Bifurcation Structure Research Articles

Bifurcation and stability for charged drops

In this paper, we investigate the Laplace’s equation for the electrical potential of charge drops on exterior domain, and overdetermined boundary conditions are prescribed. We determine the local bifurcation structure with respect to the surface tension coefficient as bifurcation parameter. Furthermore, we establish the stability and the instability near the bifurcation point.

Enhancement of thermal management for cylindrical battery module based on a novel wrench-shaped design for the cold plate

Battery modules with uniform cold plate designs suffer from the temperature non-uniformity issue caused by increased coolant temperature. To break the monotonically increasing trend of battery temperature along the flow direction, a novel wrench-shaped design is developed in this study for the cold plate applied in cylindrical battery modules. Superiority of the wrench-shaped cold plate is examined from the perspectives of temperature rise control, temperature uniformity maintenance, lightweight extent and auxiliary power consumption. Corresponding mechanism analyses reveal that the different cold plate designs in the upstream and downstream segments have changed the temperature distribution feature in the battery module. The monotonically increasing trend of the battery temperature could be effectively broken when the cooling ability of the cold plate at the junction between the upstream and downstream segments is specially enhanced. Thus, the superior heat dissipation performance of the wrench-shaped cold plate is attributed to its bifurcation structure at the junction which demonstrates satisfactorily enhanced heat dissipation effect. Extended study is also performed to identify optimal position for the bifurcation, which is located at battery 4 based on the balance between the thermal behaviors of upstream and downstream segments.

Abstract B013: T cell clonotype expansion is common in advanced renal cell carcinoma but is not associated with altered response to PD-1 blockade

Abstract The diversity of T cell receptors (TCRs) becomes increasingly restricted with advancing stage in renal cell carcinoma (RCC), with individual T cell clones becoming dramatically expanded. Further, prior studies have suggested that TCR clonality (i.e. less diversity) could be associated with improved clinical response to anti-PD-1 treatment. To further interrogate the TCR landscape of advanced RCC and its relation to both T cell phenotype and clinical outcomes, single-cell TCR sequencing (scTCR-seq) was performed in parallel with single-cell RNA sequencing (scRNA-seq) for eligible patients with clear cell or non-clear cell metastatic RCC both pre- (baseline) or post- (progression) treatment with front-line nivolumab monotherapy as part of the HCRN GU16-260 trial (NCT03117309). Clonotype assignments and diversity metrics for 19 tumor samples were determined using VDJdive (R/Bioconductor). Overall, the top 10 clonotypes accounted for 12.3% of all T cells (range by sample: 4.2-34.6%). Of note, 5 of 16 (31%) samples with at least 100 T cells displayed substantial clonotype expansion, with the top 10 clonotypes accounting for at least 20% of tumor-infiltrating T cells. Average TCR diversity (normalized Shannon entropy) was 0.55 (range: 0.38-0.91) for all 19 samples. There were no significant differences in T cell diversity both between baseline (n=10) and post-treatment (n=9) samples (p= 0.66) or between clear cell (n=14) and non-clear cell (n=5) histologies (p= 0.75). To assess the relationship between T cell specificity and phenotypic state, we investigated the distribution of the most expanded clonotypes along a trajectory of T cells. Previous trajectory analysis of T cell populations revealed a bifurcating structure with naïve T cells at the root and lineages terminating in either terminally exhausted CD8+ T cells or SLAMF7+ CD8+ T cells (Braun, ASCO GU, 2023). Of the top 20 most expanded clonotypes across all T cells, 95% were distributed almost exclusively in the terminally exhausted lineage. The TCR diversity decreased along each lineage, from naïve T cell cluster at the root (TCR diversity = 0.61) to either the terminally exhausted CD8+ T cell cluster (0.42) or the SLAMF7+ CD8+ T cell cluster (0.56). There were no significant differences in TCR diversity between patients with progressive disease (n=6) versus complete/partial response (n= 6), (p= 0.96). Similarly, there was no association between TCR diversity and either progression-free (p= 0.41) or overall (p=0.39) survival. Altogether, these data reaffirm the presence of clonal expansion in advanced RCC, with the degree varying by sample. The localization of most expanded clonotypes to a single T cell lineage suggests a low degree of phenotypic plasticity. Enrichment of these clonotypes along an exhausted lineage is consistent with prior findings of lower TCR diversity in terminally exhausted CD8+ T cells. Future directions will focus on inferring the tumor specificity of infiltrating T cells along the diverging lineages. Citation Format: Miya B. Hugaboom, Neil Ruthen, Opeyemi Jegede, Nicholas R. Schindler, Lena Wirth, Sasha Kyrysyuk, David F. McDermott, Elizabeth R. Plimack, Jeffrey A. Sosman, Naomi B. Haas, Michael E. Hurwitz, Hans J. Hammers, Sabina Signoretti, Michael B. Atkins, Catherine J. Wu, David A. Braun, Kelly Street. T cell clonotype expansion is common in advanced renal cell carcinoma but is not associated with altered response to PD-1 blockade [abstract]. In: Proceedings of the AACR Special Conference: Advances in Kidney Cancer Research; 2023 Jun 24-27; Austin, Texas. Philadelphia (PA): AACR; Cancer Res 2023;83(16 Suppl):Abstract nr B013.

