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This paper presents a control technique capable of driving a harmonically driven nonlinear system between two distinct periodic orbits. A vital component of the method is a temporary dual-frequency driving with tunable driving amplitudes. Theoretical considerations revealed two necessary conditions: one for the frequency ratio of the dual-frequency driving and another one for torsion numbers of the two orbits connected by bifurcation curves in the extended dual-frequency driving parameter space. Although the initial and the final states of the control strategy are single-frequency driven systems with distinct parameter sets (frequencies and driving amplitudes), control of multistability is also possible via additional parameter tuning. The technique is demonstrated on the symmetric Duffing oscillator and the asymmetric Toda oscillator. [ABSTRACT FROM AUTHOR]

A wide variety of stationary or moving spatially localized structures is present in evolution problems on unbounded domains, governed by higher-than-second-order reversible spatial interactions. This work provides a generic unfolding in one spatial dimension of a certain codimension-three singularity that explains the organization of bifurcation diagrams of such localized states in a variety of contexts, ranging from nonlinear optics to fluid mechanics, mathematical biology, and beyond. The singularity occurs when a cusp bifurcation associated with the onset of bistability between homogeneous steady states encounters a pattern-forming, or Turing, bifurcation. The latter corresponds to a Hamiltonian-Hopf point of the corresponding spatial dynamics problem. Such codimension-three points are sometimes called Lifshitz points in the physics literature. In the simplest case where the spatial system conserves a first integral, the system is described by a canonical fourth-order scalar system. The problem contains three small parameters: two that unfold the cusp bifurcation and one that unfolds the Turing bifurcation. Several cases are revealed, depending on open conditions on the signs of the lowest-order nonlinear terms. Taking the case in which the Turing bifurcation is subcritical, various parameter regimes are considered and the bifurcation diagrams of localized structures are elucidated. A rich bifurcation structure is revealed which involves transitions between regions of localized periodic patterns generated by standard homoclinic snaking, and regions of stationary domains of one homogeneous solution embedded in the other organized in a collapsed snaking structure. The theory is shown to unify previous numerical results obtained in models arising in nonlinear optics, fluid mechanics, and excitable media more generally. [ABSTRACT FROM AUTHOR]

A 2D piecewise linear continuous two-parameter map known as the Lozi map is a special case of the 2D border collision normal form depending on four parameters. In the present paper, we investigate how the bifurcation structure of the Lozi map is incorporated into the bifurcation structure of the 2D border collision normal form using an analytical representation of the boundaries of the largest periodicity regions related to the cycles with rotation number 1/n, n ⩾ 3. At the centre bifurcation boundary of the stability domain of the fixed point both maps are conservative which leads to a quite intricate bifurcation structure near this boundary. [ABSTRACT FROM AUTHOR]

Present study reports the application potential of a natural down fiber in the fields of oil-water separation and interfacial dye adsorption. The dripping experiment shows that the blue water droplets can maintain a spherical shape on the surface of the fiber with a contact angle of 155.8°, while the contact angle of oils was 0°, indicating that the fiber has superhydrophobic and ultralipophilic characteristics. Further studies reveal that this fiber has excellent absorption capacities for various oils with 120.11 g/g (vacuum pump oil), 102.11 g/g (heat transfer oil), 99.27 g/g (engine oil), and oil retention capacities after 24 hours with 94.51%, 94.29% and 92.14% respectively. The oil absorption ratios after ten cycles of use are respectively 52.75%, 60.30%, 64.79% of the initial value. More importantly, as an oil-water separation and interfacial adsorption material, down fiber with a separation efficiency as high as 97.7% and dye removal efficiency are 88.70% (Methylene blue) and 88.39% (Rhodamine B). Therefore, this natural fiber can be used as a recyclable oil-water separation and adsorption material to protect water resources.

Present study reports the application potential of a natural down fiber in the fields of oil-water separation and interfacial dye adsorption. The dripping experiment shows that the blue water droplets can maintain a spherical shape on the surface of the fiber with a contact angle of 155.8°, while the contact angle of oils was 0°, indicating that the fiber has superhydrophobic and ultralipophilic characteristics. Further studies reveal that this fiber has excellent absorption capacities for various oils with 120.11 g/g (vacuum pump oil), 102.11 g/g (heat transfer oil), 99.27 g/g (engine oil), and oil retention capacities after 24 hours with 94.51%, 94.29% and 92.14% respectively. The oil absorption ratios after ten cycles of use are respectively 52.75%, 60.30%, 64.79% of the initial value. More importantly, as an oil-water separation and interfacial adsorption material, down fiber with a separation efficiency as high as 97.7% and dye removal efficiency are 88.70% (Methylene blue) and 88.39% (Rhodamine B). Therefore, this natural fiber can be used as a recyclable oil-water separation and adsorption material to protect water resources. [ABSTRACT FROM AUTHOR]

