Bifurcation Evolution Research Articles

Nonlinear dynamics of planetary roller screw mechanism.

The synchronous meshing of the gear pair and the screw pair is a typical feature of the planetary roller screw mechanism. In order to fully derive and analyze the nonlinear dynamic characteristics of the system, this paper creatively incorporates the time-varying meshing stiffness of gear pair and the comprehensive transmission error into the research content. Combined with the thread contact force and friction force between the roller and the screw and between the roller and the nut, the nonlinear dynamic model of the planetary roller screw mechanism considering the meshing excitation of the gear pair is established. According to the time domain diagram, frequency domain diagram, phase plane diagram, Poincaré section diagram, three-dimensional spectrum diagram, and spatial phase diagram, the nonlinear behavior of the system is described in detail, and the bifurcation evolution process of the system under the excitation frequency parameters of the external load is revealed. In addition, based on the theory of multi-scale method and considering the variables such as meshing stiffness, meshing damping, and load fluctuation, the stability equation of the primary resonance of the system is derived. The analysis of the stability of the system under different working conditions shows that the parameters are selected within a reasonable range, which effectively reduces the primary common amplitude value and enhances the overall stability of the system. The research content improves the previous system dynamics modeling method and provides a theoretical basis for the nonlinear dynamics modeling method and parameter optimization design of the planetary roller screw mechanism.

On two-parameter bifurcation and analog circuit implementation of a Chameleon chaotic system

In this paper, the two-parameter space bifurcation of a three-dimensional Chameleon system is investigated. It is called Chameleon since the type and the number of the system equilibrium are adjustable for different parameter configurations. Aided by the computation analysis, the graphic structures of two-parameter bifurcation of the Chameleon system are characterized for the first time. With different two-parameter configurations, the bifurcation evolution shows that various self-excited and hidden attractors exist. In addition, numerical demonstration of the two-dimensional slice through the attraction basin space is presented. The results show that the basin of attraction of the typical hidden chaotic attractor does not associated with the origin, which makes the attractor difficult to be numerically localized and experimentally observed. To solve the problem, offset boost scheme is adopted to control the basin of attraction and make it touch the origin, which allows to coin the hidden attractor via configuring zero initial value and making it feasible in experimental observation. Finally, the analog circuit-assisted experiment validated the feasibility of the scheme.

Digital synchronous decomposition and period-N bifurcation size identification in dynamic systems: application to a milling process

Period-N bifurcation signal commonly contains the typical vibration and period-N bifurcation components. The period-N bifurcation size can be defined based on the period-N bifurcation component. It is challenging to extract the period-N bifurcation component with high accuracy in real-time since many frequencies may be contained in the period-N bifurcation signal. We proposed a digital synchronous decomposition (DSD) method to address this issue. Mathematical proof and numerical simulations validated that the DSD accurately decomposes the period-N bifurcation signal into the typical vibration and period-N bifurcation components in real-time. Moreover, an identification method and a bifurcation diagram were proposed to identify the period-N bifurcation size and depict period-N bifurcation evolution based on the DSD, respectively. We applied the proposed identification method to a benchmark milling process model and an actual milling process. Comparisons with the fast Fourier transform (FFT) highlight the proposed identification method's advantage. Besides, the proposed bifurcation diagram overcomes the limitation of the existing bifurcation diagrams and can quantitatively depict the period-N bifurcation severity and evolution.

Real-time observation of phase transition of bifurcation evolution in mode-locked lasers.

By using the dispersive temporal holography technology, we observe the real-time dynamics of period-doubling bifurcation evolution in an ultrafast fiber laser. The pulse properties including the phase, polarization state, chirp, pulse width, and pulse energy are fully bifurcated, which embody the bifurcation induced "phase transition." There are two types of bifurcation in the laser buildup: one to many bifurcation towards the chaos and the multiple bifurcated pulses back to the stationary state along the reversed trajectory. Both of them can be explained by the rule of the same phase transition. We conclude and illustrate the differences and connections to another common pulsation, the Hopf bifurcation. The findings can promote an understanding of the bifurcations in ultrafast lasers, and are beneficial for an improvement of laser stability.

