Nonlinear Dynamic Interactions Research Articles

Modeling of nonlinear interactions between guided waves and fatigue cracks using local interaction simulation approach

This article presents a parallel algorithm to model the nonlinear dynamic interactions between ultrasonic guided waves and fatigue cracks. The Local Interaction Simulation Approach (LISA) is further developed to capture the contact-impact clapping phenomena during the wave crack interactions based on the penalty method. Initial opening and closure distributions are considered to approximate the 3-D rough crack microscopic features. A Coulomb friction model is integrated to capture the stick-slip contact motions between the crack surfaces. The LISA procedure is parallelized via the Compute Unified Device Architecture (CUDA), which enables parallel computing on powerful graphic cards. The explicit contact formulation, the parallel algorithm, as well as the GPU-based implementation facilitate LISA’s high computational efficiency over the conventional finite element method (FEM). This article starts with the theoretical formulation and numerical implementation of the proposed algorithm, followed by the solution behavior study and numerical verification against a commercial finite element code. Numerical case studies are conducted on Lamb wave interactions with fatigue cracks. Several nonlinear ultrasonic phenomena are addressed. The classical nonlinear higher harmonic and DC response are successfully captured. The nonlinear mode conversion at a through-thickness and a half-thickness fatigue crack is investigated. Threshold behaviors, induced by initial openings and closures of rough crack surfaces, are depicted by the proposed contact LISA model.

A Linear Stochastic Field Model of Midlatitude Mesoscale Variability

AbstractA semiempirical model of midlatitude sea surface height (SSH) variability is formulated and tested against two decades of weekly global fields of merged altimeter data. The model is constrained to match approximately the observed SSH wavenumber power spectrum, but it predicts the spatiotemporal SSH field structure as a propagating, damped, linear response to a stochastic forcing field. An objective, coherent-eddy identification and tracking procedure is applied to the model and altimeter SSH fields, with a focus on eddies with lifetimes L ≥ 16 weeks. The model eddy dataset reproduces the basic global-mean characteristics of the altimeter eddy dataset, including the structure of mean amplitude and scale life cycles, the number distributions versus lifetime, and the distributions of all eddy length scale realizations. The model underpredicts the numbers of eddy realizations with large amplitudes and large scales, overpredicts the growth of mean amplitude and scale with lifetime, and modestly overpredicts the curvature of the mean amplitude life cycle and the number of eddies with intermediate lifetimes. The stochastic forcing evidently represents nonlinear dynamical interactions, implying that eddy splitting and merging events are equally likely, and that mesoscale nonlinearity is weaker than longwave linearity but as strong as shortwave dispersion. The time-reversal symmetry of the life cycles is explained by the time reversibility of the underlying stochastic model. The random SSH increment processes are effectively continuous on the derived 25-week damping time scale, with SSH-increment standard deviation σW ≈ 2.5 × 10−3 cm s−1/2.

Cable dynamics under non-ideal support excitations: Nonlinear dynamic interactions and asymptotic modelling

Cable dynamics under ideal longitudinal support motions/excitations assumes that the support׳s mass, stiffness and mechanical energy are infinite. However, for many long/slender support structures, their finite mass and stiffness should be taken into account and the cable-support dynamic interactions should be modelled and evaluated. These moving supports are non-ideal support excitations, deserving a proper coupling analysis. For systems with a large support/cable mass ratio, using the multiple scale method and asymptotic approximations, a cable-support coupled reduced model, with both cable׳s geometric nonlinearity and cable-support coupling nonlinearity included, is established asymptotically and validated numerically in this paper. Based upon the reduced model, cable׳s nonlinear responses under non-ideal support excitations(and also the coupled responses) are found, with stability and bifurcation characteristics determined. By finding the modifications caused by the support/cable mass ratio, boundary damping, and internal detuning, full investigations into coupling-induced dynamic effects on the cable are conducted. Finally, the approximate analytical results based on the reduced model are verified by numerical results from the original full model.

