Bifurcation Scenario Research Articles

Anomalous Diffusion in Random Dynamical Systems.

Consider a chaotic dynamical system generating diffusionlike Brownian motion. Consider a second, nonchaotic system in which all particles localize. Let a particle experience a random combination of both systems by sampling between them in time. What type of diffusion is exhibited by this random dynamical system? We show that the resulting dynamics can generate anomalous diffusion, where in contrast to Brownian normal diffusion the mean square displacement of an ensemble of particles increases nonlinearly in time. Randomly mixing simple deterministic walks on the line, we find anomalous dynamics characterized by aging, weak ergodicity breaking, breaking of self-averaging, and infinite invariant densities. This result holds for general types of noise and for perturbing nonlinear dynamics in bifurcation scenarios.

Dynamical system analysis of Einstein–Skyrme model in a Kantowski–Sachs spacetime

The present work deals with anisotropic (but zero heat flux) Skyrme fluid with a constant radial profile in locally rotational Kantowski–Sachs spacetime in the background of Einstein gravity. By suitably change of variables the field equations are transformed to an autonomous system. To examine the stability of the system critical points are determined. For hyperbolic critical points the analysis of the system is done using Hartman–Grobman theorem, while center manifold theory is used to analyze non-hyperbolic critical points. Also stability at infinity is analyzed to visualize the global evolution of the Universe. As a result the 3D-phase space is identified with the Poincaré 3-sphere embedded in R4. It is found that the parameters in the autonomous system play a crucial role for phase transition of the Universe. Finally possible bifurcation scenarios are discussed to identify the point of phase transition.

Revisiting the analysis of a codimension-three Takens–Bogdanov bifurcation in planar reversible systems

A family of planar nilpotent reversible systems with an equilibrium point located at the origin has been studied in the recent paper Algaba et al. (Nonlinear Dyn 87:835–849, 2017). The authors investigate the candidate for an universal unfolding of a codimension-three degenerate case which exhibits a rich bifurcation scenario. However, a codimension-two point is missed in one of the two cases considered. In this paper, we complete the bifurcation set demonstrating the existence of this new organizing center and analyzing the dynamics generated in this case. Moreover, by means of the Melnikov theory, we study analytically four different global connections present in the system under consideration. Numerical continuation of the bifurcation curves illustrates that the first-order analytical approximation is valid in a large region of the parameter space.

Three‐dimensional internal gravity‐capillary waves in finite depth

We consider three‐dimensional inviscid‐irrotational flow in a two‐layer fluid under the effects of gravity and surface tension, where the upper fluid is bounded above by a rigid lid and the lower fluid is bounded below by a flat bottom. We use a spatial dynamics approach and formulate the steady Euler equations as an infinite‐dimensional Hamiltonian system, where an unbounded spatial direction x is considered as a time‐like coordinate. In addition, we consider wave motions that are periodic in another direction z. By analyzing the dispersion relation, we detect several bifurcation scenarios, two of which we study further: a type of 00(is)(iκ0) resonance and a Hamiltonian Hopf bifurcation. The bifurcations are investigated by performing a center‐manifold reduction, which yields a finite‐dimensional Hamiltonian system. For this finite‐dimensional system, we establish the existence of periodic and homoclinic orbits, which correspond to, respectively, doubly periodic travelling waves and oblique travelling waves with a dark or bright solitary wave profile in the x direction. The former are obtained using a variational Lyapunov‐Schmidt reduction and the latter by first applying a normal form transformation and then studying the resulting canonical system of equations.

Leap-frog patterns in systems of two coupled FitzHugh-Nagumo units.

We study a system of two identical FitzHugh-Nagumo units with a mutual linear coupling in the fast variables. While an attractive coupling always leads to synchronous behavior, a repulsive coupling can give rise to dynamical regimes with alternating spiking order, called leap-frogging. We analyze various types of periodic and chaotic leap-frogging regimes, using numerical path-following methods to investigate their emergence and stability, as well as to obtain the complex bifurcation scenario which organizes their appearance in parameter space. In particular, we show that the stability region of the simplest periodic leap-frog pattern has the shape of a locking cone pointing to the canard transition of the uncoupled system. We also discuss the role of the timescale separation in the coupled FitzHugh-Nagumo system and the relation of the leap-frog solutions to the theory of mixed-mode oscillations in multiple timescale systems.