Dynamics Analysis of an Aquatic Ecological Model with Temperature Effect

Under the control framework of algae bloom in eutrophic lakes and reservoirs based on biological manipulation, the temperature variable is introduced into ecological modeling to show that it is a necessary condition for the rapid occurrence of algal blooms, and an aquatic ecological model with temperature effect is proposed to describe dynamic relationship between algae and biological manipulation predator. The mathematical theory work mainly investigates the existence and stability of some equilibrium points and some critical conditions for the occurrence of transcritical bifurcation and Hopf bifurcation. The numerical simulation mainly shows the dynamic evolution process of bifurcation dynamics, which can not only verify the validity and feasibility of these theoretical works but also analyze the influence of some key parameters on dynamic behavior evolution. Furthermore, It is worth emphasizing that temperature plays an important role in the coexistence of algae and biological manipulation predators. Moreover, the coexistence mode of algae and biological manipulation predators is discovered by means of dynamic bifurcation evolution. Finally, it is hoped that these research results can provide some reference for the study of aquatic ecosystems.

Application of Parameter Optimization to Search for Oscillatory Mass-Action Networks Using Python

Biological systems can be described mathematically to model the dynamics of metabolic, protein, or gene-regulatory networks, but locating parameter regimes that induce a particular dynamic behavior can be challenging due to the vast parameter landscape, particularly in large models. In the current work, a Pythonic implementation of existing bifurcation objective functions, which reward systems that achieve a desired bifurcation behavior, is implemented to search for parameter regimes that permit oscillations or bistability. A differential evolution algorithm progressively approximates the specified bifurcation type while performing a global search of parameter space for a candidate with the best fitness. The user-friendly format facilitates integration with systems biology tools, as Python is a ubiquitous programming language. The bifurcation–evolution software is validated on published models from the BioModels Database and used to search populations of randomly-generated mass-action networks for oscillatory dynamics. Results of this search demonstrate the importance of reaction enrichment to provide flexibility and enable complex dynamic behaviors, and illustrate the role of negative feedback and time delays in generating oscillatory dynamics.

Feature recognition of small amplitude hunting signals based on the MPE-LTSA in high-speed trains

Hunting stability is an important factor for high-speed trains to achieve safe operation, which can be monitored by on-board instruments. When analysing measured online tracking data of high-speed trains, the authors have observed that small amplitude hunting tend to appear. When these signals show growth of lateral vibration to high enough amplitude, train derailment would happen. Research on the bifurcation evolution of small amplitude hunting of high-speed trains has been rarely reported so far. In this paper, chaotic features of the data are extracted and the results show that lateral acceleration signals from the bogie frame has strong nonlinear characteristics. Then a commonly used method based on frequency distribution characteristics of bogie vibration energy is first used to separate different states of hunting. However, the results are not satisfactory. So a feature extraction method based on Multiscale Permutation Entropy (MPE) and Local Tangent Space Alignment (LTSA) is proposed to distinguish the different states of complex signals. The proposed method is applied to extract features of the small amplitude hunting signals at high-speed of 320–350 km/h. The results show that the MPE-LTSA method can identify the bifurcation evolution of small amplitude hunting signals much more effectively than the method based on the MPE-ISOMAP (Isometric Feature Mapping) and MPE-PCA (Principle Component Analysis). The method can be used in other feature recognition for the complex chaotic signals.

Existence, bifurcation, and geometric evolution of quasi-bilayers in the multicomponent functionalized Cahn–Hilliard equation

Multicomponent bilayer structures arise as the ubiquitous plasma membrane in cellular biology and as blends of amphiphilic copolymers used in electrolyte membranes, drug delivery, and emulsion stabilization within the context of synthetic chemistry. We present the multicomponent functionalized Cahn-Hilliard (mFCH) free energy as a model which allows competition between bilayers with distinct composition and between bilayers and higher codimensional structures, such as co-dimension two filaments and co-dimension three micelles. We construct symmetric and asymmetric homoclinic bilayer profiles via a billiard limit potential and show that co-dimensional bifurcation is driven by the experimentally observed layer-by-layer pearling mechanism. We investigate the stability and slow geometric evolution of multicomponent bilayer interfaces within the context of an [Formula: see text] gradient flow of the mFCH, addressing the impact of aspect ratio of the amphiphile (lipid or copolymer unit) on the intrinsic curvature and the codimensional bifurcation. In particular we derive a Canham-Helfrich sharp interface energy whose intrinsic curvature arises through a Melnikov parameter associated to amphiphile aspect ratio.