An Efficient Algorithm for Nonlinear Analysis of Vehicle/Track Interaction Problems

In this paper, a new algorithm for solving the vehicle/track dynamic interaction problem is developed, aimed at reducing the computational cost. The algorithm called Advanced Solver Algorithm (ASA) uses the full Newton–Raphson incremental-iterative method in conjunction with the Newmark integration scheme to solve the equilibrium equations of the coupled vehicle/track system in time domain. Considering the track as a beam resting on a viscoelastic foundation and each vehicle as a wagon with ten degrees of freedom, the governing differential equations of motion of the vehicle/track system were derived. The wheel/rail contact was considered as a nonlinear Hertz spring and consequently the vehicle/track nonlinear dynamic interaction problem was solved. A comparison between the results of the ASA and those of the most advanced algorithm available was made to evaluate the efficiency of the ASA. It is confirmed that using the ASA can result in 40–70 % of reduction in computational cost.

Stochastic and deterministic dynamics of intrinsically irregular firing in cortical inhibitory interneurons.

Most cortical neurons fire regularly when excited by a constant stimulus. In contrast, irregular-spiking (IS) interneurons are remarkable for the intrinsic variability of their spike timing, which can synchronize amongst IS cells via specific gap junctions. Here, we have studied the biophysical mechanisms of this irregular spiking in mice, and how IS cells fire in the context of synchronous network oscillations. Using patch-clamp recordings, artificial dynamic conductance injection, pharmacological analysis and computational modeling, we show that spike time irregularity is generated by a nonlinear dynamical interaction of voltage-dependent sodium and fast-inactivating potassium channels just below spike threshold, amplifying channel noise. This active irregularity may help IS cells synchronize with each other at gamma range frequencies, while resisting synchronization to lower input frequencies.

Radio-over-fiber DSB-to-SSB conversion using semiconductor lasers at stable locking dynamics.

In radio-over-fiber systems, optical single-sideband (SSB) modulation signals are preferred to optical double-sideband (DSB) modulation signals for fiber distribution in order to mitigate the microwave power fading effect. However, typically adopted modulation schemes generate DSB signals, making DSB-to-SSB conversion necessary before or after fiber distribution. This study investigates a semiconductor laser at stable locking dynamics for such conversion. The conversion relies solely on the nonlinear dynamical interaction between an input DSB signal and the laser. Only a typical semiconductor laser is therefore required as the key conversion unit, and no pump or probe signal is necessary. The conversion can be achieved for a broad tunable range of microwave frequency up to at least 60 GHz. In addition, the conversion can be carried out even when the microwave frequency, the power of the input DSB signal, or the frequency of the input DSB signal fluctuates over a wide range, leading to high adaptability and stability of the conversion system. After conversion, while the microwave phase quality, such as linewidth and phase noise, is mainly preserved, a bit-error ratio down to 10-9 is achieved for a data rate up to at least 8 Gb/s with a detection sensitivity improvement of more than 1.5 dB.

Limit-Cycle Analysis of Three-Dimensional Flexible Shaft/Rigid Rotor/Autobalancer System With Symmetric Rigid Supports