Stability and bifurcation analysis of interacting f(T) cosmology

The present work deals with dynamical system analysis of Interacting f(T) cosmology. Einstein field equations are second order non-linear differential equations. So it is very difficult to solve them analytically. We can draw the vector field and analyze the stability of the universe in different phase by dynamical system analysis. By suitable transformation of variables the Einstein field equations are converted to an autonomous system. The critical points are determined and the stability of the equilibrium points are examined by center manifold theory. Possible bifurcation scenarios have also been explained.

Spiral waves in externally excited neuronal network: Solvable model with a monotonically differentiable magnetic flux.

Spiral waves are particular spatiotemporal patterns connected to specific phase singularities representing topological wave dislocations or nodes of zero amplitude, witnessed in a wide range of complex systems such as neuronal networks. The appearance of these waves is linked to the network structure as well as the diffusion dynamics of its blocks. We report a novel form of the Hindmarsh-Rose neuron model utilized as a square neuronal network, showing the remarkable multistructure of dynamical patterns ranging from characteristic spiral wave domains of spatiotemporal phase coherence to regions of hyperchaos. The proposed model comprises a hyperbolic memductance function as the monotone differentiable magnetic flux. Hindmarsh-Rose neurons with an external electromagnetic excitation are considered in three different cases: no excitation, periodic excitation, and quasiperiodic excitation. We performed an extensive study of the neuronal dynamics including calculation of equilibrium points, bifurcation analysis, and Lyapunov spectrum. We have found the property of antimonotonicity in bifurcation scenarios with no excitation or periodic excitation and identified wide regions of hyperchaos in the case of quasiperiodic excitation. Furthermore, the formation and elimination of the spiral waves in each case of external excitation with respect to stimuli parameters are investigated. We have identified novel forms of Hindmarsh-Rose bursting dynamics. Our findings reveal multipartite spiral wave formations and symmetry breaking spatiotemporal dynamics of the neuronal model that may find broad practical applications.

Discrete and data packet delays as determinants of switching stability in wireless sensor networks

• To understand transmission dynamics of malicious signals in WSN, an energy efficient e-epidemic delay model is proposed. • Malicious signals can be controlled, system stabilizes with small delay and exhibits oscillatory behavior for large delay. • Data packet and discrete delays are responsible for stability switching and occurrence of chaotic dynamics respectively. • Occurrence of double Hopf bifurcation indicates about regular functionality and robust security of sensor network. • We examine how the stability regions and bifurcation scenario changes with the transmission rate. An attempt has been made to understand the transmission dynamics of malicious signals in wireless sensor networks . An energy efficient e-epidemic model with data packet transmission delay has been considered. Linear stability analysis is performed for all the equilibrium points, whose characteristic equations involve the time delay. Global stability and Hopf bifurcation analyses are carried out for the endemic equilibrium point of the delay system. Attention has been paid to the direction of Hopf bifurcation and the stability of the resulting periodic solutions. Numerical study exhibits double Hopf bifurcation dynamics and it causes stability switching i.e., instability to stability and back to instability or the reverse transition of the solution of the considered system. Finally, numerical simulations provide useful observations for different delays and they show an interesting bifurcation scenario. The impact of the control parameters β and τ on the system dynamics have been investigated. Our results suggest that the data packet delay and discrete delay are responsible for the stability switching and the occurrence of chaotic dynamics respectively. The presence of chaotic dynamics indicates fragile security system of the network. Looking into the simulation results, we have indicated the most effective control measures to control the propagation of malicious signals.

Optically trapped polariton condensates as semiclassical time crystals

We analyse nonequilibrium phase transitions in microcavity polariton condensates trapped in optically induced annular potentials. We develop an analytic model for annular optical traps, which gives an intuitive interpretation for recent experimental observations on the polariton spatial mode switching with variation of the trap size. In the vicinity of polariton lasing threshold we then develop a nonlinear mean-field model accounting for interactions and gain saturation, and identify several bifurcation scenarios leading to formation of high angular momentum quantum vortices. For experimentally relevant parameters we predict the emergence of spatially and temporally ordered polariton condensates (time crystals), which can be witnessed by frequency combs in the polariton lasing spectrum or by direct time-resolved optical emission measurements. In contrast to previous realizations, our polaritonic time crystal is spontaneously formed from an incoherent excitonic bath and does not inherit its frequency from any periodic driving field.