Bar dynamics and bifurcation evolution in a modelled braided sand‐bed river

AbstractMorphodynamics in sand‐bed braided rivers are associated with simultaneous evolution of mid‐channel bars and channels on the braidplain. Bifurcations around mid‐channel bars are key elements that divide discharge and sediment. This, in turn, may control the evolution of connected branches, with effects propagating to both upstream and downstream bifurcations. Recent works on bifurcation stability and development hypothesize major roles of secondary flow and gradient advantage. However, this has not been tested for channel networks within a fully developed dynamic braided river. A reason for this is a lack of detailed measurements with sufficient temporal and spatial length, covering multiple bifurcations. Therefore we used a physics‐based numerical model to generate a dataset of bathymetry, flow and sediment transport of an 80 km river reach with self‐formed braid bars and bifurcations. The study shows that bar dissection due to local transverse water surface gradients is the dominant bifurcation initiation mechanism, although conversion of unit bars into compound bars dominates in the initial stage of a braided river. Several bifurcation closure mechanisms are equally important. Furthermore, the study showed that nodal point relations for bifurcations are unable to predict short‐term bifurcation evolution in a braided river. This is explained by occurrence of nonlinear processes and non‐uniformity within the branches, in particular migrating bars and larger‐scale backwater‐effects, which are not included in the nodal point relations. Planform morphology, on the other hand, has predictive capacity: bifurcation angle asymmetry and bar‐tail limb shape are indicators for near‐future bifurcation evolution. Remote sensing data has predictive value, for which we developed a conceptual model for interactions between bars, bifurcations and channels in the network. We conducted a preliminary test of the conceptual model on satellite images of the Brahmaputra. Copyright © 2015 John Wiley & Sons, Ltd.

Near-bed and surface flow division patterns in experimental river bifurcations

Understanding channel bifurcation mechanics is of great importance for predicting and managing multichannel river processes and avulsion in distributary river deltas. To date, research on river channel bifurcations has focused on factors determining the stability and evolution of bifurcations. It has recently been shown that, theoretically, the nonlinearity of the relation between sediment transport and flow discharge causes one of the two distributaries of a (slightly) asymmetrical bifurcation to grow and the other to shrink. The positive feedback introduced by this effect results in highly asymmetrical bifurcations. However, there is a lack of detailed insight into flow dynamics within river bifurcations, the consequent effect on bed load flux through bifurcating channels, and thus the impact on bifurcation stability over time. In this paper, three key parameters (discharge ratio, width-to-depth ratio, and bed roughness) were varied in order to examine the secondary flow field and its effect on flow partitioning, particularly near-bed and surface flow, at an experimental bifurcation. Discharge ratio was controlled by varying downstream water levels. Flow fields were quantified using both particle image velocimetry and ultrasonic Doppler velocity profiling. Results show that a bifurcation induces secondary flow cells upstream of the bifurcation. In the case of unequal discharge ratio, a strong increase in the secondary flow near the bed causes a larger volume of near-bed flow to enter the dominant channel compared to surface and depth-averaged flow. However, this effect diminishes with larger width-to-depth ratio and with increased bed roughness. The flow structure and division pattern will likely have a stabilizing effect on river channel bifurcations. The magnitude of this effect in relation to previously identified destabilizing effects is addressed by proposing an adjustment to a widely used empirical bed load nodal-point partition equation. Our finding implies that river bifurcations can be stable under a wider range of conditions than previously thought. Key Points Secondary flow in symmetrical bifurcations causes strong near-bed flow curvature A disproportional amount of near-bed flow enters the dominant downstream channel Flow curvature adds a stabilizing feedback on bifurcation evolution.