In recent years, there has been much interest in the use of automatic balancing devices (ABD) in rotating machinery. Autobalancers consist of several freely moving eccentric balancing masses mounted on the rotor, which, at certain operating speeds, act to cancel rotor imbalance. This “automatic balancing” phenomenon occurs as a result of nonlinear dynamic interactions between the balancer and rotor wherein the balancer masses naturally synchronize with the rotor with appropriate phase to cancel the imbalance. However, due to inherent nonlinearity of the autobalancer, the potential for other undesirable nonsynchronous limit-cycle behavior exists. In such situations, the balancer masses do not reach their desired synchronous balanced positions resulting in increased rotor vibration. To explore this nonsynchronous behavior of ABD, the unstable limit-cycle analysis of three-dimensional (3D) flexible shaft/rigid rotor/ABD/rigid supports described by the modal coordinates has been investigated here. Essentially, this paper presents an approximate harmonic analytical solution to describe the limit-cycle behavior of ABD–rotor system interacting with flexible shaft, which has not been fully considered by ABD researchers. The modal shape of flexible shaft is determined by using well-known fixed–fixed boundary condition due to symmetric rigid supports. Here, the whirl speed of the ABD balancer masses is determined via the solution of a nonlinear characteristic equation. Also, based upon the analytical limit-cycle solutions, the limit-cycle stability of three primary design parameters for ABD is assessed via a perturbation and Floquet analysis: the size of ABD balancer mass, the ABD viscous damping, and the relative axial location of ABD to the imbalance rotor along the shaft. The coexistence of the stable balanced synchronous condition and undesirable nonsynchronous limit-cycle is also studied. It is found that for certain combinations of ABD parameters and rotor speeds, the nonsynchronous limit-cycle can be made unstable, thus guaranteeing asymptotic stability of the synchronous balanced condition at the supercritical shaft speeds between each flexible mode. Finally, the analysis is validated through numerical simulation. The findings in this paper yield important insights for researchers wishing to utilize ABD in flexible shaft/rigid rotor systems and limit-cycle mitigation.

Limit-Cycle Analysis of Planar Rotor/Autobalancer System Influenced by Alford's Force

In recent years, there has been much interest in the use of so-called automatic balancing devices (ABDs) in rotating machinery. Essentially, ABDs or “autobalancers” consist of several freely moving eccentric balancing masses mounted on the rotor, which, at certain operating speeds, act to cancel rotor imbalance at steady-state. This “automatic balancing” phenomenon occurs as a result of nonlinear dynamic interactions between the balancer and rotor, wherein the balancer masses naturally synchronize with the rotor with appropriate phase and cancel the imbalance. However, due to inherent nonlinearity of the autobalancer, the potential for other, undesirable, nonsynchronous limit-cycle behavior exists. In such situations, the balancer masses do not reach their desired synchronous balanced steady-state positions resulting in increased rotor vibration. In this paper, an approximate analytical harmonic solution for the limit cycles is obtained for the special case of symmetric support stiffness together with the so-called Alford's force cross-coupling term. The limit-cycle stability is assessed via Floquet analysis with a perturbation. It is found that the stable balanced synchronous conditions coexist with undesirable nonsynchronous limit cycles. For certain combinations of bearing parameters and operating speeds, the nonsynchronous limit-cycle can be made unstable guaranteeing global asymptotic stability of the synchronous balanced condition. Additionally, the analytical bifurcation of the coexistence zone and the pure balanced synchronous condition is derived. Finally, the analysis is validated through numerical time- and frequency-domain simulation. The findings in this paper yield important insights for researchers wishing to utilize ABDs on rotors having journal bearing support.

Untangling Brain-Wide Dynamics in Consciousness by Cross-Embedding.

Brain-wide interactions generating complex neural dynamics are considered crucial for emergent cognitive functions. However, the irreducible nature of nonlinear and high-dimensional dynamical interactions challenges conventional reductionist approaches. We introduce a model-free method, based on embedding theorems in nonlinear state-space reconstruction, that permits a simultaneous characterization of complexity in local dynamics, directed interactions between brain areas, and how the complexity is produced by the interactions. We demonstrate this method in large-scale electrophysiological recordings from awake and anesthetized monkeys. The cross-embedding method captures structured interaction underlying cortex-wide dynamics that may be missed by conventional correlation-based analysis, demonstrating a critical role of time-series analysis in characterizing brain state. The method reveals a consciousness-related hierarchy of cortical areas, where dynamical complexity increases along with cross-area information flow. These findings demonstrate the advantages of the cross-embedding method in deciphering large-scale and heterogeneous neuronal systems, suggesting a crucial contribution by sensory-frontoparietal interactions to the emergence of complex brain dynamics during consciousness.