Perturbations of Hindmarsh-Rose neuron dynamics by fractional operators: Bifurcation, firing and chaotic bursts

Studying & understanding the bursting dynamics of membrane potential in neurobiology is captivating in applied sciences, with many features still to be uncovered. In this study, 2D and 3D neuronal activities given by models of Hindmarsh-Rose (HR) neurons with external current input are analyzed numerically with Haar wavelet method, proven to be convergent through error analysis. Our numerical analysis considers two control parameters: the external current $ I^{\text{ext}} $ and the derivative order $ \gamma $, on top of the other seven usual parameters $ a, b, c, d, \nu_1, \nu_2 $ and $ x_{\text{rest}}. $ Bifurcation scenarios for the model show existence of equilibria, both stable and unstable of type saddle and spiral. They also reveal existence of stable limit cycle toward which the trajectories get closer. Numerical approximations of solutions to the 2D model reveals that equilibria remains the same in all cases, irrespective of control parameters'values, but the observed repeated sequences of impulses increase as $ \gamma $ decreases. This inverse proportionability reveals a system likely to be $ \gamma- $controlled with $ \gamma $ varying from 1 down to 0. A similar observation is done for the 3D HR neuron model where regular burst is observed and turns into period-adding chaotic bifurcation (burst with uncountable peaks) as $ \gamma $ changes from 1 down to 0.

Multiphonic modeling using impulse pattern formulation

Multiphonics, the presence of multiple pitches within the sound of wind-instruments can be produced in several ways. Either complex fingerings are used or the blowing pressure is very low or very high. Such multiphonics can be modeled by the Impulse Pattern Formulation (IPF) proposed previously [R. Bader, Nonlinearities and Synchronization in Musical Acoustics and Music Psychology (2013)]. This top-down method assumes musical instruments to work with impulses which are produced at a generator, travel through the instrument, are reflected at various positions, are exponentially damped, and finally trigger or at least interact with succeeding impulses produced by the generator. While modeling sounds produced at blowing-threshold, the IPF fully captures transitions between regular periodicity at nominal pitch, bifurcation scenarios, and noise, just like regular instruments do, when multiphonics appear in the transition regime. Using IPF, complex fingerings translate to multiple reflection points at open finger holes with different reflection strengths. Here, complex multiphonics can be modeled. The IPF can also synthesize multiphonic sounds when applying the typical impulse form of wind instruments.Multiphonics, the presence of multiple pitches within the sound of wind-instruments can be produced in several ways. Either complex fingerings are used or the blowing pressure is very low or very high. Such multiphonics can be modeled by the Impulse Pattern Formulation (IPF) proposed previously [R. Bader, Nonlinearities and Synchronization in Musical Acoustics and Music Psychology (2013)]. This top-down method assumes musical instruments to work with impulses which are produced at a generator, travel through the instrument, are reflected at various positions, are exponentially damped, and finally trigger or at least interact with succeeding impulses produced by the generator. While modeling sounds produced at blowing-threshold, the IPF fully captures transitions between regular periodicity at nominal pitch, bifurcation scenarios, and noise, just like regular instruments do, when multiphonics appear in the transition regime. Using IPF, complex fingerings translate to multiple reflection points at open fing...

Bifurcation Formulas and Algorithms of Constructing Central Manifolds of Discrete Dynamical Systems

Ones of the main questions in theory of local bifurcations and its applications are questions about direction of bifurcations (sub- or supercriticality) and on stability of the solutions arising in neighborhood of a nonhyperbolic equilibrium point or cycle dynamic system. We consider problems of local bifurcations in dynamical systems with discrete time. New features are proposed to orientation of bifurcations and properties stability of bifurcation solutions for problems on basic scenarios of bifurcations. We also propose new algorithms for constructing central manifolds of the corresponding problems, allowing to obtain new bifurcation formulas, in particular, formulas to calculate Lyapunov quantities. Proposed algorithms and formulas are based on the common operator method the study of problems on local bifurcations and allow under the new conditions effective qualitative analysis of bifurcations in terms of the initial equations.