We develop a simple financial market model in which a market maker adjusts the price with respect to orders placed by chartists and fundamentalists. A novel feature of our model is that fundamentalists optimistically (pessimistically) believe in a relatively high (low) fundamental value when the financial market is increasing (decreasing). As it turns out, the dynamics of our model is driven by a two-dimensional discontinuous piecewise linear map for which we provide an in-depth analytical and numerical investigation. Among other things, we obtain in explicit form the boundaries of the periodicity regions associated with attracting cycles with rotation number 1/n, n ≥ 3. These boundaries correspond to border collision bifurcations of the related cycles. We show that the periodicity regions are organized in a specific period adding structure, and some of the regions may overlap. Several examples of coexisting cycles and their basins of attraction are also presented. Economically, our results offer a new explanation for the boom-bust behavior of actual financial markets. [ABSTRACT FROM AUTHOR]

We consider the well-known Lozi map, which is a 2D piecewise linear map depending on two parameters. This map can be considered as a piecewise linear analog of the Hénon map, allowing to simplify the rigorous proof of the existence of a chaotic attractor. The related parameter values belong to a part of the parameter plane where the map has two saddle fixed points. In the present paper, we investigate a different part of the parameter plane, namely, the vicinity of the curve related to a center bifurcation of the fixed point. A distinguishing property of the Lozi map is that it is conservative at the parameter value corresponding to this bifurcation. As a result, the bifurcation structure close to the center bifurcation curve is quite complicated. In particular, an attracting fixed point (focus) can coexist with various attracting cycles, as well as with chaotic attractors, and the number of coexisting attractors increases as the parameter point approaches the center bifurcation curve. The main result of the present paper is related to the rigorous description of this bifurcation structure. Specifically, we obtain, in explicit form, the boundaries of the main periodicity regions associated with the pairs of complementary cycles with rotation number 1 / n. Similar approach can be applied to other periodicity regions. Our study contributes also to the border collision bifurcation theory since the Lozi map is a particular case of the 2D border collision normal form. [ABSTRACT FROM AUTHOR]

Highlights • A complete modeling of a periodically driven plate with boundary conditions correction in subsonic ow is developed. • A transformation upon the Legendre polynomials is applied for converting the boundary conditions with correction. • The systems with or without correction show the similar global bifurcation structure and scaling property. • The correction can cause reversals which increase the diversity and complexity of the local bifurcation structure. Abstract When a plate is subjected to long-term vibration the original boundary conditions are no longer ideal and necessary correction should be considered during its dynamical modeling. This paper aims at the modeling of the boundary conditions correction and its influence on the nonlinear dynamics of a harmonic excited plate in subsonic flow. The plate boundary conditions with correction are non-autonomous and converted into autonomous by using a coordinate transformation upon the orthogonal Legendre polynomials. The fluid force is separated into the reactive and resistive parts. The reactive fluid force, due to plate motion, is derived from the bound and wake vorticity in Glauert’s expansion; the resistive force, which is independent on plate motion, is evaluated in drag coefficient. The governing nonlinear partial differential equation is discretized in space and time domains by using the Galerkin method. Results show that the present fluid model is in good agreement with other theories archived and is reliable. The unforced system loses its stability by divergence following a pitchfork-like bifurcation featured by the appearance of new bifurcated stable equilibrium points due to nonlinearity after instability. The boundary conditions correction can stabilize the system and does not change the bifurcation structure and scaling property. The symmetry-breaking/restoring bifurcations play an important role in the change of different period-1 motions, and the nature of such bifurcations is associated with the collision or division of attractors. Subharmonic motions and chaos appear alternatively, and the transition between chaos and periodic motions is generally in accompany with period-doubling bifurcations, explosion and condensation of attractors. [ABSTRACT FROM AUTHOR]

The present study focuses on the possibilities of dynamic stabilization of a gas-vapour bubble below Blake’s critical threshold (Blake in the onset of cavitation in liquids, 1949) by harmonic forcing. In bubble dynamics, in terms of the ambient pressure, this threshold is known as a special limit where bubbles tend to grow infinity due to the non-strictly dissipative nature of the governing equations. The employed model is the harmonically excited Rayleigh-Plesset equation that is a nonlinear, second-order ordinary differential equation. Partial results have already been published in the literature (Hegedűs in Ultrasonics 54(4):1113, 2014), Hegedűs in Phys Lett A 380(9-10):1012, 2016). Throughout this paper, however, the investigated parameter space is significantly extended: excitation properties (pressure amplitude and frequency), ambient pressure, bubble size and liquid viscosity (amount of dissipation). The numerical results have indicated that domains where stable oscillations exist can always be found below Blake’s threshold. However, from application point of view, it is mandatory to raise the dissipation rate of the system to significantly increase the extent of these domains making the process of dynamic stabilization robust. [ABSTRACT FROM AUTHOR]