Life of a bifurcation in a gravel‐bed braided river

ABSTRACTChannel bifurcation is a key element in braided rivers, determining the water and sediment distribution and hence controlling the morphological evolution. Recent theoretical and experimental findings, as well as field observations, showed that bifurcations in gravel‐bed braided rivers are often asymmetrical and highly unstable. In this paper field data are presented on a bifurcation in the Tagliamento River, northeast Italy. The planform configuration of the bifurcation and its temporal evolution was monitored by an automatic digital camera during a series of seven floods with different magnitudes. This remote sensing technique allowed a high temporal resolution (pictures were acquired every hour) that was proved to be essential in a highly dynamic system as the one considered here. Digitized maps of the channels provided information on the location of the bifurcation, the width of the anabranches, the angle between them, along with the occurrence and migration of sediment bars. Data were acquired at two different water levels, giving the possibility to compare low and high flow conditions. The monitored bifurcation is largely unstable and shows sudden changes in the water distribution, mainly driven by the bar migrating in the upstream channel and entering the distributaries. A relationship between width asymmetry and flood magnitude was observed, confirming previous analyses. Moreover, recent theoretical findings were applied, in order to test the possibility to estimate general trends in bifurcation evolution. The analysis pointed out the relevance of a correct assessment of the characteristic temporal scales, as the bifurcation evolves on a timescale similar to that of bar migration and flood duration. Understanding the interactions between these processes is therefore crucial in order to increase the ability to model and predict the morphological evolution of a braided network. Copyright © 2012 John Wiley & Sons, Ltd.

Stability of backwater‐influenced river bifurcations: A study of the Mississippi‐Atchafalaya system

In this paper I use numerical modeling to show that the hydraulic backwater profile creates a feedback that may stabilize river bifurcations. The numerical model simulates flow and sediment transport in the Mississippi‐Atchafalaya River system without the Old River Control Structure. The results show that bifurcation evolution strongly depends on the discharge upstream of the bifurcation. At upstream discharges greater than 12600 m3 s−1 the Atchafalaya River discharge increases through time at the expense of the Mississippi River. Interestingly, at upstream discharges lower than 12600 m3 s−1 the opposite occurs and the Mississippi River discharge increases at the expense of the Atchafalaya River. The capture direction changes because the backwater profile of each river varies enough at high and low discharge to invert the water surface slope ratio. These results suggest that the capture direction would change at high and low flow, which would have a stabilizing effect by preventing the runaway growth of one channel. Accounting for this, I calculate that in the absence of the Old River Control Structure capture would not happen catastrophically, but rather the Atchafalaya River would capture the Mississippi River in ∼300 years from present day.

Flow structures at an idealized bifurcation: a numerical experiment

ABSTRACTRiver bifurcations are key nodes within braided river systems controlling the flow and sediment partitioning and therefore the dynamics of the river braiding process. Recent research has shown that certain geometrical configurations induce instabilities that lead to downstream mid‐channel bar formation and the formation of bifurcations. However, we currently have a poor understanding of the flow division process within bifurcations and the flow dynamics in the downstream bifurcates, both of which are needed to understand bifurcation stability. This paper presents results of a numerical sensitivity experiment undertaken using computational fluid dynamics (CFD) with the purpose of understanding the flow dynamics of a series of idealized bifurcations. A geometric sensitivity analysis is undertaken for a range of channel slopes (0.005 to 0.03), bifurcation angles (22° to 42°) and a restricted set of inflow conditions based upon simulating flow through meander bends with different curvature on the flow field dynamics through the bifurcation. The results demonstrate that the overall slope of the bifurcation affects the velocity of flow through the bifurcation and when slope asymmetry is introduced, the flow structures in the bifurcation are modified. In terms of bifurcation evolution the most important observation appears to be that once slope asymmetry is greater than 0.2 the flow within the steep bifurcate shows potential instability and the potential for alternate channel bar formation. Bifurcation angle also defines the flow structures within the bifurcation with an increase in bifurcation angle increasing the flow velocity down both bifurcates. However, redistributive effects of secondary circulation caused by upstream curvature can very easily counter the effects of local bifurcation characteristics. Copyright © 2011 John Wiley & Sons, Ltd.