Biomarkers, the control panel and personalized COPD medicine.

Chronic obstructive pulmonary disease (COPD) is a complex and heterogeneous disease, so its clinical management needs to be personalized as much as possible. 'Complex' means that COPD has several components with non-linear dynamic interactions, whereas 'heterogeneous' indicates that not all these components are present in all patients or, in a given patient, at all time points. This complexity and heterogeneity explains and justifies the need for personalizing the assessment and treatment of patients with COPD. We propose here that the implementation of a 'control panel' will facilitate the deployment of personalized COPD medicine in clinical practice. Such a control panel should provide the practicing clinician complementary and relevant information on treatable clinical characteristics in a single patient at a given time point. For the purpose of this discussion, we consider these variables to be 'biomarkers'. Which treatable clinical characteristics should the COPD control panel include has not yet been formally validated. The review below suggests and discusses which ones might be considered in the future and should be viewed as a working proposal.

Low Doses of Traditional Nanophytomedicines for Clinical Treatment: Manufacturing Processes and Nonlinear Response Patterns.

The purpose of the present paper is to (a) summarize evidence for the nanoparticle nature and biological effects of traditional homeopathically-prepared medicines at low and ultralow doses; (b) provide details of historically-based homeopathic green manufacturing materials and methods, relating them to top-down mechanical attrition and plant-based biosynthetic processes in modern nanotechnology; (c) outline the potential roles of nonlinear dose-responses and dynamical interactions with complex adaptive systems in generating endogenous amplification processes during low dose treatment. Possible mechanisms of low dose effects, for which there is evidence involving nanoparticles and/or homeopathically-manufactured medicines, include hormesis, time-dependent sensitization, and stochastic resonance. All of the proposed mechanisms depend upon endogenous nonlinear amplification processes in the recipient organism in interaction with the salient, albeit weak signal properties of the medicine. Conventional ligand-receptor mechanisms relevant to higher doses are less likely involved. Effects, especially for homeopathically-prepared nanophytomedicines, include bidirectional host state-dependent changes in function. Homeopathic clinicians report successful treatment of serious infections and cancers. Preclinical biological evidence is consistent with such claims. Controlled biological data on homeopathically-prepared medicines indicate modulation of gene expression and biological signaling pathways regulating cell cycles, immune reactions, and central nervous system function from studies on cells, animals, and human subjects. As a 200-year old system of traditional medicine used by millions of people worldwide, homeopathy offers a pulsed low dose treatment strategy and strong safety record to facilitate progress in translational nanomedicine with plants and other natural products. In turn, modern nanotechnology methods can improve homeopathic manufacturing procedures, characterize nanoparticle end-products, and describe interactions of homeopathic nanophytomedicines with living systems at the nanoparticle and even individual organism level of detection. Faster progress toward safe and effective personalized nanophytomedicine treatments can result.

Low Frequency Stimulation (Quenching) Affects Amygdala Kindling-Induced Gene Expression in Rat Brain

s / Brain Stimulation 8 (2015) 412e427 414 Columbia University, New York City, NY, USA Background: Electroconvulsive therapy (ECT) efficacy and cognitive side effects remain influenced by several parameters including electrodes position and configuration, the applied current intensity, duration, and polarity. We report on latest development in FEAST and its mechanisms of actions. Method: Patients were treated with FEAST using a modified MECTA spECTrum 5000Q device (MECTA Corp, Tualatin, Oregon) and following the same anesthesia protocol. Titration and 2 treatment sessions at 6 times seizure threshold (6*ST) involved a simultaneous 64 channel EEG recording (Neuroscan, Compumedics). We derived the non-linear dynamic interaction models from modified neuronal population activity models whose dynamics can reproduce basic features of ECT-induced seizures within local areas and across distant cortical areas. We applied the Square-Root Cubature Kalman filter in three EEG states: baseline under general anesthesia, ictal and post-ictal. This yielded the functional connectivity between right and left frontal and parietal regions. In addition, we computed the global power relative to baseline for ictal and post-ictal phases. Results: To date, we acquired 28 recordings from 10 patients with major depressive disorder (4 females, age 1⁄4 44.5 10 years). These included 8 titration sessions (28.7 10 mC), 10 6*ST direct polarity and 10 6*ST reversed polarity (172.8 59.48 mC). Right frontal region showed a significant difference in relative ictal power changes from baseline between titration and treatment sessions (-0.16 0.07, -0.26 0.09, p1⁄40.037). Post-ictal parameters showed an enhanced suppression in parietal regions with reversed polarity. Conclusions: This innovative research highlights the regional relationships of ictal and post-ictal activity with FEAST. FEAST is clearly initiating seizure activity in the frontal region (right > left). Ongoing analyses are focusing on regional interactions and detailed power spectra. Future work will focus on comparing FEAST with more classic ECT modalities and relationship to clinical outcomes.

