Sequence Of Period-doubling Bifurcations Research Articles

Ballistic to diffusive transition for swimmers in a periodic vortex array.

We study the transport of rigid ellipsoidal swimmers in a periodic vortex array via numerical simulation and dynamical systems analysis. Via ensemble simulations, we show the counterintuitive result that slower swimming speeds can generate fast ballistic transport, while faster swimming speeds generate chaotic and diffusive transport, which is inherently slower in the long run. To explain this, we use the symmetry of the flow to construct a time-reversible Poincaré return map on a two-dimensional surface of sectionin phase space. For sufficiently small swimming speeds, we find stable periodic orbits on the surface of sectionsurrounded by invariant tori, similar to Kolmogorov-Arnold-Moser curves. Trajectories within these tori are ballistic. As the swimming speed is increased, the periodic orbits undergo a sequence of period-doubling bifurcations that destroys the ballistic tori. These bifurcations exactly match the ballistic to diffusive transition from the ensemble simulations. Additional ensemble simulations are used to test the robustness of these results to noise. The ballistic behavior is destroyed as the strength of rotational diffusion increases. However, we estimate that the ballistic tori might still be seen in experiments.

Thermocapillary convection in a cuboid pool with a sidewall of different temperature sections

Thermocapillary convection in a liquid cuboid pool with a sidewall of different temperature sections is investigated using 3D numerical simulation. One half section of the sidewall is heated but the other cooled. Numerical simulation is performed for Marangoni number (Ma) from 100 to 4.0 × 106 with Prandtl number (Pr) of 28.6 and aspect ratios (Ah and Aw) of 0.43. A temperature field with a weak thermocapillary convection flow is dominated by conduction for small Ma. As Ma increases, the flow becomes stronger and reveals a complex transition route to a chaotic state with a series of bifurcations. It has been demonstrated that the symmetry of the temperature field breaks between Ma = 102 and 103. A Hopf bifurcation occurs between Ma = 8.0 × 105 and 9.0 × 105 in which the flow becomes periodic. As Ma increases further, a sequence of period-doubling bifurcations occurs between Ma = 2.78 × 106 and 2.95 × 106. Once Ma exceeds 3.0 × 106, the flow becomes chaotic. Typical thermocapillary convection flows in the pool are characterized with topological analyses, spectral analyses, attractors, Lyapunov exponents, and fractal dimensions. Furthermore, the heat and mass transfer in the cuboid pool have been investigated.

On bifurcations of chaotic attractors in a pulse width modulated control system

This paper discusses bifurcational phenomena in a control system with pulse-width modulation of the first kind. We show that the transition from a regular dynamics to chaos occurs in a sequence of classical supercritical period doubling and border collision bifurcations. As a parameter is varied, one can observe a cascade of doubling of the cyclic chaotic intervals, which are associated with homoclinic bifurcations of unstable periodic orbits. Such transition are also refereed as merging bifurcation (known also as merging crisis). At the bifurcation point, the unstable periodic orbit collides with some of the boundaries of a chaotic attractor and as a result, the periodic orbit becomes a homoclinic. This condition we use for obtain equations for bifurcation boundaries in the form of an explicit dependence on the parameters. This allow us to determine the regions of stability for periodic orbits and domains of the existence of four-, two- and one-band chaotic attractors in the parameter plane.

Parametric interaction of modes in the presence of quadratic or cubic nonlinearity

The purpose of this work is a study of the dynamics of the systems of ordinary differential equations of the second order, which is obtained using the Lagrange formalism. These systems describe the parametric interaction of oscillators (modes) in the presence of a general quadratic or cubic nonlinearity. Also, we compare the dynamics of the systems of ordinary differential equations of the second order and dynamics of the Vyshkind–Rabinovich and Rabinovich–Fabrikant models in order to determine the possibilities of the latter models when modeling coupled oscillators of the above type. Methods. The study is based on the numerical solution using the methods of the theory of the obtained analytically differential equations. Results. For both systems of second-order differential equations, is was presented a chart of in the parameter plane, a graphs of Lyapunov exponents at the value of the parameter that specifies the dissipation of oscillators, a time dependences of the generalized coordinates of oscillators and its amplitudes, portraits of attractors, a projection of the attractors on a phase planes of oscillators. A comparison with the dynamics of the Vyshkind–Rabinovich and Rabinovich–Fabrikant models is carried out. These models are three-dimensional real approximations of the above systems obtained by the method of slowly varying amplitudes. Conclusion. The study of the constructed systems showed that in the parameter space there are regions corresponding to both various regular regimes, such as the equilibrium position, limit cycle, two-frequency tori, and chaotic regimes. For both systems, it was shown that the transition to chaos occurs as a result of a sequence of period doubling bifurcations of the tori. In addition, a comparison of the dynamics of the constructed systems with the dynamics of the Vyshkind–Rabinovich and Rabinovich–Fabrikant models allows us to assert that if the Vyshkind–Rabinovich model predicts the dynamics of the corresponding initial system well enough, then the Rabinovich–Fabrikant model does not have such a property.