An experimental study of discharge partitioning and flow structure at symmetrical bifurcations

ABSTRACTRecent research has examined the factors controlling the geometrical configuration of bifurcations, determined the range of stability conditions for a number of bifurcation types and assessed the impact of perturbations on bifurcation evolution. However, the flow division process and the parameters that influence flow and sediment partitioning are still poorly characterized. To identify and isolate these parameters, three‐dimensional velocities were measured at 11 cross‐sections in a fixed‐walled experimental bifurcation. Water surface gradients were controlled, and systematically varied, using a weir in each distributary.As may be expected, the steepest distributary conveyed the most discharge (was dominant) while the mildest distributary conveyed the least discharge (was subordinate). A zone of water surface super‐elevation was co‐located with the bifurcation in symmetric cases or displaced into the subordinate branch in asymmetric cases. Downstream of a relatively acute‐angled bifurcation, primary velocity cores were near to the water surface and against the inner banks, with near‐bed zones of lower primary velocity at the outer banks. Downstream of an obtuse‐angled bifurcation, velocity cores were initially at the outer banks, with near‐bed zones of lower velocities at the inner banks, but patterns soon reverted to match the acute‐angled case. A single secondary flow cell was generated in each distributary, with water flowing inwards at the water surface and outwards at the bed. Circulation was relatively enhanced within the subordinate branch, which may help explain why subordinate distributaries remain open, may play a role in determining the size of commonly‐observed topographic features, and may thus exert some control on the stability of asymmetric bifurcations. Further, because larger values of circulation result from larger gradient disadvantages, the length of confluence–diffluence units in braided rivers or between diffluences within delta distributary networks may vary depending upon flow structures inherited from upstream and whether, and how, they are fed by dominant or subordinate distributaries. Copyright © 2011 John Wiley & Sons, Ltd.

Evolution of a bifurcation in a meandering river with adjustable channel widths, Rhine delta apex, The Netherlands

ABSTRACTRivers may dramatically change course on a fluvial plain. Such an avulsion temporarily leads to two active channels connected at a bifurcation. Here we study the effect of dynamic meandering at the bifurcation and the effect of channel width adjustment to changing discharge in both downstream branches on the evolution of a bifurcation and coexisting channels. As an example, we reconstructed the last major avulsion at the Rhine delta apex.We combined historical and geological data to reconstruct a slowly developing avulsion process spanning 2000 years and involving channel width adjustment and meandering at the bifurcation. Based on earlier idealised models, we developed a one‐dimensional model for long‐term morphodynamic prediction of upstream channel and bifurcates connected at the bifurcation node. The model predicts flow and sediment partitioning at the node, including the effect of migrating meanders at the bifurcation and channel width adjustment.Bifurcate channel width adaptation to changing discharge partitioning dramatically slows the pacing of bifurcation evolution because the sediment balance for width adjustment and bed evolution are coupled. The model further shows that meandering at the bifurcation modulates channel abandonment or enlargement periodically. This explains hitherto unrecognised reactivation signals in the sedimentary record of the studied bifurcation meander belts, newly identified in our geological reconstruction. Historical maps show that bifurcation migration due to meander bend dynamics increases the bifurcation angle, which increases the rate of closure of one bifurcate.The combination of model and reconstruction identifies the relevant timescales for bifurcation evolution and avulsion duration. These are the time required to fill one downstream channel over one backwater length, the time to translate one meander wavelength downstream and, for strong river banks, the adaptation timescale to adjust channel width. The findings have relevance for all avulsions where channel width can adjust to changing discharge and where meandering occurs. Copyright © 2011 John Wiley & Sons, Ltd.

Bifurcation dynamics and avulsion duration in meandering rivers by one‐dimensional and three‐dimensional models

