Multiple Bifurcations Research Articles

On the Nonlinear Dynamical System of Amplitude Equations Corresponding to Intersections of Bifurcations in the Flow Between Permeable Cylinders with Radial and Axial Flows

In the present work, we study various modes that arise after a circular Couette flow loses its stability in the presence of an axial flow due to a constant axial pressure gradient and a radial flow through the permeable walls of the cylinders. The basic object of our investigation is a nonlinear system of amplitude equations describing multiple flow bifurcations between permeable cylinders. Different bifurcations will be investigated theoretically with application of numerical methods.

Bifurcation analysis of laminar isothermal planar opposed-jet flow

For a fixed geometric configuration, hydrodynamic instabilities and bifurcation processes of laminar isothermal planar opposed-jet flows with symmetric and slightly asymmetric inlet boundary conditions are investigated numerically using a high-resolution approach based on spectral element method. In current configuration, when inlet boundary conditions are symmetric, in the range of the Reynolds number considered (Re ≤ 200), multiple new symmetry-breaking bifurcations are observed and four new flow patterns are identified. Their hydrodynamic characteristics are analyzed, in particular their symmetries. In addition, the case that inlet boundary conditions are slightly asymmetric is investigated. It is found that bifurcation processes are extremely sensitive to this small symmetry-breaking imperfection and much different from those in the symmetric case. Furthermore, model equations are constructed by symmetry consideration to explain the numerical results based on hydrodynamic equations.

Analysis of vibrational resonance in bi-harmonically driven plasma.

The phenomenon of vibrational resonance (VR) is examined and analyzed in a bi-harmonically driven two-fluid plasma model with nonlinear dissipation. An equation for the slow oscillations of the system is analytically derived in terms of the parameters of the fast signal using the method of direct separation of motion. The presence of a high frequency externally applied electric field is found to significantly modify the system's dynamics, and consequently, induce VR. The origin of the VR in the plasma model has been identified, not only from the effective plasma potential but also from the contributions of the effective nonlinear dissipation. Beside several dynamical changes, including multiple symmetry-breaking bifurcations, attractor escapes, and reversed period-doubling bifurcations, numerical simulations also revealed the occurrence of single and double resonances induced by symmetry breaking bifurcations.

Bifurcation and stability of a Mimura–Tsujikawa model with nonlocal delay effect

In this paper, we investigate a Mimura–Tsujikawa model with nonlocal delay effect under the homogeneous Neumann boundary condition. By using Lyapunov–Schmidt reduction, we investigate the existence, multiplicity, stability, and Hopf bifurcation of nontrivial steady‐state solutions bifurcating from the nonzero steady‐state solution. Moreover, we illustrate our general results by applications to models with a one‐dimensional spatial domain. Copyright © 2016 John Wiley & Sons, Ltd.

A small animal model for early cerebral aneurysm pathology

Prior studies, using systemic hypertension and elastase infusion, have induced cerebral aneurysm (CA) formation in mice. However, the CAs induced were rapidly formed, relatively large, and often ruptured. These features are not completely representative of human CAs. We set out to develop a mouse model representative of the early pathological features of human CA. Twenty male C57/BL6 mice were placed in a stereotactic frame. Low dose elastase solution (2μl/min) was manually injected into the right basal cistern. Human angiotensin II (0.11μl/h) was infused subcutaneously. Mice were observed for 2–3weeks prior to euthanasia. Early CA histopathological features including endothelial change (EC) and internal elastic lamina degeneration (IELD) were systematically sought at major cerebral arterial bifurcations. Brains were harvested from 11 of 15 mice, yielding 27 bifurcations. Sub-arachnoid haemorrhage (SAH) without CA formation was observed in one brain. Macroscopic CA without SAH was observed in another brain. Early CA features were observed in 8/11 (73%) brains. All bifurcations with IELD demonstrated EC: where EC was absent, IELD was also absent. EC severity appeared to correlate with IELD severity. EC and IELD were both severe within the CA. Using lower dose elastase solution than previously employed, we developed a model of early CA pathology. Our model demonstrated that the spectrum of known early CA pathology can be created at multiple bifurcations in mice, with EC severity appearing to correlate with IELD severity. This model permits the study of factors which potentially advance or retard the progression of CA formation.

