system that modulates variability of amplitude and return time of the occasional large events and makes a difference from the homoclinic chaos. The slow-fast timescale of the low-dimensional model plays an important role on the onset of occasional extremely large events. Such extreme events are characterized by their heights when they exceed a threshold level measured by a mean-excess function. The probability density of events' height displays multimodal distribution with an upper-bounded tail. We identify the dependence structure of interevent intervals to understand the predictability of return time of such extreme events using autoregressive integrated moving average model and box-plot analysis.

Influence of the higher-order effects on the solutions of the complex cubic–quintic Ginzburg–Landau equation

ABSTRACT The influence of intrapulse Raman scattering (IRS), self-steepening (SS) and third-order of dispersion (TOD) on the solutions of the complex cubic–quintic Ginzburg–Landau equation is studied by means of bifurcation analysis of the dynamical model. The numerical bifurcation diagram of the amplitude of the solution with respect to the nonlinear gain has revealed a cascade of bifurcations that leads to chaotic behaviour. We have found that the IRS leads to the larger shifts in positions and widths of the zones with different types of solutions in comparison to the influence of SS and TOD. Using numerical bifurcation diagrams of the amplitude of the solution with respect to the parameter describing IRS the following types of transformations have been identified: (a) from chaotic into two-periodic solution; (b) from two-periodic into a limit cycle; and (c) from a limit cycle into a stationary solution.

Bernoulli Mapping with Holeanda Saddle-Node Scenario of the Birth of Hyperbolic Smale–Williams Attractor

One-dimensional Bernoulli mapping with hole is suggested to describe the regularities of the appearance of a chaotic set under the saddle-node scenario of the birth of the Smale–Williams hyperbolic attractor. In such a mapping, a non-trivial chaotic set (with non-zero Hausdorff dimension) arises in the general case as a result of a cascade of period-adding bifurcations characterized by geometric scaling both in the phase space and in the parameter space. Numerical analysis of the behavior of models demonstrating the saddle-node scenario of birth of a hyperbolic chaotic Smale–Williams attractor shows that these regularities are preserved in the case of multidimensional systems. Limits of applicability of the approximate 1D model are discussed.

Numerical Analysis of Laminar–Turbulent Transition by Methods of Chaotic Dynamics

This paper summarizes the results of studies of the laminar–turbulent transition in some fluid and gas dynamics problems obtained by applying numerical methods and methods of chaotic dynamics. The following problems are analyzed: 2D and 3D Kolmogorov problems in a periodic domain, 3D Rayleigh–Benard convection in rectangular domains, 3D backward-facing step flow, and development of 3D Rayleigh–Taylor and Kelvin–Helmholtz instabilities in viscous compressible flows. An analysis confirms that instabilities develop via cascades of subcritical or supercritical bifurcations. In all systems, a universal scenario of the transition to chaos (Feigenbaum–Sharkovskii–Magnitskii scenario) is found along with other scenarios of chaotization of dynamical systems.

Dynamics of Rubber Chaplygin Sphere under Periodic Control

This paper examines the motion of a balanced spherical robot under the action of periodically changing moments of inertia and gyrostatic momentum. The system of equations of motion is constructed using the model of the rolling of a rubber body (without slipping and twisting) and is nonconservative. It is shown that in the absence of gyrostatic momentum the equations of motion admit three invariant submanifolds corresponding to plane-parallel motion of the sphere with rotation about the minor, middle and major axes of inertia. The above-mentioned motions are quasi-periodic, and for the numerical estimate of their stability charts of the largest Lyapunov exponent and charts of stability are plotted versus the frequency and amplitude of the moments of inertia. It is shown that rotations about the minor and major axes of inertia can become unstable at sufficiently small amplitudes of the moments of inertia. In this case, the so-called “Arnol’d tongues” arise in the stability chart. Stabilization of the middle unstable axis of inertia turns out to be possible at sufficiently large amplitudes of the moments of inertia, when the middle axis of inertia becomes the minor axis for a part of a period. It is shown that the nonconservativeness of the system manifests itself in the occurrence of limit cycles, attracting tori and strange attractors in phase space. Numerical calculations show that strange attractors may arise through a cascade of period-doubling bifurcations or after a finite number of torus-doubling bifurcations.

Optical feedback induced oscillation bursts in two-state quantum-dot lasers.

We investigate the impact of short optical feedback on a two-state quantum dot laser. A region in the feedback parameter space is identified, where the laser emission periodically alternates between oscillation bursts from the quantum dot ground and excited state, i.e. two-color anti-phase oscillation bursts. We compare these results to the low-frequency fluctuations and regular pulse packages of single-color semiconductor lasers and show via an in-depth bifurcation analysis, that the two-color oscillation bursts originate from a torus-bifurcation of a two-state periodic orbit. A cascade of further period-doubling bifurcations produces chaotic dynamics of the burst envelope. Our findings showcase the rich dynamics and complexity, which can be generated via the interaction of electronic and photonic time scales in quantum dot lasers with optical feedback.

Critical Point for Bifurcation Cascades and Featureless Turbulence.

In this Letter we show that a bifurcation cascade and fully sustained turbulence can share the phase space of a fluid flow system, resulting in the presence of competing stable attractors. We analyze the toroidal pipe flow, which undergoes subcritical transition to turbulence at low pipe curvatures (pipe-to-torus diameter ratio) and supercritical transition at high curvatures, as was previously documented. We unveil an additional step in the bifurcation cascade and provide evidence that, in a narrow range of intermediate curvatures, its dynamics competes with that of sustained turbulence emerging through subcritical transition mechanisms.

Chaotic attractors in Atkinson–Allen model of four competing species

We study the occurrence of chaos in the Atkinson–Allen model of four competing species, which plays the role as a discrete-time Lotka–Volterra-type model. We show that in this model chaos can be generated by a cascade of quasiperiod-doubling bifurcations starting from a supercritical Neimark–Sacker bifurcation of the unique positive fixed point. The chaotic attractor is contained in a globally attracting invariant manifold of codimension one, known as the carrying simplex. Biologically, our study implies that the invasion attempts by an invader into a trimorphic population under Atkinson–Allen dynamics can lead to chaos.

Chaos Induced by Snap-Back Repeller in a Two Species Competitive Model

In this paper, we investigate the complex dynamics of two-species Ricker-type discrete-time competitive model. We perform a local stability analysis for the fixed points and we will discuss about its persistence for boundary fixed points. This system inherits the dynamics of one-dimensional Ricker model such as cascade of period-doubling bifurcation, periodic windows and chaos. We explore the existence of chaos for the equilibrium points for a specific case of this system using Marotto theorem and proving the existence of snap-back repeller. We use several dynamical systems tools to demonstrate the qualitative behaviors of the system.