A novel method to control multistability of nonlinear oscillators by applying dual-frequency driving is presented. The test model is the Keller-Miksis equation describing the oscillation of a bubble in a liquid. It is solved by an in-house initial-value problem solver capable to exploit the high computational resources of professional graphics cards. During the simulations, the control parameters are the two amplitudes of the acoustic driving at fixed, commensurate frequency pairs. The high-resolution bi-parametric scans in the control parameter plane show that a period-2 attractor can be continuously transformed into a period-3 one (and vice versa) by proper selection of the frequency combination and by proper tuning of the driving amplitudes. This phenomenon has opened a new way to drive the system to a desired, pre-selected attractor directly via a non-feedback control technique without the need of the annihilation of other attractors. Moreover, the residence in transient chaotic regimes can also be avoided. The results are supplemented with simulations obtained by the boundary-value problem solver AUTO, which is capable to compute periodic orbits directly regardless of their stability, and trace them as a function of a control parameter with the pseudo-arclength continuation technique. [ABSTRACT FROM AUTHOR]

We consider the stabilization of a multi-tethered lighter-than-air sphere undergoing vortex-induced vibrations in uniform flow. Motivated by recent engineering applications, we extend a single-tether model coupled with a three-dimensional wake oscillator to incorporate a four-tether configuration with general rigid-body rotations for the light sphere. The fundamental bifurcation structure of the coupled model includes periodic self-excited limit cycles, which culminate with quasiperiodic solutions with increased reduced velocity. Stabilization of limit-cycle oscillations is achieved for optimal mass ratio and tether length parameters which are obtained by asymptotic analysis of reduced-order models. The resulting reduction in transverse and rotational amplitudes is at the expense of a decreasing range of the periodic bifurcation region for large tether inclination angles, whereas in addition to rotational amplitude reduction, regularization of nonstationary solutions occurred for a medium inclination angle. [ABSTRACT FROM AUTHOR]

[Display omitted] • We propose theoretical formulas to calculate the surge impedance of the inclined downlead that is circumscribed by composite. There is no formula to express this case previously. • We build an accurate Transient Electromagnetic model in PSCAD/EMPDC for a Y-shaped composite pylon considering coupling effect between down-leads, parasitic capacitance among conductors, leader progress model and frequency dependence ground model. • We propose improved empirical formulas derived from CIGRE brochure that based on lattice diagram method to predict the lightning performance of transmission towers. It shows improved calculation accuracy on composite pylon compared with original CIGRE method. • The Monte Carlo method is employed to verify the superiority of the new empirical formulas. This paper studies the transient responses to a lightning surge terminating on towers with a bifurcation structure and presents new empirical formulas for estimating the respective lightning performance. The multi-stage segmentation is adopted for down-lead modeling, and mutual coupling and parasitic capacitance are considered. By discussing the influence degree of these factors, we propose a simplified down-lead system for formula expression. These formulas are derived from the lattice diagram method, and the original lattice diagram method recommended by CIGRE is also compared. Taking the simulation results as standard, the results derived from the proposed method show a good match with the simulation values, due to the consideration of the down-lead bifurcation configuration. The critical current (I c) at the full range of front time is obtained, and the errors for the Y-shaped pylon by the new method are much less than that of the CIGRE method when the lightning front time is in the 97.5 % confidence interval. Finally, the Monte Carlo method is employed to verify the superiority of the new empirical formulas. © 2017 Elsevier Inc. All rights reserved. [ABSTRACT FROM AUTHOR]

In this paper we formulate an initial-boundary-value-problem describing the three-dimensional motion of a cantilever in a Magnetic Resonance Force Microscopy setup. The equations of motion are then reduced to a modal dynamical system using a Galerkin ansatz and the respective nonlinear forces are expanded to cubic order. The direct application of the asymptotic multiple scales method to the truncated quadratic modal system near a 2:1 internal resonance revealed conditions for periodic and quasiperiodic energy transfer between the transverse in-plane and out-of-plane modes of the MRFM cantilever. However, several discrepancies are found when comparing the asymptotic results to numerical simulations of the full nonlinear system. Therefore, we employ the reconstitution multiple scales method to a modal system incorporating both quadratic and cubic terms and derive an internal resonance bifurcation structure that includes multiple coexisting in-plane and out-of-plane solutions. This structure is verified and reveals a strong dependency on initial conditions in which orbital instabilities and complex out-of-plane non-stationary motions are found. The latter are investigated via numerical integration of the corresponding slowly-varying evolution equations which reveal that breakdown of quasiperiodic tori is associated with symmetry-breaking and emergence of irregular solutions with a dense spectral content. [ABSTRACT FROM AUTHOR]

