Quasi-periodic Research Articles

The Phase Transition of Covariant Lyapunov Vector Precisely Locates a Stability Reversal of Quasi-Periodic Response

Abstract Assessing the stability of quasi-periodic (QP) response is crucial, as the bifurcation of QP response is usually accompanied by a stability reversal. The largest Lyapunov exponent (LLE), as an important indicator for chaotic motion, can also be used for the stability analysis of QP response. The precise location of a stability reversal, however, is tough to achieve as a poor convergence rate would be usually encountered when solving the LLE. Herein a straightforward and precise approach is suggested to identify the critical point when a stability reversal happens. Our approach is based on an explicit differential equation that provides the LLE straightforwardly via numerical integration, and the corresponding covariant Lyapunov vector is simultaneously obtained. The major finding consists in the phase transition of the covariant Lyapunov vector, which can happen much early before the LLE reaches a relatively convergent value. More importantly, the phase transition can serve as a strong indicator to locate a stability reversal of the QP response qualitatively. Numerical examples are provided to verify of the effectiveness and wide applicability the presented approach.

Observations and Model of Subauroral Sporadic E Layer Irregularities Driven by Turning Shears and Dynamic Instability

AbstractObservations of coherent scatter from patchy sporadic E layers in the subauroral zone made with a 30‐MHz coherent scatter radar imager are presented. The quasiperiodic (QP) echoes are similar to what has been observed at middle latitudes but with some differences. The echoes arise from bands of scatterers aligned mainly northwest to southeast and propagating to the southwest. A notable difference from observations at middle latitudes is the appearance of secondary irregularities or braids oriented obliquely to the primary bands and propagating mainly northward along them. We present a spectral simulation of the patchy layers that describes neutral atmospheric dynamics with the incompressible Navier Stokes equations and plasma dynamics with an extended MHD model. The simulation is initialized with turning shears in the form of an Ekman spiral. Ekman‐type instability deforms the sporadic E layer through compressible and incompressible motion. The layer ultimately exhibits both the QP bands and the braids, consequences mainly of primary and secondary neutral dynamic instability. Vorticity due to dynamic instability is an important source of structuring in the sporadic E layer.

Controlling chaos and codimension-two bifurcation in a discrete fractional-order Brusselator model

This paper explores the qualitative behavior of a discrete fractional–order Brusselator model. We analyze the local dynamics of the model around its fixed point and determine its topological classification. We perform the bifurcation analysis for both codimension-one and codimension-two cases to examine the system behavior near critical parameter values. Using normal form theory and center manifold theorem (CMT), we prove that the model exhibits period-doubling bifurcation around its interior fixed point. We also study the existence and direction of Neimark–Sacker bifurcation using normal form theory. For codimension-two bifurcation, we show that the model undergoes 1:2, 1:3, and 1:4 resonances by applying normal form theory and suitable affine transformations. The system displays a rich variety of bifurcations, including quasi–periodicity, periodic orbits, chaotic behavior, and resonance bifurcation. Furthermore, the existence of chaos is discussed in the sense of Marotto, and a novel chaos control method is proposed for discrete Brusselator model using an extended pole–placement approach. This modified approach is more suitable for codimension-two bifurcation situations. Numerical simulations are used to illustrate the theoretical discussion.

Quantitative Green’s function estimates for lattice quasi-periodic Schrödinger operators

In this paper, we establish quantitative Green's function estimates for some higher dimensional lattice quasi-periodic (QP) Schr\"odinger operators. The resonances in the estimates can be described via a pair of symmetric zeros of certain functions and the estimates apply to the sub-exponential type non-resonant conditions. As the application of quantitative Green's function estimates, we prove both the arithmetic version of Anderson localization and the $(\frac 12-)$-H\"older continuity of the integrated density of states (IDS) for such QP Schr\"odinger operators. This gives an affirmative answer to Bourgain's problem in\cite{Bou00}.