Dynamics, Analysis and Implementation of a Multiscroll Memristor-Based Chaotic Circuit

This article introduces a novel four-dimensional autonomous multiscroll chaotic circuit which is derived from the actual simplest memristor-based chaotic circuit. A fourth circuit element — another inductor — is introduced to generate the complex behavior observed. A systematic study of the chaotic behavior is performed with the help of some nonlinear tools such as Lyapunov exponents, phase portraits, and bifurcation diagrams. Multiple scroll attractors are observed in Matlab, Pspice environments and also experimentally. We also observe the phenomenon of antimonotonicity, periodic and chaotic bubbles, multiple periodic-doubling bifurcations, Hopf bifurcations, crises and the phenomenon of intermittency. The chaotic dynamics of this circuit is realized by laboratory experiments, Pspice simulations, numerical and analytical investigations. It is observed that the results from the three environments agree to a great extent. This topology is likely convenient to be used to intentionally generate chaos in memristor-based chaotic circuit applications, given the fact that multiscroll chaotic systems have found important applications as broadband signal generators, pseudorandom number generators for communication engineering and also in biometric authentication.

Mini-Workshop: Mathematics of Differential Growth, Morphogenesis, and Pattern Selection

Living structures are highly heterogeneous systems that consist of distinct regions made up of characteristic cell types with a specific structural organization. During evolution, development, disease, or environmental adaptation each region may grow at its own characteristic rate. Differential growth creates a balanced interplay between tension and compression and plays a critical role in biological function. In plant physiology, typical every-day examples include the petioles of celery, caladium, or rhubarb with a slower growing compressive outer surface and a faster growing tensile inner core. In developmental biology, differential growth is critical to the organogenesis of various structures including the gut, the heart, and the brain. From a structural point of view, these phenomena are close associated with instabilities, of twisting, looping, folding, and wrinkling. From a mathematical point of view, the governing equations of organogenesis are highly nonlinear and often characterized through multiple bifurcation points. Bifurcation is critical in symmetry breaking, pattern formation, and selection of shape. While biologists are studying differential growth, morphogenesis, and pattern selection merely by observation, our goal in this workshop is to explore, discuss, and advance the fundamental theory of differential growth to characterize morphogenesis and pattern selection by mathematical modeling. This workshop will bring together scientists with similar interests and complementary backgrounds in applied mathematics, mathematical biology, developmental biology, plant biology, dynamical systems, biophysics, biomechanics, and clinical sciences. We will identify common features of growth phenomena in living systems with the overall objectives to establish a unified mathematical theory for growing systems and to identify the necessary mathematical tools to address challenging questions in biology and medicine.

Multiple Bifurcations and Dynamics of a Discrete Time Predator-Prey System with Group Defense and Non-Monotonic Functional Response

In this paper, using the forward Euler method, we present a discrete predator-prey model with group defense and non-monotonic functional response. We study the stability of fixed points and analyze the bifurcations phenomena. Using the ideas of center manifold theorem, bifurcation theory and normal form methods we prove that the system undergoes fold bifurcation, flip bifurcation, Neimark-Sacker bifurcation and codimension two (i.e. 1:1 resonance) bifurcation. At the same time, complex dynamics and chaotic behaviors are justified numerically via computing the maximum Lyapunov exponent. Furthermore, we show that for some parameter values the system exhibits the so-called “paradox of enrichment” phenomenon.

Strength and weaknesses of modeling the dynamics of mode-locked lasers by means of collective coordinates

We address the efficiency of theoretical tools used in the development and optimization of mode-locked fiber lasers. Our discussion is based on the practical case of modeling the dynamics of a dispersion-managed fiber laser. One conventional approach uses discrete propagation equations, followed by the analysis of the numerical results through a collective coordinate projection. We compare the latter with our dynamical collective coordinate approach (DCCA), which combines both modeling and analysis in a compact form. We show that for single pulse dynamics, the DCCA allows a much quicker solution mapping in the space of cavity parameters than the conventional approach, along with a good accuracy. We also discuss the weaknesses of the DCCA, in particular when multiple pulsing bifurcations occur.

Multiple bifurcations in the periodic orbit around Eros

We investigate the multiple bifurcations in periodic orbit families in the potential field of a highly irregular-shaped celestial body. Topological cases of periodic orbits and four kinds of basic bifurcations in periodic orbit families are studied. Multiple bifurcations in periodic orbit families consist of four kinds of basic bifurcations. We found both binary period-doubling bifurcations and binary tangent bifurcations in periodic orbit families around asteroid 433 Eros. The periodic orbit family with binary period-doubling bifurcations is nearly circular, with almost zero inclination, and is reversed relative to the body of the asteroid 433 Eros. This implies that there are two stable regions separated by one unstable region for the motion around this asteroid. In addition, we found triple bifurcations which consist of two real saddle bifurcations and one period-doubling bifurcation. A periodic orbit family generated from an equilibrium point of asteroid 433 Eros has five bifurcations, which are one real saddle bifurcation, two tangent bifurcations, and two period-doubling bifurcations.