The main purpose of the present survey is to contribute to the theory of dynamical systems defined by one-dimensionalpiecewise monotone maps. We recall some definitions known from the theory ofsmoothmaps, which are applicable to piecewise smooth ones, and discuss the notions specific for the considered class of maps. To keep the presentation clear for the researchers working in other fields, especially in applications, many examples are provided. We focus mainly on the notions and concepts which are used for the investigation of various kinds of attractors of a map and related bifurcation structures observed in its parameter space. [ABSTRACT FROM PUBLISHER]

The bifurcation structure of a periodically driven spherical gas/vapour bubble is examined by means of methods of nonlinear analysis. The study of Behnia et al. (Ultrasonics 49(8):605, 2009) revealed that the bifurcation structures with the pressure amplitude of the excitation as control parameter are structurally similar provided that $$R_\mathrm{E} \omega $$ is kept constant. In the present paper, this problem is revisited. Analytical and numerical investigations of the bubble oscillator, which is the Keller-Miksis equation, are presented. It is shown that the validity range of Behnia's condition is governed by the viscosity and the surface tension, and holds only for relatively large bubbles. In water, the effect of viscosity is negligible, and the surface tension plays significant role at bubble size lower than approximately $$5\,\upmu \mathrm {m}$$ . [ABSTRACT FROM AUTHOR]

The topology of the stable periodic orbits of a harmonically driven bubble oscillator, the Rayleigh–Plesset equation, in the space of the excitation parameters (pressure amplitude and frequency) has been revealed numerically. This topology is governed by a hierarchy of two-sided Farey trees initiated from a unique primary structure defined also by a simple asymmetric Farey tree. The sub-topology of each of these building blocks is driven by a homoclinic tangency of a periodic saddle. This self-similar organisation is a suitable basis for a general description, since it is in good agreement with partial results obtained in other periodically forced oscillators and iterated maps. The applied ambient pressure in the model is near but still below Blake's critical threshold. Therefore, this paper is also a straightforward continuation of the work of Hegedűs [1] , who first found numerical evidence for the existence of stable, period 1 solutions beyond Blake's threshold. The present findings are crucial for the extension of the available numerical results from period 1 to arbitrary periodicity. [ABSTRACT FROM AUTHOR]

In mathematical modelling, several different functional forms can often be used to fit a data set equally well, especially if the data is sparse. In such cases, these mathematically different but similar looking functional forms are typically considered interchangeable. Recent work, however, shows that similar functional responses may nonetheless result in significantly different bifurcation points for the Rosenzweig–MacArthur predator–prey system. Since the bifurcation behaviours include destabilizing oscillations, predicting the occurrence of such behaviours is clearly important. Ecologically, different bifurcation behaviours mean that different predictions may be obtained from the models. These predictions can range from stable coexistence to the extinction of both species, so obtaining more accurate predictions is also clearly important for conservationists. Mathematically, this difference in bifurcation structure given similar functional responses is called structural sensitivity. We extend the existing work to find that the Leslie–Gower–May predator–prey system is also structurally sensitive to the functional response. Using the Rosenzweig–MacArthur and Leslie–Gower–May models, we then aim to determine if there is some way to obtain a functional description of data so that different functional responses yield the same bifurcation structure, i.e., we aim to describe data such that our model is not structurally sensitive. We first add stochasticity to the functional responses and find that better similarity of the resulting bifurcation structures is achieved. Then, we analyse the functional responses using two different methods to determine which part of each function contributes most to the observed bifurcation behaviour. We find that prey densities around the coexistence steady state are most important in defining the functional response. Lastly, we propose a procedure for ecologists and mathematical modellers to increase the accuracy of model predictions in predator–prey systems. • Structurally sensitive models may lead to vastly different model predictions. • Bifurcation structures are studied for two models using three functional responses. • Fitting around the coexistence steady state decreases structural sensitivity. • We propose a method to improve model predictions in predator–prey systems. [ABSTRACT FROM AUTHOR]