Effect of a modulated acoustic field on the dynamics of a vibrating charged bubble

We investigate the effect of amplitude-modulated acoustic irradiation on the dynamics of a charged bubble vibrating in a liquid. We show that the potential V(x) of the bubble, and the number and stability of its equilibria, depend on the magnitude of the charge it carries. Under high-frequency amplitude-modulation, a modulation threshold, Gth, was found for the onset of increased bubble amplitude oscillations. For some pressure field values, charge can facilitate the control of chaotic dynamics via reversed period-doubling bifurcation sequences. There is evidence for peak-shouldering and shock waves. The Mach number increases rapidly with the drive amplitude G. In the supersonic regime, for G>1.90Pa, the high-frequency modulation raises both Blake’s and the transient cavitation thresholds. We found a decrease in the bubble’s maximum charge threshold, and threshold modulation amplitude for the occurrence Vibrational resonance (VR). VR occurs due to the modulated oscillatory pressure field, and the influence on VR of the electrostatic charge, and other parameters of the system are investigated. In contrast to the cases of VR reported earlier, where the amplitude G of the high-frequency driving is typically much higher than the amplitude of the low-frequency driving (Ps), the VR resonance peaks occur here at relatively low G values (0<G<10Pa) compared to the acoustic driving pressure Ps∼105 Pa. The optimal parameter values for enhanced response could be useful in acoustic cavitation applications.

SPECTROSCOPY AND DYNAMICS OF NONLINEAR PROCESSES IN RELATIVISTIC BACKWARD-WAVE TUBE WITH ACCOUNTING FOR EFFECTS OF SPACE CHARGE, DISSIPATION AND WAVE REFLECTIONS

An effective universal, approach to solving the problems of quantitative modelling and analysis of the fundamental characteristics of spectroscopy and dynamics of the nonlinear processes in relativistic microwave electronics devices, in particular, a relativistic backward wave tube (RBWT) has been developed and implemented. It has been performed modelling, analysis, and prediction of chaotic dynamics of RBWT with simultaneous consideration of not only relativistic effects, but also the effects of dissipation, presence of space charge, wave reflections at the ends of the decelerating system, etc in a wide range of changes at different values of control parameters, which are characteristic for the distributed relativistic electron-wave self-oscillating systems. From physical viewpoint the transition to chaos in the dynamics of the studied RBWT occurs according to the scenario everywhere in the sequence of period-doubling bifurcations, but with the growth of relativism, the dynamics become fundamentally complicated with the alternation of quasi-harmonic and chaotic regimes (including the appearance of the "beak" effect on the bifurcation diagram) and the transition everywhere intermittency to the high-D attractor.

Numerical simulations of interfacial and elastic instabilities

<h2>Abstract</h2> The interplay of elasticity and capillarity in free-surface flows of viscoelastic fluids gives rise to intriguing flow instabilities. We present novel numerical simulations and analyse the physics related to the appearance of surface distortions on polymer melt extrudates, often referred to as "sharkskin" instability, and the abrupt increase in the rise velocity of a bubble in a viscoelastic solution. Regarding the extrusion process, the transition from smooth to wavy extrudate with increasing flow rate is attributed to a Hopf bifurcation, followed by a sequence of period-doubling bifurcations, which eventually lead to elastic turbulence under creeping flow. We suggest that the inception of these waves is related with intense stretch of polymer chains and subsequent recoil at the region where the melt detaches from the solid wall of the die. The ‘velocity discontinuity' of the rising bubble is attributed to a hysteresis loop, triggered by the change of shape of the rear pole into a tip that favors the formation of intense stress gradients, which reduce the fluid viscosity and facilitate the bubble translation. Finally, we discuss the numerical methodology that has allowed us to obtain stable numerical solutions in highly deformed meshes and for high values of the Weissenberg number.