Evidence for Electron Precipitation Diffused by Rising‐Tone Quasiperiodic Whistler Waves in Extremely Low L−Shells Observed by CSES

AbstractBased on the observations by the sun‐synchronous circular orbit China Seismo‐Electromagnetic Satellite(CSES), a typical case of the rising‐tone quasiperiodic (QP) emissions with a large period of around 2 min was reported to appear on the dayside on 23 October 2021. The frequency of QP waves ranges from 2,000 ∼ 3,000 Hz and the wave spectral structure consists of four elements with right‐handed polarization. Three elements of them are located at extremely low L‐shells from 2 ∼ 3.5. The periodic precipitating fluxes of 100 ∼ 400 keV electrons were found in the conjugate region also observed by CSES, which appeared in the same magnetic field lines with the elements of QP waves, respectively. By numerical simulations of quasi‐linear diffusion theory, we confirm that QP emissions can effectively precipitate energetic electrons near the loss cone into the atmosphere in the inner radiation belt. To our best knowledge, this is the first evidence that the rising‐tone QP whistler waves scatter electrons in the inner radiation belt. This new finding will help to deepen our understanding of the electron precipitation dynamics in radiation belt physics.

Performance of potential evapotranspiration models across different climatic stations in New South Wales, Australia

Study regionNew South Wales, southeast Australia Study focusEstimating potential evapotranspiration (ETp) rates, detecting its temporal trends and analysing its interannual oscillation are critical for long-term assessment of water availability and regional drought. This study aimed to evaluate the comprehensive performance of 12 simplified models in characterising ETp against the benchmark Penman model across different climate sites in southeast Australia. This study used Taylor skill score (S), normalised root mean square error (nRMSE) and relative mean bias error (rMBE) to estimate models’ capability in estimating ETp rates. Then, this study adopted Mann-Kendall test and continuous wavelet transform (CWT) to test temporal trends and periodicity of ETp estimated by all models. New hydrology insights for the regionJensen-Haise model was capable to produce fair (nRMSE ≤ 30%) estimates of daily ETp across all stations. Models except Mak1 were generally able to produce reasonable estimates of ETp at larger time scale. In addition, we found that the 12 alternative ETp models generally agreed with the Penman model on the primary (9.6–12.4-year) and quasi (2.6–3.9-year) periods of ETp, but they did not show matchable ability in detecting ETp temporal trends. The comprehensive investigation on models’ performance will shed light on models’ selection in estimation of drought and hydrological cycle.

Characteristics and longitudinal extent of VLF quasi-periodic emissions using multi-point ground-based observations

Quasi-periodic (QP) emissions are a type of magnetospheric ELF/VLF waves characterized by a periodic intensity modulation ranging from tens of seconds to several minutes. Here, we present 63 QP events observed between January 2017 and December 2018. Initially detected at the VLF receiver in Kannuslehto, Finland (KAN, MLAT = 67.7°N, L = 5.5), we proceeded to check whether these events were simultaneously observed at other subauroral receivers. To do so we used the following PWING stations: Athabasca (ATH, MLAT = 61.2°N, L = 4.3, Canada), Gakona (GAK, MLAT = 63.6°N, L = 4.9, Alaska), Husafell (HUS, MLAT = 64.9°N, L = 5.6, Iceland), Istok (IST, MLAT = 60.6°N, L = 6.0, Russia), Kapuskasing (KAP, MLAT = 58.7°N, L = 3.8, Canada), Maimaga (MAM, MLAT = 58.0°N, L = 3.6, Russia), and Nain (NAI, MLAT = 65.8°N, L = 5.0, Canada). We found that: (1) QP emissions detected at KAN had a relatively longer observation time (1–10 h) than other stations, (2) 11.3% of the emissions at KAN were observed showing one-to-one correspondence at IST, and (3) no station other than IST simultaneously observed the same QP emission as KAN. Since KAN and IST are longitudinally separated by 60.6°, we estimate that the maximum meridional spread of conjugated QP emissions should be close to 60° or 4 MLT. Comparison with geomagnetic data shows half of the events are categorized as type II, while the rest are mixed (type I and II). This study is the first to clarify the longitudinal spread of QP waves observed on the ground by analyzing simultaneous observations over 2 years using multiple ground stations.Graphical

A universal phase condition in solving quasi-periodic responses with multiple self-excited fundamental frequencies

Solving quasi-periodic (QP) responses of nonlinear dynamical systems, particularly when multiple self-excited fundamental frequencies have to be determining, has been a challenging task. The presence of unknown frequencies usually leads to an under-determined problem, where the number of unknowns exceeds that of equations, as obtained through harmonic balancing or difference techniques, for instance. Therefore, it is necessary to introduce supplementary equations to ensure the well-posedness of the solving process. To the best of our knowledge, some existing methods are only applicable to cases with a single unknown frequency or they require prior numerical results attained, for example, by time-marching integration techniques. In this paper, we propose a new phase condition that is unconditionally valid for QP solutions with multiple unknown frequencies. This condition reveals an inherent feature of QP motion in a multi-dimensional hyper-time domain and provides a theoretical basis for solving QP responses in such a domain. To demonstrate its effectiveness, we apply the phase condition to both the finite difference and the harmonic balance methods. Numerical examples illustrate that, by employing the phase condition, both methods can solve stable as well as unstable QP solutions accurately. As the phase condition holds universally, it is suitable to be incorporated into other solution techniques based on either frequency or time-domain analysis.