Multiple limit cycles bifurcation from the degenerate singularity for a class of three-dimensional systems

In this paper, bifurcation of small amplitude limit cycles from the degenerate equilibrium of a three-dimensional system is investigated. Firstly, the method to calculate the focal values at nilpotent critical point on center manifold is discussed. Then an example is studied, by computing the quasi-Lyapunov constants, the existence of at least 4 limit cycles on the center manifold is proved. In terms of degenerate singularity in high-dimensional systems, our work is new.

Decrease in Erythrocyte Deformability Promotes the Cell Accumulation near the Vessel Wall in Small Arterioles

This study demonstrated that the deformability‐impaired red blood cells (RBCs) preferentially accumulate near the vessel walls in arterioles in vivo. Chemically hardened RBCs (hRBCs) with fluorescent labeling from the donor rat were infused into the recipient rat circulatory system while maintaining the total blood volume. Fluorescent imaging analysis was performed to quantify the radial (R) distribution of the labeled hRBCs in small arterioles (~76 μm of diameter). In vivo experimental results showed that majority of the hRBCs (65%) were accumulated near the vessel wall (0.7R–0.9R). This near‐wall accumulation of hRBCs partly contributed to the decrease in the mean cell‐free layer width, and significantly increased the variation of cell‐free layer width (P < 0.05). Dynamic particle simulation by modeling individual RBCs was in agreement with the in vivo experiments by predicting that the hRBCs exhibit relatively weakened tank‐treading motion as compared with normal RBCs. Consequently, this leads to an attenuation in the inward migration of the cells toward the flow center. Our findings conclusively proposed that the preferential near‐wall accumulation of hRBCs in vivo could be attributed by the attenuated dispersion of hRBCs with reduced tank‐treading motion once they entered into the near‐wall zone. In addition, frequent sudden changes of lateral displacement of hRBCs while passing through multiple bifurcations partly contributed to their near‐wall accumulation despite the relatively shorter cell travel distance in vivo.Support or Funding InformationThis work was supported by National Medical Research Council (NMRC)/Cooperative Basic Research Grant (CBRG) No./0078/2014

Multiple bifurcation trees of period-1 motions to chaos in a periodically forced, time-delayed, hardening Duffing oscillator

In this paper, bifurcation trees of periodic motions in a periodically forced, time-delayed, hardening Duffing oscillator are analytically predicted by a semi-analytical method. Such a semi-analytical method is based on the differential equation discretization of the time-delayed, nonlinear dynamical system. Bifurcation trees for the stable and unstable solutions of periodic motions to chaos in such a time-delayed, Duffing oscillator are achieved analytically. From the finite discrete Fourier series, harmonic frequency-amplitude curves for stable and unstable solutions of period-1 to period-4 motions are developed for a better understanding of quantity levels, singularity and catastrophes of harmonic amplitudes in the frequency domain. From the analytical prediction, numerical results of periodic motions in the time-delayed, hardening Duffing oscillator are completed. Through the numerical illustrations, the complexity and asymmetry of period-1 motions to chaos in nonlinear dynamical systems are strongly dependent on the distributions and quantity levels of harmonic amplitudes. With the quantity level increases of specific harmonic amplitudes, effects of the corresponding harmonics on the periodic motions become strong, and the certain complexity and asymmetry of periodic motion and chaos can be identified through harmonic amplitudes with higher quantity levels.

Multiple Bifurcations of a Cylindrical Dynamical System

Abstract This paper focuses on multiple bifurcations of a cylindrical dynamical system, which is evolved from a rotating pendulum with SD oscillator. The rotating pendulum system exhibits the coupling dynamics property of the bistable state and conventional pendulum with the ho- moclinic orbits of the first and second type. A double Andronov-Hopf bifurcation, two saddle-node bifurcations of periodic orbits and a pair of homoclinic bifurcations are detected by using analytical analysis and nu- merical calculation. It is found that the homoclinic orbits of the second type can bifurcate into a pair of rotational limit cycles, coexisting with the oscillating limit cycle. Additionally, the results obtained herein, are helpful to explore different types of limit cycles and the complex dynamic bifurcation of cylindrical dynamical system.

Symmetric periodic solutions of delay-coupled optoelectronic oscillators

Delay-coupled optoelectronic oscillators are considered. These structures are based on mutually coupled oscillators which oscillate at the same frequency. By taking the time delay as a bifurcation parameter, the stability of the zero equilibrium and the existence of Hopf bifurcations induced by delay are investigated, and then stability switches for the trivial solution are found. Conditions ensuring the stability and direction of the Hopf bifurcation are determined by applying the normal form theory and the center manifold theorem. Using the symmetric functional differential equation theories combined with the representation theory of Lie groups, the multiple Hopf bifurcations of the equilibrium are demonstrated. In particular, we find that the spatio-temporal patterns of bifurcating periodic oscillations will alternate according to the change of the propagation time delay in the coupling. The existence of multiple branches of bifurcating periodic solutions and their spatio-temporal patterns are obtained. Some numerical simulations are used to illustrate the effectiveness of the obtained results.