Bifurcation Trees of (1:2)- Asymmetric Periodic Motions with Corresponding Infinite Homoclinic Orbits in the Lorenz System

In this paper, bifurcation dynamics and infinite homoclinic orbits of (1:2)- asymmetric periodic motions in the Lorenz systems are studied through the discrete mapping method. The infinite homoclinic orbits pertaining to the unstable periodic motions on the bifurcation trees of (1:2)-asymmetric periodic motions to chaos are determined. The stability and bifurcations of periodic motions are determined through the eigenvalue analysis. A bifurcation tree of (1:2) asymmetric periodic motions, varying with the Rayleigh number, is presented by discrete nodes, and the corresponding harmonic frequency-amplitude characteristics of periodic motions on the bifurcation tree are discussed through finite Fourier series analysis. A sequence of period-doubling bifurcations is observed on the bifurcation tree of the (1:2)- asymmetric periodic motion to chaos. The scenario of (1:2)- asymmetric period-1, period-2, period-4 motions to chaos scenario is illustrated. Homoclinic orbits are as appearance or vanishing of unstable periodic motions on the bifurcation tree. Illustrations of periodic motions and homoclinic orbits are completed for intuitive demonstrations of the corresponding topological structures. This study extends the initial study of the bifurcation tree from the (1:1)-symmetric periodic motions to asymmetric periodic motions then to chaos in the Lorenz system, and the corresponding infinite homoclinic orbits induced by all unstable periodic motions can be determined. Such results can enrich the understanding of bifurcation dynamics of asymmetric periodic motions to chaos in the Lorenz system.

Origin of the Sharkskin Instability: Nonlinear Dynamics.

The appearance of surface distortions on polymer melt extrudates, often referred to as sharkskin instability, is a long-standing problem. We report results of a simple physical model, which link the inception of surface defects with intense stretch of polymer chains and subsequent recoil at the region where the melt detaches from the solid wall of the die. The transition from smooth to wavy extrudate is attributed to a Hopf bifurcation, followed by a sequence of period doubling bifurcations, which eventually lead to elastic turbulence under creeping flow. The predicted flow profiles exhibit all the characteristics of the experimentally observed surface defects during polymer melt extrusion.

How Do You Feel About Going &lt;i&gt;Green&lt;/i&gt;? Modelling Environmental Sentiments in a Growing Open Economy

Contrary to the near global consensus among the scientific community, public perceptions of climate change are known to differ between nations and to have fluctuated over time. To study such a stylised fact, this article develops a small-scale agent-based model that allows for feedback effects between sentiments, environmental regulation and macroeconomic outcomes in an open economy set-up. Conditional on the level of interaction between agents, two different basins of attraction emerge: one with the majority of the population supporting climate change mitigation policies, and another with most agents opposing environmental regulation. We demonstrate that complex dynamics might occur via a sequence of period-doubling bifurcations. Employment series are more persistent than sentiments, resulting in relatively high volatility in the latter and smooth long-waves in the labour market. A sufficiently robust response of sentiments to emissions may lead to the disappearance of the Pareto inferior equilibria, allowing for a unique green steady-state.

Global Dynamics of Delayed Sigmoid Beverton–Holt Equation

In this paper, certain dynamic scenarios for general competitive maps in the plane are presented and applied to some cases of second-order difference equation xn+1=fxn,xn−1, n=0,1,…, where f is decreasing in the variable xn and increasing in the variable xn−1. As a case study, we use the difference equation xn+1=xn−12/cxn−12+dxn+f, n=0,1,…, where the initial conditions x−1,x0≥0 and the parameters satisfy c,d,f&gt;0. In this special case, we characterize completely the global dynamics of this equation by finding the basins of attraction of its equilibria and periodic solutions. We describe the global dynamics as a sequence of global transcritical or period-doubling bifurcations.

Dynamics of Convective Thermal Explosion in Porous Media

In this paper, we study complex dynamics of the interaction between natural convection and thermal explosion in porous media. This process is modeled with the nonlinear heat equation coupled with the nonstationary Darcy equation under the Boussinesq approximation for a fluid-saturated porous medium in a rectangular domain. Numerical simulations with the Radial Basis Functions Method (RBFM) reveal complex dynamics of solutions and transitions to chaos after a sequence of period doubling bifurcations. Several periodic windows alternate with chaotic regimes due to intermittence or crisis. After the last chaotic regime, a final periodic solution precedes transition to thermal explosion.