Self-organized criticality, plasticity and sensorimotor coupling. Explorations with a neurorobotic model in a behavioural preference task.

During the last two decades, analysis of 1/ƒ noise in cognitive science has led to a considerable progress in the way we understand the organization of our mental life. However, there is still a lack of specific models providing explanations of how 1/ƒ noise is generated in coupled brain-body-environment systems, since existing models and experiments typically target either externally observable behaviour or isolated neuronal systems but do not address the interplay between neuronal mechanisms and sensorimotor dynamics. We present a conceptual model of a minimal neurorobotic agent solving a behavioural task that makes it possible to relate mechanistic (neurodynamic) and behavioural levels of description. The model consists of a simulated robot controlled by a network of Kuramoto oscillators with homeostatic plasticity and the ability to develop behavioural preferences mediated by sensorimotor patterns. With only three oscillators, this simple model displays self-organized criticality in the form of robust 1/ƒ noise and a wide multifractal spectrum. We show that the emergence of self-organized criticality and 1/ƒ noise in our model is the result of three simultaneous conditions: a) non-linear interaction dynamics capable of generating stable collective patterns, b) internal plastic mechanisms modulating the sensorimotor flows, and c) strong sensorimotor coupling with the environment that induces transient metastable neurodynamic regimes. We carry out a number of experiments to show that both synaptic plasticity and strong sensorimotor coupling play a necessary role, as constituents of self-organized criticality, in the generation of 1/ƒ noise. The experiments also shown to be useful to test the robustness of 1/ƒ scaling comparing the results of different techniques. We finally discuss the role of conceptual models as mediators between nomothetic and mechanistic models and how they can inform future experimental research where self-organized critically includes sensorimotor coupling among the essential interaction-dominant process giving rise to 1/ƒ noise.

Nonlinear interactions in deformable container cranes

Nonlinear dynamic interactions in harbour quayside cranes due to a two-to-one internal resonance between the lowest bending mode of the deformable boom and the in-plane pendular mode of the container are investigated. To this end, a three-dimensional model of container cranes accounting for the elastic interaction between the crane boom and the container dynamics is proposed. The container is modelled as a three-dimensional rigid body elastically suspended through hoisting cables from the trolley moving along the crane boom modelled as an Euler-Bernoulli beam. The reduced governing equations of motion are obtained through the Euler-Lagrange equations employing the boom kinetic and stored energies, derived via a Galerkin discretisation based on the mode shapes of the two-span crane boom used as trial functions, and the kinetic and stored energies of the rigid body container and the elastic hoisting cables. First, conditions for the onset of internal resonances between the boom and the container are found. A higher order perturbation treatment of the Taylor expanded equations of motion in the neighbourhood of a two-to-one internal resonance between the lowest boom bending mode and the lowest pendular mode of the container is carried out. Continuation of the fixed points of the modulation equations together with stability analysis yields a rich bifurcation behaviour, which features Hopf bifurcations. It is shown that consideration of higher order terms (cubic nonlinearities) beyond the quadratic geometric and inertia nonlinearities breaks the symmetry of the bifurcation equations, shifts the bifurcation points and the stability ranges, and leads to bifurcations not predicted by the low order analysis.