High-precision semi-analytical solution for the quasi-periodic nanobeam system based on the weight time-domain minimum residual method

Nanobeams have been widely used in various micro-electromechanical systems because of their small size and high sensitivity. At the nanoscale, quasi-periodic (QP) vibration often occurs in nanobeams due to the surface effects such as surface elasticity, residual surface tension and geometric nonlinearity. According to the QP motion properties of the nanobeam system, a weight time-domain minimum residual method (WTMRM) is presented in this paper by introducing the weight vector into the original time-domain minimum residual method (TMRM), and the high-precision semi-analytical QP (SAQP) solution is obtained by the WTMRM. This method mainly includes four aspects. Firstly, according to the properties of QP motion of the nanobeam system, a new SAQP solution constructed by the dominant frequency and frequency interval is presented. Then, the problem of solving the SAQP solution is converted into a nonlinear minimum optimization issue by deducing the presumptive solution and substituting the state variables back to the original control equations, and this optimization issue is resolved by the enhanced response sensitivity approach (ERSA) iteratively. Next, the weight vector is introduced to deal with the contradiction of the residual between the control equation and the initial condition constraints. Finally, the numerical results of the Runge–Kutta (RK) method are compared with those of the proposed method. The comparison results show that the precision of the SAQP solution achieved by the WTMRM is much higher than the existing results.

Influence of pivot support stiffness on dynamic characteristics of vertical rotor system

The pivot-jewel bearing pair is the only dynamic and static contact component of the flexible vertical rotor system, and its support stiffness directly affects the dynamic response of the rotor system. A non-linear dynamic model of the vertical rotor system is established, and the effect of different elastic pivot stiffnesses on the dynamic behavior of the rotor system is investigated in combination with experiments. The results show that with the increase of rotational speed, the system has the dynamic characteristics of alternating period 1 and quasi period, and when the speed reaches 550 r/s, 1/3 times low frequency whirl occurs and when the speed rises to 675 r/s, the low frequency amplitude is higher than the rotational frequency amplitude. With the increase of the elastic pivot stiffness, the amplitude of the low frequency whirl at the lower end of the rotor decreases continuously but the proportion of the “width” gradually increases. The rotational speed corresponding to the large amplitude quasi periodic motion of the rotor is delayed from 550 r/s to 650 r/s. The research results can provide a theoretical basis for the optimal design of the bearing subsystem of the vertical rotor system.

Multiwavelength Temporal Variability of the Blazar PKS 1510–089

We perform correlation and periodicity search analyses on long-term multiband light curves of the flat-spectrum radio quasar PKS 1510−089 observed by the space-based Fermi-Large Area Telescope in γ-rays, the SMARTS and Steward Observatory telescopes in optical and near-infrared (NIR), and the 13.7 m radio telescope in Metsähovi Radio Observatory between 2008 and 2018. The z-transform discrete correlation function method is applied to study the correlation and possible time lags among these multiband light curves. Among all pairs of wavelengths, the γ-ray versus optical/NIR and optical versus NIR correlations show zero time lags; however, both the γ-ray and optical/NIR emissions precede the radio radiation. The generalized Lomb–Scargle periodogram, weighted wavelet z-transform, and REDFIT techniques are employed to investigate the unresolved core emission–dominated 37 GHz light curve and yield evidence for a quasi period around 1540 days, although given the length of the whole data set it cannot be claimed to be significant. We also investigate the optical/NIR color variability and find that this source shows a simple redder-when-brighter behavior over time, even in the low-flux state.