Nonlinear Dynamics of a Magnetically Supported Flexible Rotor in Auxiliary Bearings

The results of a numerical simulation on the nonlinear dynamics of a flexible rotor mounted on magnetic and auxiliary bearings are presented in this work. The focus of this work was to investigate the effect of shaft flexibility on the rotor response during load sharing operation between the magnetic and auxiliary bearings. For the range of parameters considered herein, results from the numerical simulation showed that the stiffness ratio, which represented the contact stiffness of the auxiliary bearing, was a more effective parameter, as compared to the Coulomb sliding friction coefficient, for the suppression of sub-synchronous and non-synchronous vibrations in the rotor’s response. Sub-synchronous vibrations of period-2, period-4, period-8 and period-16 were observed in the response of the rotor. Non-synchronous vibration which was also seen in the rotor’s response was determined to be chaos. For the case of the rotor response with variation of the stiffness ratio, the route to chaos was found to be due to a sequence of period-doubling bifurcations, where the synchronous or period-1 response bifurcated into period-2, period-4, period-8 and period-16 responses, culminating into chaotic vibration. Vibrations which are of sub-synchronous and non-synchronous nature are best avoided in the rotor’s response during operation, whereby the load is shared between magnetic and auxiliary bearings, as they produce stress reversals. Such stresses can potentially cause fatigue failure of the rotors and their associated structures.

Transition to turbulence in driven active matter.

A Lorenz-like model was set up recently to study the hydrodynamic instabilities in a driven active matter system. This Lorenz model differs from the standard one in that all three equations contain nonlinear terms. The additional nonlinear term comes from the active matter contribution to the stress tensor. In this work, we investigate the nonlinear properties of this Lorenz model both analytically and numerically. The significant feature of the model is the passage to chaos through a complete set of period-doubling bifurcations above the Hopf point for Schmidt numbers above a critical value. Interestingly enough, at these Schmidt numbers a strange attractor and stable fixed points coexist beyond the homoclinic point. At the Hopf point, the strange attractor disappears leaving a high-period periodic orbit. This periodic state becomes the expected limit cycle through a set of bifurcations and then undergoes a sequence of period-doubling bifurcations leading to the formation of a strange attractor. This is the first situation where a Lorenz-like model has shown a set of consecutive period-doubling bifurcations in a physically relevant transition to turbulence.

Spatial Memory and Taxis-Driven Pattern Formation in Model Ecosystems

Mathematical models of spatial population dynamics typically focus on the interplay between dispersal events and birth/death processes. However, for many animal communities, significant arrangement in space can occur on shorter timescales, where births and deaths are negligible. This phenomenon is particularly prevalent in populations of larger, vertebrate animals who often reproduce only once per year or less. To understand spatial arrangements of animal communities on such timescales, we use a class of diffusion–taxis equations for modelling inter-population movement responses between N ge 2 populations. These systems of equations incorporate the effect on animal movement of both the current presence of other populations and the memory of past presence encoded either in the environment or in the minds of animals. We give general criteria for the spontaneous formation of both stationary and oscillatory patterns, via linear pattern formation analysis. For N=2, we classify completely the pattern formation properties using a combination of linear analysis and nonlinear energy functionals. In this case, the only patterns that can occur asymptotically in time are stationary. However, for N ge 3, oscillatory patterns can occur asymptotically, giving rise to a sequence of period-doubling bifurcations leading to patterns with no obvious regularity, a hallmark of chaos. Our study highlights the importance of understanding between-population animal movement for understanding spatial species distributions, something that is typically ignored in species distribution modelling, and so develops a new paradigm for spatial population dynamics.

Wake dynamics of low-Reynolds-number flow around a two-dimensional airfoil

We perform two-dimensional simulations of unsteady flow separation around a NACA0015 airfoil. We consider five different angles of attack, 10°, 12.5°, 15°, 17.5°, and 20°, with the Reynolds number varying from 100 to 1300. The central aim is to study the wake dynamics under the low-Reynolds-number condition, when the flow is most likely experiencing transitions between different states. For the low angles of attack of α = 10° and α = 12.5°, we find different branches on the Strouhal-Reynolds number relationship curves. Specifically, for α = 10°, we identify four distinct branches, marked with L1 to L4, in which the second discontinuity between L3 and L4 is proved to be hysteretic, characterized by the wake deflection. The branches L1 and L2 differ from the other two in the near-body flow topology, with the “trailing-edge vortex” mode for the first two, while the “separation vortex” mode for the last two. For the high angles of attack, α = 15°, 17.5°, and 20°, we cannot find marked discontinuity on the Strouhal-Reynolds number relationship curve. We further demonstrate that the flow transits to a chaotic state by a sequence of successive period-doubling bifurcations at α = 20°. Our results suggest that the operation of micro unmanned air vehicles in very low Reynolds numbers should be treated with caution because the flow is very sensitive to the Reynolds number, as well as the angle of attack.