In the last decade many industrial applications have emerged based on the rapidly developing ultrasonic technology such as ultrasonic pasteurization, alteration of the viscosity of food systems, and mixing immiscible liquids. The fundamental physical basis of these applications is the prevailing extreme conditions (high temperature, pressure and even shock waves) during the collapse of acoustically excited bubbles. By applying the sophisticated numerical techniques of modern bifurcation theory, the present study intends to reveal the regions in the excitation pressure amplitude–ambient temperature parameter plane where collapse-like motion of an acoustically driven gas bubble in highly viscous glycerine exists. We report evidence that below a threshold temperature the bubble model, the Keller–Miksis equation, becomes an overdamped oscillator suppressing collapse-like behaviour. In addition, we have found periodic windows interspersed with chaotic regions indicating the presence of transient chaos, which is important from application point of view if predictability is required. [ABSTRACT FROM AUTHOR]

The objective of this study is to clarify the wrinkle behavior of a flat rectangular membrane undergoing shear displacement. To achieve this goal, an equilibrium path tracking method using a finite element method is developed. This method includes a bifurcation path tracking analysis that searches for bifurcation solutions. This method establishes an image of the membrane behavior by calculating a series of successive equilibrium states before and after bifurcation buckling. Generally in experiments, flat rectangular membranes have shear displacement imposed on top or bottom edges, while the left and right sides have free boundaries. At large values of shear displacement, the wrinkles cover the entire membrane, but the free boundaries result in uneven shapes and distributions. Further increase in shear results in small wrinkles, referred to as collapsed sections, generated on existing wrinkles. As collapsed sections grow, new wrinkles are generated. However, the universality of this wrinkle generation mechanism may be affected by the free boundaries. By applying cyclic boundary conditions, effects of free boundaries, which include uneven wrinkle shape and distribution, can be eliminated. In addition, by changing the membrane aspect ratio, the effects of geometry are also evaluated. For all membranes, the wrinkle generation from collapsed sections is observed and its independence from free boundaries and aspect ratio is shown. By analyzing stress and displacement fields, the formation of collapsed sections is explained. In addition, for the cyclic boundary conditions, the change in aspect ratio results in almost the same bifurcation structure. Therefore, the wrinkle behavior evaluation in this study can be useful in predicting wrinkle behavior. [ABSTRACT FROM AUTHOR]

The bifurcation structure of the oscillations of ultrasound contrast agents (UCAs) was studied as a function of the driving pressure for excitation frequencies that were determined using the UCAs pressure-dependent resonances $$(f_\mathrm{s})$$ . It was shown that when excited by the $$(f_\mathrm{s})$$ , the UCA can undergo a saddle-node bifurcation (SNB) to higher amplitude oscillations. The driving pressure at which the SNB occurs is controllable and depends on the $$(f_\mathrm{s})$$ magnitude. Utilizing the appropriate $$(f_\mathrm{s})$$ , the scattering cross section of the UCAs can significantly be enhanced (e.g., $$\sim $$ ninefold) while at the same time avoiding potential UCA destruction (by limiting the radial expansion ratio $$<$$ 2). This offers significant advantages for optimizing UCA-mediated imaging and therapeutic ultrasound applications. [ABSTRACT FROM AUTHOR]

An external feedback control is inserted in a nonlinear continuum formulation of a noncontact AFM model. The aim of the feedback is to keep the system response to an operationally suitable one, thus allowing reliable measurement of the sample surface by avoiding possible unstable microcantilever sensor motions. The study of the weakly nonlinear system dynamics about the desired fixed point close to primary resonance is carried out via multiple-scale asymptotics, whose outcomes are validated via numerical simulations of the original system equations of motion. The latter include controllable periodic dynamics and additional periodic and distinct quasiperiodic solutions that appear beyond the asymptotic stability thresholds. The results highlight the effectiveness of the applied feedback control technique and also enable the derivation of a comprehensive system bifurcation structure highlighting the stability thresholds for robust controllable AFM dynamics. [ABSTRACT FROM AUTHOR]

Bifurcation structure and scaling properties of a nonlinear subsonic panel with a harmonic excitation are investigated numerically and theoretically in this paper. The panel under study is two-dimensional and simply supported considering Kelvin's model of structural damping and geometric nonlinearity generating by large deflection based on the von Kármán's theory. Galerkin method is used to derive the subsonic aerodynamic pressure and transform the governing partial equations to a series of ordinary differential equations. Numerical calculations were conducted to the analysis on the bifurcation structure of the nonlinear panel system and the distributions of complex responses regions are shown in a two-parameter space. The interesting scaling properties of the bifurcation structure are discussed in terms of a discrete mapping theoretically based on linear approximation. The proposed approximate analytical method shows a good agreement with the numerically found scaling properties. [ABSTRACT FROM AUTHOR]