Gaussian processes for radial velocity modeling

Context. Instrumental radial velocity (RV) precision has reached a level where the detection of planetary signals is limited by the ability to understand and simultaneously model stellar astrophysical “noise.” A common method for mitigating the effects of stellar activity is Gaussian process (GP) regression. Aims. In this study we present an analysis of the performance and properties of the quasi-periodic (QP) GP kernel, which is the multiplication of the squared-exponential kernel by the exponential-sine-squared kernel, based on an extensive set of synthetic RVs, into which the signature of activity was injected. Methods. The stellar activity within our synthetic data sets was simulated using astrophysically motivated models with different spot distributions and spot lifetimes rotating on the surface of a modeled late-type star. We used dynamic nested sampling to fit different model sets, including QP–GPs, Keplerian models, white noise models, and combinations of these, to synthetic RV time series data that in some cases included additional injected planetary signals. Results. We find that while the QP–GP rotation parameter matches the simulated rotation period of the star, the length scale cannot be directly connected to the spot lifetimes on the stellar surface. Regarding the setup of the priors for the QP–GP, we find that it can be advantageous to constrain the QP–GP hyperparameters in different ways depending on the application and the goal of the analysis. We find that a constraint on the length scale of the QP–GP can lead to a significant improvement in identifying the correct rotation period of the star, while a constraint on the rotation hyperparameter tends to lead to improved planet detection efficiency and more accurately derived planet parameters. Even though for most of the simulations the Bayesian evidence performed as expected, we identified not far-fetched cases where a blind adoption of this metric would lead to wrong conclusions. Conclusions. We conclude that modeling stellar astrophysical noise by using a QP–GP considerably improves detection efficiencies and leads to precise planet parameters. Nevertheless, there are also cases in which the QP–GP does not perform optimally, for example RV variations dynamically evolving on short timescales or a mixture of a very stable activity component and random variations. Knowledge of these limitations is essential for drawing correct conclusions from observational data.

Quasiperiodic ELF/VLF Emissions Associated With Corresponding Pulsations of the Geomagnetic Field

AbstractWe present a comparison between properties of quasiperiodic (QP) extra low frequency/very low frequency emissions observed by the low‐altitude DEMETER spacecraft and ultra‐low frequency (ULF) geomagnetic field pulsations measured on the ground by the Canadian Array for Realtime Investigations of Magnetic Activity system of flux‐gate magnetometers and by the Sodankylä Geophysical Observatory magnetometer. Altogether, we have analyzed 398 QP events observed at the times when DEMETER was close to the ground‐based magnetometers. The modulation periods of the analyzed QP events were larger than 10 s and their frequency bandwidths were larger than 200 Hz. For a part of QP emissions with modulation periods about 30 s, there was a good agreement between the modulation periods and peak frequencies of ULF magnetic field pulsations measured on the ground. These QP emissions appear to be closely associated with coincident geomagnetic pulsations (QP1 type), and they represent ∼18% of the total number of analyzed QP events. No corresponding geomagnetic pulsations were identified in the remaining 82% of QP events (QP2 type). The intensity of QP1 events does not seem to correlate with the intensity of geomagnetic field pulsations, while the intensity of QP2 events increases with the integral intensity of geomagnetic field pulsations. Based on the observed association between QP emissions and geomagnetic field pulsations, we estimate the radial distance of the generation region of QP1 emissions to L ∼ 7.

Describing function method with pointwise balancing in two-dimensional regularized time domain for quasi-periodic responses

This paper proposes a new semi-analytical approach, called as describing function method with pointwise balancing (DFPB), to solve quasi-periodic (QP) responses of nonlinear dynamical systems especially those with non-smooth nonlinearities. This method is to describe the QP motion by generalized Fourier series with multiple regularized time scales. By discretizing the solution in the regularized time domain, the original equation is transformed into over-determined algebraic equations governing Fourier series coefficients via pointwise balancing. The coefficients are determined by solving the balance equations via the Newton–Raphson iteration algorithm. In addition, the Tikhonov regularization is utilized to enhance the convergence of the iteration scheme, so that the least squares solution of the over-determined equations can be obtained efficiently. On the one hand, the pointwise balancing avoids repeated transformations of non-smooth functions between time- to frequency-domain. On the other hand, the regularized time scales make it convenient to discretize the system equations regardless of unknown fundamental frequencies. With these procedures, the DFPB is well-designed to solve non-smooth systems for QP response with frequencies to be determined. In the numerical examples, the effectiveness and accuracy are elucidated in detail through both smooth and non-smooth models. The DFPB results are in excellent agreement with those attained by the Runge–Kutta method. In addition, it is worthy of pointing out the DFPB can track both stable and unstable QP responses.