Effects of thermal noise on the transitional dynamics of an inextensible elastic filament in stagnation flow.

We investigate the dynamics of a single inextensible elastic filament subject to anisotropic friction in a viscous stagnation-point flow, by employing both a continuum model represented by Langevin type stochastic partial differential equations (SPDEs) and a dissipative particle dynamics (DPD) method. Unlike previous works, the filament is free to rotate and the tension along the filament is determined by the local inextensible constraint. The kinematics of the filament is recorded and studied with normal modes analysis. The results show that the filament displays an instability induced by negative tension, which is analogous to Euler buckling of a beam. Symmetry breaking of normal modes dynamics and stretch-coil transitions are observed above the threshold of the buckling instability point. Furthermore, both temporal and spatial noise are amplified resulting from the interaction of thermal fluctuations and nonlinear filament dynamics. Specifically, the spatial noise is amplified with even normal modes being excited due to symmetry breaking, while the temporal noise is amplified with increasing time correlation length and variance.

Modelling, Simulation and Experimental Validation of Nonlinear Dynamic Interactions in an Aramid Rope System

Vibration phenomena taking place in lifting and hoist installations may influence the dynamic performance of their components. For example, in an elevator system they may affect ride quality of a lift car. Lateral and longitudinal vibrations of suspension ropes and compensating cables may result in an adverse dynamic behaviour of the entire installation. Thus, there is a need to develop reliable mathematical and computer simulation models to predict the dynamic behaviour of suspension rope and compensating cable systems. The aim of this paper is to develop a model of an aramid suspension rope system in order to predict nonlinear modal interactions taking place in the installation. A laboratory model comprising an aramid suspension rope, a sheave/ pulley assembly and a rigid suspended mass has been studied. Experimental tests have been conducted to identify modal nonlinear couplings in the system. The dynamic behaviour of the model has been described by a set of nonlinear partial differential equations. The equations have been solved numerically. The numerical results have been validated by experimental tests. It has been shown that the nonlinear couplings may lead to adverse modal interactions in the system.

Adaptive Decentralized PI Controller for Two Conical Tank Interacting Level System

The implementation of control algorithms for the Multi-Input/Multi-Output systems is often complicated due to variations in process dynamics that occur because of change in operating point and the characteristics of nonlinear dynamic coupling. In this paper, the authors proposed regime-based multi-model adaptive control strategy for decoupling-based decentralized PI controller for a Two Conical Tank Interacting Level System (TCTILS) which is a Two-Input Two-Output interacting nonlinear system that exhibits nonlinear dynamics and dynamic interaction. The controller design is divided into two parts: firstly, a decoupling matrix is designed in order to minimize the interaction effects for each pair of linearized operating regimes. Then, the decentralized PI controller is designed for the TCTILS using stability boundary equations method and real-coded GA for each pair of linearized operating regimes. The stability analysis of the TCTILS with the designed set of decentralized PI controllers and decouplers is investigated for each pair of linearized operating regimes by using multivariable Nyquist plot method. Using simulation and experimental studies, the effectiveness and practicality of the proposed scheme is demonstrated on TCTILS experimental setup. The results reveal encouraging input tracking and disturbance rejection capabilities. The regime-based multi-model adaptive control strategy minimizes interaction effects effectively and meet the design specifications for each loop independently in each linearized operating regime.