Dynamic analysis of a novel jerk system with composite tanh-cubic nonlinearity: chaos, multi-scroll, and multiple coexisting attractors

It is well known that the dynamics of very simple physical systems can be quite complex if a sufficient amount of nonlinearity is present. In this contribution, a novel jerk system with smooth composite tanh-cubic nonlinearity is proposed and investigated. Interestingly, the new nonlinearity takes advantage of the classical smooth cubic polynomial in the sense that it induces more complex and interesting dynamics (e.g. five equilibria instead of three in the case of a (traditional) cubic nonlinearity, multi-scroll, and multistability). The fundamental properties of the model are discussed including equilibria and stability, phase portraits, Poincare sections, bifurcation diagrams and Lyapunov exponent’s plots. Period doubling bifurcation, antimonotonicity, chaos, hysteresis, and coexisting bifurcations are reported. In particular, a rare phenomenon is found in which two different pairs of coexisting limit cycles born from the Hopf bifurcation follow each a different sequence of period-doubling bifurcations, then merge to form a three-scroll chaotic attractor as a parameter is smoothly changed. As another major result of this work, several windows in the parameter space are depicted in which the novel jerk system develops the striking behaviour of multiple coexisting attractors (i.e. coexistence of three, four, six, or eight disjoint periodic and chaotic attractors for the same parameters set) due to the combined effects of three families of parallel bifurcation branches and hysteresis. An electronic analogue of the new jerk system is designed and simulated in PSPICE. As far as the authors’ knowledge goes, no example of such a simple and ‘elegant’ 3D autonomous system with this combination of features is reported in the relevant literature.

Multi-Gbit/s optical phase chaos communications using a time-delayed optoelectronic oscillator with a three-wave interferometer nonlinearity.

We propose a chaos communication scheme based on a chaotic optical phase carrier generated with an optoelectronic oscillator with nonlinear time-delay feedback. The system includes a dedicated non-local nonlinearity, which is a customized three-wave imbalanced interferometer. This particular feature increases the complexity of the chaotic waveform and thus the security of the transmitted information, as these interferometers are characterized by four independent parameters which are part of the secret key for the chaos encryption scheme. We first analyze the route to chaos in the system, and evidence a sequence of period doubling bifurcations from the steady-state to fully developed chaos. Then, in the chaotic regime, we study the synchronization between the emitter and the receiver, and achieve chaotic carrier cancellation with a signal-to-noise ratio up to 20 dB. We finally demonstrate error-free chaos communications at a data rate of 3 Gbit/s.

Dynamic analysis of nonlinear variable frequency water supply system with time delay

In this paper, we study the mathematical model about variable frequency water supply system in the process of applying glue to particleboard. Based on the original linear ordinary differential model, the effects of time delay and the nonlinear factor are considered. Then, we obtain the delayed nonlinear differential equation associated with variable frequency water supply system. Further, we consider the existence and stability of the equilibria and the existence of several types of bifurcations in this functional differential equation. Next, we derive the normal forms of Hopf bifurcation and Bogdanov–Takens bifurcation by using the multiple time scales method and the center manifold reduction method, respectively, and analyze the classifications of local dynamics. Finally, by using Matlab software, we obtain numerical simulations with estimated values of parameters and show the existence of stable equilibrium, stable periodic-1, periodic-2, and periodic-4 solutions, and a complex chaotic attractor from a sequence of period-doubling bifurcations.

Dissipative solitons with extreme spikes: bifurcation diagrams in the anomalous dispersion regime

Dissipative solitons with extreme spikes (DSESs), previously thought to be rare solutions of the complex cubic–quintic Ginzburg–Landau equation, occupy in fact a significant region in its parameter space. The variation of any of its five parameters results in a rich structure of bifurcations. We have constructed several bifurcation diagrams that reveal periodic and chaotic dynamics of DSESs. There are various routes to the chaotic behavior of DSESs, including a sequence of period-doubling bifurcations. It is well known that the complex cubic–quintic Ginzburg–Landau equation can serve as a master equation for the description of passively mode-locked lasers. Our results may lead to the observation of DSESs in laser systems.