Non-linear dynamics and stability of rolling bearing–axle coupling system of a railway freight wagon with the change of axle rotating speed

PurposeThe purpose of this paper is to investigate the non-linear characteristics and stability of the rolling bearing–axle coupling system under the excitation of the axle/wheel speed of railway freight cars, so as to put forward a rationale for judging the vibration law and running stability of railway freight wagon.Design/methodology/approachConsidering the effects of eccentric force of the railway wagon axle, the non-linear resistance of the wagon and non-linear support forces of axle box rolling bearings, a centralized mass model of rolling bearing-axle coupling system of railway freight wagon is presented on the basis of the theory of rotor dynamics and non-linear dynamics. Then the Runge-Kutta method is adopted to solve the non-linear response of the proposed system, and numerical simulation including bifurcation diagrams, axis trajectory curves, phase plane plots, Poincaré sections and amplitude spectras are analysed when the axle rotating speed is changed. Meantime, the relation curve between Floquet multiplier and axle rotating speed, which affects the stability of coupling system, is plotted by numerical method based on the Floquet theory and method.FindingsThe simulation results of the dynamic model reveal the abundant dynamic behaviour of the coupling system when the axle rotating speed changes, including single period, quasi period, multi-period and chaotic motion, as well as the evolution law from multi-period motion to chaotic motion. And especially, the bearing–axle coupling system is in stable state with a single period motion when the axle rotating speed changes from 410 rpm to 510 rpm, in which the running speed of railway freight wagon is changed from 62 km/h to 80 km/h, the vibration displacement of the coupling system in X direction is between 1.2 mm and 1.8 mm, and the vibration displacement of the coupling system in Y direction is between 1.0 mm and 1.45 mm. Meanwhile, the influence law of axle rotating speed on the stability is obtained by comparing the bifurcation diagram and Floquet multiplier graph of the coupling system.Originality/valueThe numerical simulation data obtained in this study can provide a theoretical evidence for designing the running speed of railway freight wagon, utilizing or controlling the non-linear dynamic behaviours of the proposed coupling system, and ensuring the stability of railway freight wagons.

On the fine inner structure of whistler mode quasiperiodic emissions

Whistler mode quasiperiodic (QP) emissions are electromagnetic waves at frequencies between about 0.5 and 4 kHz observed in the Earth’s inner magnetosphere, which are characterized by their nearly periodic time modulation of the wave intensity. Although they have been studied using both ground-based and satellite instruments for already a few decades, their origin remains unclear. Recent studies of these emissions revealed that, on top of the main modulation period, some of the events exhibit an additional fine inner intensity modulation. We use high-resolution electromagnetic wave measurements obtained by the low-altitude DEMETER spacecraft to systematically study the presence/absence of such fine inner structure and the corresponding fine inner modulation periods. Altogether, as many as 251 events are analyzed. Out of these, the clear fine inner modulation is observed for 71 events, only unclear fine inner modulation is observed for 63 events, and the fine inner modulation is completely absent for 117 events. We show that the fine inner structure tends to occur primarily for QP events with shorter modulation periods. The fine inner modulation periods are on the order of few seconds, corresponding to bounce times of whistler mode waves in between the hemispheres. They typically stay quite constant within a single event, and they tend to be lower at larger geomagnetic latitudes (L-shells). Interestingly, the fine inner modulation periods are positively correlated with the main QP modulation periods. The results obtained are of interest for understanding possible mechanisms responsible for the generation of QP events.

Midlatitude Sporadic E‐Layer Horizontal Structuring Modulated by Neutral Instability and Mixing in the Lower Thermosphere

AbstractObservations of 30‐MHz coherent backscatter from sporadic‐E ionization layers were obtained with a VHF imaging radar located in Ithaca, New York. The volume probed by the radar lies at relatively high magnetic latitudes, on the northern edge of the mid‐latitude region and underneath the ionospheric trough. Banded, quasi‐periodic (QP) echoes observed from Ithaca are similar to those found in lower midlatitude regions. The Doppler shifts observed are smaller and, so far, do not appear to reach the threshold for Farley‐Buneman instability. However, many of the echoes exhibit fine‐scale structure, with secondary bands or braids oriented obliquely to the primary bands. Secondary bands have been seen only rarely at lower middle latitudes. In previous observations, the QP scattering has been linked to unstable neutral wind shears. Neutral wind shear commonly found in the lower thermosphere could play a key role in the formation of these irregularities and explain some morphological features of the resulting plasma density irregularities and the radar echoes. We consider whether neutral instability and turbulence in the lower thermosphere is the likely cause for some of the structuring in the sporadic‐ E layers. Results of 3D numerical simulations of atmospheric dynamics in the mesosphere to lower thermosphere support the proposition. In particular, we focus on Ekman‐type instabilities that, like the more common Kelvin‐Helmholtz instabilities, are inflection point instabilities, although specifically associated with turning shears, and result in convective rolls aligned close to the mean wind direction, with smaller‐scale secondary waves aligned normal to the primary structures.