A reduced multimodal thermoelastic model of a circular Mindlin plate

A nonlinear thermoelastic model of a circular plate is presented in the paper. The model, based on the Mindlin plate theory, is extended by taking into account nonlinear geometrical terms. Partial differential equations of plate’s dynamics are derived for a fully coupled thermal and mechanical fields. Then the model is reduced to a set of ordinary differential equations taking into account the first three natural modes and assuming a constant thermal field. The influence of elevated temperature on the resonance curves and the mode involvement due to nonlinear and thermal couplings is presented. The analysis shows that the increased temperature may lead to various bifurcation scenarios. The buckling phenomenon and post-buckling nonlinear regular and chaotic oscillations are studied.

Vortex pairing in jets as a global Floquet instability: modal and transient dynamics

The spontaneous pairing of rolled-up vortices in a laminar jet is investigated as a global secondary instability of a time-periodic spatially developing vortex street. The growth of subharmonic perturbations, associated with vortex pairing, is analysed both in terms of modal Floquet instability and in terms of transient growth dynamics. The article has the double objective to outline a toolset for the global analysis of time-periodic flows, and to leverage such an analysis for a fresh view on the vortex pairing phenomenon. Axisymmetric direct numerical simulations (DNS) of jets with single-frequency inflow forcing are performed, in order to identify combinations of the Reynolds and Strouhal numbers for which vortex pairing is naturally observed. The same DNS calculations are then repeated with an added time-delay control term, which artificially suppresses pairing, so as to obtain time-periodic unpaired base flows for linear stability analysis. It is demonstrated that the natural occurrence of vortex pairing in nonlinear DNS coincides with a linear subharmonic Floquet instability of the underlying unpaired vortex street. However, DNS results suggest that the onset of pairing involves much stronger temporal growth of subharmonic perturbations than that predicted by modal Floquet analysis, as well as a spatial distribution of these fast-growing perturbation structures that is inconsistent with the unstable Floquet mode. Singular value decomposition of the phase-shift operator (the operator that maps a given perturbation field to its state one flow period later) is performed for an analysis of optimal transient growth in the vortex street. Non-modal mechanisms near the jet inlet are thus found to provide a fast route towards the limit-cycle regime of established vortex pairing, in good agreement with DNS observations. It is concluded that modal Floquet analysis accurately predicts the parameter regime where sustained vortex pairing occurs, but that the bifurcation scenario under typical conditions is dominated by transient growth phenomena.

Bifurcations from steady to quasi-periodic flows in a laterally heated cavity filled with low Prandtl number fluids

This study deals with the transition toward quasi-periodicity of buoyant convection generated by a horizontal temperature gradient in a three-dimensional parallelepipedic cavity with dimensions $4\times 2\times 1$ (length $\times$ width $\times$ height). Numerical continuation techniques, coupled with an Arnoldi method, are used to locate the steady and Hopf bifurcation points as well as the different steady and periodic flow branches emerging from them for Prandtl numbers ranging from 0 to 0.025 (liquid metals). Our results highlight the existence of two steady states along with many periodic cycles, all with different symmetries. The bifurcation scenarios consist of complex paths between these different solutions, giving a succession of stable flow states as the Grashof number is increased, from steady to periodic and quasi-periodic. The change of these scenarios with the Prandtl number, in connection with the crossing of bifurcation points, was carefully analysed.

Opportunities for Probabilistic Forecasts of Tropical Cyclone Formations and Tracks on Intraseasonal Timescales