Integrated signaling pathway and gene expression regulatory model to dissect dynamics of Escherichia coli challenged mammary epithelial cells

Cells transform external stimuli, through the activation of signaling pathways, which in turn activate gene regulatory networks, in gene expression. As more omics data are generated from experiments, eliciting the integrated relationship between the external stimuli, the signaling process in the cell and the subsequent gene expression is a major challenge in systems biology. The complex system of non-linear dynamic protein interactions in signaling pathways and gene networks regulates gene expression.The complexity and non-linear aspects have resulted in the study of the signaling pathway or the gene network regulation in isolation. However, this limits the analysis of the interaction between the two components and the identification of the source of the mechanism differentiating the gene expression profiles. Here, we present a study of a model of the combined signaling pathway and gene network to highlight the importance of integrated modeling.Based on the experimental findings we developed a compartmental model and conducted several simulation experiments. The model simulates the mRNA expression of three different cytokines (RANTES, IL8 and TNFα) regulated by the transcription factor NFκB in mammary epithelial cells challenged with E. coli. The analysis of the gene network regulation identifies a lack of robustness and therefore sensitivity for the transcription factor regulation. However, analysis of the integrated signaling and gene network regulation model reveals distinctly different underlying mechanisms in the signaling pathway responsible for the variation between the three cytokine's mRNA expression levels. Our key findings reveal the importance of integrating the signaling pathway and gene expression dynamics in modeling. Modeling infers valid research questions which need to be verified experimentally and can assist in the design of future biological experiments.

Underlying Low-Order Dynamics of Nonlinear Interaction among Northern Hemisphere Teleconnection Patterns and Its Association with the AO/NAM

Abstract Studies on the nonlinear natures of the spatiotemporal structure of the Arctic Oscillation/Northern Hemisphere annular mode (AO/NAM) in the context of nonlinear interaction among the North Atlantic Oscillation (NAO), the Pacific–North American pattern (PNA), and the stratospheric polar vortex (SPV) are performed. The non-Gaussianity of the multivariate probability density function (PDF) in the phase space spanned by their indices is examined first. Five local maxima potentially related to circulation regimes are identified from the so-called angular PDF. One opposite pair of these regimes is found to correspond to the positive and negative phases of the AO/NAM. Since the authors are not sure that, due to uncertainty as suggested by statistical tests, some of the above regimes are non-Gaussian, the nonlinearity of phase-space tendency is employed as an assistant measure to identify them as nonlinear modes. It seems phase-space tendency traditionally estimated from time difference failed to be effective because of its dependence on Δt. To overcome this drawback a low-order stochastic dynamical model is established empirically from the indices. The investigation on the basic deterministic dynamics of this model suggests that the existence of regimes, such as those associated with the AO/NAM, can primarily be explained by its nonlinear deterministic part. However, two problems still remain unsolved: 1) one of the local maxima was almost not identified and 2) life cycles of the basic deterministic dynamics are too long to be related to the low-frequency variability. By introducing a multilevel approach of modeling, further insight into the residual noise of the above stochastic model can address these two issues quite well.

Nonlinear dynamic Interactions between flow-induced galloping and shell-like buckling

For an elastic system that is non-conservative but autonomous, subjected for example to time-independent loading by a steadily flowing fluid (air or water), a dangerous bifurcation, such as a sub-critical bifurcation, or a cyclic fold, will trigger a dynamic jump to one or more remote stable attractors. When there is more than one candidate attractor, the one onto which the structure settles can then be indeterminate, being sensitive to infinitesimally small variations in starting conditions or parameters.In this paper we develop and study an archetypal model to explore the nonlinear dynamic interactions between galloping at an incipient sub-critical Hopf bifurcation of a structure with shell-like buckling behaviour that is gravity-loaded to approach a sub-critical pitch-fork bifurcation. For the fluid forces, we draw on the aerodynamic coefficients determined experimentally by Novak for the flow around a bluff body of rectangular cross-section. Meanwhile, for the structural component, we consider a variant of the propped-cantilever model that is widely used to illustrate the sub-critical pitch-fork: within this model a symmetry-breaking imperfection makes the behaviour generic.The compound bifurcation corresponding to simultaneous galloping and buckling is the so-called Takens-Bodganov Cusp. We make a full unfolding of this codimension-3 bifurcation for our archetypal model to explore the adjacent phase-space topologies and their indeterminacies.