Circularly distributed multipliers with deterministic moduli assessing the stability of quasiperiodic response.

The stability and bifurcation of a periodic solution of a dynamical system can be handled well by using the Floquet multipliers of the perturbed system with periodic coefficients. However, for a quasiperiodic (QP) response as a natural extension of a periodic one, it is much more difficult to do it quantitatively. Therefore, proposed here is an approach for defining and obtaining effective multipliers for QP stability. The proposed approach is based on a series of auxiliary variables via which the perturbed system with QP coefficients is transformed approximately into a constant one, whereupon the multipliers are obtained efficiently by performing eigenvalue analysis on the constant coefficient matrix. The major finding involves circularly distributed multipliers with deterministic moduli, with the QP response being stable if all the moduli are less than or equal to unity; otherwise it is unstable. When the QP response degenerates to periodic due to the reducibility of fundamental frequencies, the proposed approach exactly provides the Floquet multipliers for the periodic solution. From this respect, the obtained multipliers can be considered to some extent as being a generalization for QP response of the Floquet multipliers for a periodic solution.

Geochemical characteristics of hydrogen, oxygen, carbon isotopes and REE of cenozoic dolomites in Well Xike 1, Xisha Islands, South China sea and the significance for dolomitization in island-reef areas

Due to the special tectonic background and complex geological evolution characteristics of the South China Sea, reef dolomite reservoirs (sush as Well Xike 1) are widely developed. Based on the drilling core data of Well Xike 1, the structure and geochemical characteristics of dolomite reservoirs, including carbon, oxygen, hydrogen isotopes and REE were systematically studied using geochemical and petrological methods. It is found that the geochemical characteristics of REE show that the main diagenetic environment of dolomites is a low-temperature alkaline semi open oxidation environment; the carbon and oxygen isotopes of the dolomites are generally lack of correlation, the δD value is significantly lower than the hydrogen isotope value of seawater. Meanwhile, the oxygen isotope value of deep dolomites is negatively biased, which may be due to the increase or decrease of pore water temperature caused by deep thermal convection that related to the regional tectonic movements of the South China Sea. The δ18O value is also consistent with the geological reality of increasing saddle dolomite content in deep dolomites. The distribution of the δ13C value indicates that the dolomite inherited the carbon of the original limestone during dolomitization, while the characteristic of the δD value shows that it may be affected by the mixing of atmospheric precipitation and concentrated seawater in the quasi contemporaneous period. Based on the comprehensive analysis of the geochemical characteristics of the Well Xike 1, it is considered that the higher diagenetic temperature could be an important factor leading to the huge differences between the diagenetic model of deep and shallow dolomites. The geochemical characteristics of the shallow dolomites show that it is mainly reflux infiltration dolomitization under the micro evaporation and concentration sea water environments, while the deep dolomite is transformed by the hot water fluids in the epigenetic diagenetic evolution stage.

Route to extreme events in a parametrically driven position-dependent nonlinear oscillator.

We explore the dynamics of a damped and driven Mathews-Lakshmanan oscillator type model with position-dependent mass term and report two distinct bifurcation routes to the advent of sudden, intermittent large-amplitude chaotic oscillations in the system. We characterize these infrequent and recurrent large oscillations as extreme events (EE) when they are significantly greater than the pre-defined threshold height. In the first bifurcation route, the system exhibits a bifurcation from quasiperiodic (QP) attractor to chaotic attractor via strange non-chaotic (SNA) attractor as a function of damping parameter. In the second route, the chaotic attractor in the form of EE has emerged directly from the QP attractor. Hence, to the best of our knowledge, this is the first study to report the birth of EE from these two distinct bifurcation routes. We also discuss that EE are emerged due to the sudden expansion of the chaotic attractor via interior crisis in the system. Regions of different dynamical states are distinguished using the Lyapunov exponent spectrum. Further, SNA and QP dynamics are determined using the singular spectrum analysis and 0-1 test. The region of EE is characterized using the threshold height.