ABSTRACT An opportunity that now exists for probabilistic forecasts of western North Pacific tropical cyclone formations and tracks on timescales of 7-30 days is demonstrated from the 51-member European Center for Medium-range Weather Forecasts 32-day ensemble predictions made once a week during the 2008 and 2009 seasons. A methodology has been developed to match the ensemble member vortices to form ensemble storms, and a weighted-mean vector motion technique is applied to generate the track. An objective validation relative to the Joint Typhoon Warning Center best-tracks is supplemented with a subjective evaluation of the quality of the agreement. Success of the approach was first demonstrated with three intense typhoons during the 2008 season. Although some successful predictions were also achieved with some moderate typhoons and tropical storms, the early and late season typhoons and the tropical depressions were not predictable on 7-30 day timescales. Evaluation of the performance during the more active and typical 2009 season again indicated success in predicting typhoons and most of the weak tropical storms and tropical depressions. The exceptions were some multiple storm events and track bifurcation scenarios due to midlatitude interactions. Among the challenges to produce an operational product are the requirements to make the entire procedure more automated, more objective, and with forecaster-friendly displays. A documentation of the reliability and of the limitations of the predictions will be helpful in gaining forecaster acceptance.

Two scenarios of advective washing-out of localized convective patterns under frozen parametric disorder

The effect of spatial localization of states in distributed parameter systems under frozen parametric disorder is well known as the Anderson localization and thoroughly studied for the Schrödinger equation and linear dissipation-free wave equations. Some similar (or mimicking) phenomena can occur in dissipative systems such as the thermal convection ones. Specifically, many of these dissipative systems are governed by a modified Kuramoto–Sivashinsky equation, where the frozen spatial disorder of parameters has been reported to lead to excitation of localized patterns. Imposed advection in the modified Kuramoto–Sivashinsky equation can affect the localized patterns in a non-trivial way; it changes the localization properties and suppresses the pattern. The latter effect is considered in this paper by means of both numerical simulation and model reduction, which turns out to be useful for a comprehensive understanding of the bifurcation scenarios in the system. Two possible bifurcation scenarios of advective suppression (‘washing-out’) of localized patterns are revealed and characterized.

Selecting dynamics of the quantum dot light emitting diode with a small optical feedback strength

In this work a four-variable dimensionless model of a quantum dot light emitting diode (QD-LED) under optical feedback effect is studied. Depending on the optical feedback (OFB) strength a complex bifurcation scenarios for the photon intensity were completely determined by the variation of the OFB in the QD as well as time series, inter-spike interval (ISI) and phase plane of all dynamic variables of the QD-LED. Our results show that small OFB lead to a good controllability of QD-LED dynamics as evidenced by the bifurcation diagram. The delay time leads to delay control of dynamics. Moreover, bias has a clear and greater effect than that of OFB. By adding a grating mirror as a special technique to suppress unwanted frequencies, there is an important transition between periodic and chaotic states. Delayed feedback and turn-on dynamics are studied.

Traveling chimera states for coupled pendula

We investigate the phenomenon of traveling chimera states in the ring of self-excited coupled pendula suspended on the horizontally oscillating wheel. The bifurcation scenario of chimera creation and destruction is discussed, and the influence of the suspension’s parameters on possible behavior is studied. We describe the properties of the investigated states, analyzing the traveling time as well as the dynamics of the pendula, depending on their position within different pattern’s regions. The energy transfer method has been used to present how the units cooperate with each other, making the chimera arising possible. Unlike other studies on traveling chimera states, we examine the typical mechanical system including simple topology of coupling, which suggests that the described behavior can be observed naturally in the models of coupled dynamical oscillators.

Nonlinear Analysis of Hysteretic Modulation-Based Sliding Mode Controlled Quadratic Buck–Boost Converter

In this paper, the dynamics of hysteresis current-controlled quadratic buck-boost converter is investigated in detail. The system model is derived based on the sliding mode approach and also in its dimensionless form for algebraic brevity. The stability of the system is disclosed with the aid of the movement of eigenvalues. Onset of Hopf bifurcation is identified when the complex conjugate eigenvalue pair crosses the imaginary axis of the complex plane. The stability boundary is drawn to benefit the power electronics engineer for a stable and reliable design. The computer simulation of the switched model is performed using MATLAB/Simulink software to uncover the sequential occurrence of nonlinear behavior exhibited due to Hopf bifurcation for variation in control parameters and the input voltage. The phase portrait disclosing the subtle periodicity is plotted at different operating points to elicit that the stable period-1 attractor bifurcates to the quasi-periodic orbit and finally to a limit cycle. The precise dynamic of the phase portrait is also captured using the Poincare section. Experimental outputs are presented for confirming the low-frequency bifurcation scenario witnessed in the simulated and analytical results.