Competition between drift and spatial defects leads to oscillatory and excitable dynamics of dissipative solitons.

We have reported in Phys. Rev. Lett. 110, 064103 (2013)PRLTAO0031-900710.1103/PhysRevLett.110.064103 that in systems which otherwise do not show oscillatory dynamics, the interplay between pinning to a defect and pulling by drift allows the system to exhibit excitability and oscillations. Here we build on this work and present a detailed bifurcation analysis of the various dynamical instabilities that result from the competition between a pulling force generated by the drift and a pinning of the solitons to spatial defects. We show that oscillatory and excitable dynamics of dissipative solitons find their origin in multiple codimension-2 bifurcation points. Moreover, we demonstrate that the mechanisms leading to these dynamical regimes are generic for any system admitting dissipative solitons.

Complex dynamics in biological systems arising from multiple limit cycle bifurcation

ABSTRACTIn this paper, we study complex dynamical behaviour in biological systems due to multiple limit cycles bifurcation. We use simple epidemic and predator–prey models to show exact routes to new types of bistability, that is, bistability between equilibrium and periodic oscillation, and bistability between two oscillations, which may more realistically describe the real situations. Bifurcation theory and normal form theory are applied to investigate the multiple limit cycles bifurcating from Hopf critical point.

0371: Bifurcation techniques with ABSORB® bioresorbable vascular scaffold: optical frequency-domain imaging and micro-computed tomography assessment

We aimed to determine the correlation between optical frequency-domain imaging (OFDI) and micro-computed tomography (mCT) in the quantitative and qualitative assessment of ABSORB® bioresorbable vascular scaffold (BVS) in a bench test of multiple bifurcation technique. BVS were deployed in bifurcation silicon phantoms divided into two benches. Bench A is a LAD-diagonal bifurcation with a 40° angle and bench B, a left main bifurcation with a 70° angle. Finet’s law was respected. Different bifurcation techniques (provisional T stenting, kissing balloon, mini crush and culotte) were performed with a total of 16 procedures, 8 for each bench. All procedures were imaged by OFDI and mCT. BVS area, lumen area, number of struts, maximal stent thickness, disruption, lumen protrusion and deformation were the parameters collected. The analysis was performed in each segment, proximal, bifurcation and distal. In bench A, we found no differences between OFDI and mCT for each proximal, bifurcation and distal segments concerning BVS area, lumen area, number of struts, maximal stent thickness, fractures, lumen protrusion, BVS deformation or BVS fragments. In the proximal segment, length was higher in mCT (13.2mm vs 10.4mm; p<0.0001) and there was more malapposition detected in mCT (75% vs 0%, p=0.007). In bench B, for the proximal segment, lumen area was higher (13.7mm 2 vs 13.mm 2 , p=0.01) and we found more BVS malapposition in mCT (100% vs 0%, p =0.0002). All the other parameters were similar. In each bench, the apposition defect measured by mCT remained minimal. We demonstrated an excellent correlation between OFDI and mCT which is considered as the gold standard to evaluate stent in bench tests. In vitro techniques can now be validated in vivo using OFDI.

Group Equivariant Singularity Theory in the Bifurcation of Interfacial Wilton Ripples

An investigation is made into the capillary-gravity waves which arise on the interface of two fluids and which are formed by the interaction of the first two harmonics of the motion. The problem is transformed into a nonlinear operator equation between suitable function spaces which is shown to be invariant under certain group actions. The infinite dimensional problem is reduced, by the classical procedure of Lyapunov–Schmidt, to a finite system of polynomial equations, known as the bifurcation equations. Because these equations inherit the symmetry properties of the original operator, it is possible to make quite specific statements concerning their structure, thus rendering their analysis easier. Solutions to the equations are sought, both in the cases of exact and of near-resonance. A wide variety of solutions is found depending on the values of the parameters: both simple, multiple and secondary bifurcations may occur, and in addition, there may exist isolated solution curves.

Multiple bifurcations and local energy minimizers in thermoelastic martensitic transformations

Thermoelastic martensitic transformations in shape memory alloys can be modeled on the basis of nonlinear elastic theory. Microstructures of fine phase mixtures are local energy minimizers of the total energy. Using a one-dimensional effective model, we have shown that such microstructures are inhomogeneous solutions of the nonlinear Euler–Lagrange equation and can appear upon loading or unloading to certain critical conditions, the bifurcation conditions. A hybrid numerical method is utilized to calculate the inhomogeneous solutions with a large number of interfaces. The characteristics of the solutions are clarified by three parameters: the number of interfaces, the interface thickness, and the oscillating amplitude. Approximated analytical expressions are obtained for the interface and inhomogeneity energies through the numerical solutions.