At river bifurcations, water and sediment are divided over two branches. The dynamics of the bifurcation determine the long‐term evolution (centuries) of the downstream branches, potentially leading to avulsion, but the dynamics are poorly understood. The long‐term evolution can only be studied by one‐dimensional models because of computational costs. For such models, a relation describing the sediment division is necessary, but only few relations are available and these remain poorly tested so far. We study the division of sediment and the morphodynamics on a timescale of decades to centuries by idealized three‐dimensional modeling of bifurcations with upstream meanders and dominantly bed load transport. An upstream meander favors one bifurcate with more sediment and the other with more water, leading to destabilization. The bifurcations commonly attain a highly asymmetrical division of discharge and sediment after a few decades to a few centuries, depending on combinations of the relevant parameters. Although past work on avulsions focused on slope advantage, we found that bifurcations can be quasibalanced by opposing factors, such as a bifurcate connected to the inner bend with a downstream slope advantage. Nearly balanced bifurcations develop much slower than unbalanced bifurcations, which explains the observed variation in avulsion duration in natural systems. Which branch becomes dominant and the timescale to attain model equilibrium are determined by the length of the downstream bifurcates, the radius of the upstream bend, a possible gradient advantage for one bifurcate and, notably, the width–depth ratio. The latter determines the character of the bars which may result in overdeepening and unstable bars. The distance between the beginning of the upstream bend and the bifurcation determines the location of such bars and pools, which may switch the dominant bifurcate. In fact, when the bifurcation is quasibalanced by opposing factors, any minor disturbance or a different choice of roughness or sediment transport predictor may switch the dominant bifurcate. The division of sediment is nearly the same as the division of flow discharge in most runs until the discharge division becomes very asymmetrical, so that a bifurcate does not close off entirely. This partly explains the sustained existence of residual channels and existence of anastomosing rivers and the potential for reoccupation of old channel courses. We develop a new relation for sediment division at bifurcations in one‐dimensional models incorporating the effect of meandering. The flow and sediment divisions predicted by two existing relations and the new relation for one‐dimensional models are in qualitative agreement with the three‐dimensional model. These one‐dimensional relations are however of limited value for wider rivers because they lack the highly three‐dimensional bar dynamics that may switch the direction of bifurcation evolution. The potential effects of bed sediment sorting, bank erosion, and levee formation on bifurcation stability and avulsion duration are discussed.

A one‐dimensional model of bifurcations in gravel bed channels with erodible banks

In a recent paper, Bolla Pittaluga et al. proposed a physically based formulation for the nodal point conditions to be adopted at a channel bifurcation in the context of a one‐dimensional approach. They employed such conditions to study the equilibrium configurations of a simple bifurcation and their stability, assuming erodible bed and fixed banks. With the present work we extend the model proposed by Bolla Pittaluga et al. to the case of channels with erodible banks, i.e., channels which may adapt their width to the actual flow conditions. Such an extension is of a particular interest for river bifurcations in gravel bed braided streams, in which bed evolution and bank erosion processes occur over comparable timescales. Moreover, to study the morphological evolution of a bifurcation, we employ a different approach with respect to Bolla Pittaluga et al. and introduce a “local analysis” which does not account for the influence exerted on channel morphology by downstream conditions on longer timescales. For values of the controlling parameters typical of gravel bed braided streams, the model shows that the stable equilibrium solutions of a bifurcation are invariably characterized by a strongly unbalanced partition of water and sediment discharges in the two branches. Numerical simulations, based on a simplified model of the bifurcation evolution, allow us to investigate the role of the mechanism of generation of a bifurcation on its final equilibrium configurations. It is shown that bifurcations which form through the incision of a new, initially narrow, channel may lead to significantly different equilibrium configurations with respect to bifurcations which generate from a central deposition in a wide channel. In spite of its simplicity the model seems to retain the most relevant effects which govern the behavior of gravel bed channel bifurcations.

The effect of automatic voltage regulation on the bifurcation evolution in power systems

This paper discusses the relationships between types of bifurcations in power systems and their expected occurrence for voltage regulated and unregulated synchronous machines. A time-scale decomposition is performed to identify critical dynamics in the slow and fast subsystems. For single machine systems, the existence of an unstable limit cycle prior to Hopf bifurcation is analysed in the context of the region of attraction.

Bifurcation evolution across metal-metal contacts sustaining high charge injection rates

In real world even the simplest of the electrical components may exhibit unpredictable characteristics, and it seems that the theory of chaos touches all disciplines. In this work, the contact potential instabilities induced by the high electronic injection rates between metal-to-metal contacts are experimentally investigated in an I-V phase space using state-of-the-art data logging systems. The mechanically contacted metals are energized by a sinusoidal power source, and their response is systematically studied within a 50-Hz cycle. The obtained results convincingly demonstrate their chaotic nature. Two entirely different instability types have been observed for such systems. Their classification may lead to a better understanding of negative differential resistance (NDR) formation, which is frequently observed across highly injecting interfaces. An equivalent circuit for the examined dynamic system is provided, and an effective test for the industrial no-load switching contacts is proposed.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>