Bifurcation instructed design of multistate machines

We propose a design paradigm for multistate machines where transitions from one state to another are organized by bifurcations of multiple equilibria of the energy landscape describing the collective interactions of the machine components. This design paradigm is attractive since, near bifurcations, small variations in a few control parameters can result in large changes to the system's state providing an emergent lever mechanism. Further, the topological configuration of transitions between states near such bifurcations ensures robust operation, making the machine less sensitive to fabrication errors and noise. To design such machines, we develop and implement a new efficient algorithm that searches for interactions between the machine components that give rise to energy landscapes with these bifurcation structures. We demonstrate a proof of concept for this approach by designing magnetoelastic machines whose motions are primarily guided by their magnetic energy landscapes and show that by operating near bifurcations we can achieve multiple transition pathways between states. This proof of concept demonstration illustrates the power of this approach, which could be especially useful for soft robotics and at the microscale where typical macroscale designs are difficult to implement.

The occurrence mechanisms of extreme events in a class of nonlinear Duffing-type systems under random excitations.

The occurrence mechanisms of extreme events under random disturbances are relatively complex and not yet clear. In this paper, we take a class of generalized Duffing-type systems as an example to reveal three mechanisms for the occurrence of extreme events. First, it is intuitive that a very large excitation can generate extreme events, such as the Lévy noise. In such a case, extreme excitation works, while it does not require much about the systems. Second, when a system has a bifurcation structure, if the difference of the branches at the bifurcation point is large, a randomly varying bifurcation parameter can lead to extreme events. Finally, when a system has rare attractors, a random impulse excitation, such as Poisson white noise, is able to cause the system to escape from one general attractor into rare attractors. Such a kind of special regime switching behavior can lead to extreme events. These results reveal the possible mechanisms of extreme events in a class of nonlinear Duffing-type systems and provide guidance for further prediction and avoidance of extreme events.

Stability and Bautin bifurcation of four-wheel-steering vehicle system with driver steering control.

In this paper, the stability and Bautin bifurcation of a four-wheel-steering (4WS) vehicle system, by considering driver steering control, are investigated. By using the central manifold theory and projection method, the first and second Lyapunov coefficients are calculated to predict the type of Hopf bifurcation of the vehicle system. The topological structure of Bautin bifurcation, a degenerate Hopf bifurcation of the 4WS vehicle system, is presented in parameter space, and it reveals the dynamics of the vehicle system of different choices of control parameters. The influences of system parameters on critical values of the bifurcation parameter are also analyzed. It is shown that with the increase in the frontal visibility distance of the driver control strategy coefficient and the cornering stiffness coefficients of rear wheels, the critical speed increases. Nevertheless, the critical speed decreases with the increase in the distance from the center of gravity of the vehicle to the front axles, Driver's perceptual time delay, and cornering stiffness coefficients of the front wheels.

Dynamical behavior analysis of the Van der Pol oscillator with sine nonlinearity subjected to non-sinusoidal periodic excitations by the bifurcation structures

The present work studies the dynamical behavior of the Van der Pol oscillator with sine nonlinearity subjected to the effects of non-sinusoidal excitations through the analysis of bifurcation structures. In this work, the classical Van der Pol oscillator is modified by replacing the linear term with the nonlinear term The idea is to see the impact of the strength of this function on the appearance of chaotic dynamics for small and large values of the nonlinear dissipation term ε. To do this, studies using numerical simulation and analogue simulation are proposed. Firstly, using nonlinear analysis tools, a similarity in the bifurcation sequences is observed despite a difference in the ranks at which the particular behaviours and bifurcation points are obtained. This study is confirmed by their respective maximum Lyapunov exponents. Secondly, a real implementation using microcontroller technology, motivated by the use of an artificial pacemaker, is carried out. For a more practical case, the excitation signal is generated by another microcontroller. The study shows a similarity with the results obtained numerically. Thirdly, in order to show that the model can be derived from mathematical modelling, an electronic simulation using OrCAD-Pspice software is also proposed. The results obtained are in qualitative agreement.

An efficient fog collector inspired by Tillandsia and cactus spines

When the direction of the fog flow changes, the fog collection efficiency of the common fog collector relying on Janus will be affected. Inspired byTillandsia and cacti, three kinds of fog collectors with different bifurcation structures that are not affected by the direction of fog flow are designed. The fog collecting efficiency of sample 3 can reach 2.26 times that of the modified cylinder, and the fog collecting efficiency is even close to 0.9 g·h−1·cm−2. In addition, the fog collecting efficiency of sample 3 can reach 12.17 times that of the unmodified cylinder. In addition, the sample can still maintain a high fog collecting level after ten fog collecting cycles.

Coexistence-segregation dichotomy in the full cross-diffusion limit of the stationary SKT model

This paper studies the Lotka-Volterra competition model with cross-diffusion terms under homogeneous Dirichlet boundary conditions. We consider the asymptotic behavior of positive steady-states as equal two cross-diffusion coefficients tend to infinity (so-called the full cross-diffusion limit). The first result shows that, at the full cross-diffusion limit, the set of positive steady-state solutions can be classified into two types: the small coexistence or the complete segregation. The small coexistence can be characterized by the set of positive solutions of a nonlinear elliptic equation, while the complete segregation can be characterized by the set of nodal solutions of another nonlinear elliptic equation. The second result concerns the global bifurcation structure of the 1d model with large cross-diffusion terms to show that the branch of small coexistence bifurcates from the trivial solution and many branches of complete segregation bifurcate from solutions on the branch of small coexistence. Finally, in order to verify the theoretical results, we chase some bifurcation branches numerically via the continuation software pde2path.

Emergence of collapsed snaking related dark and bright Kerr dissipative solitons with quartic-quadratic dispersion.

We theoretically investigate the dynamics, bifurcation structure, and stability of dark localized states emerging in Kerr cavities in the presence of positive second- and fourth-order dispersion. In this previously unexplored regime, dark states form through the locking of uniform wave fronts, or domain walls, connecting two coexisting stable uniform states, and undergo a generic bifurcation structure known as collapsed homoclinic snaking. We characterize the robustness of these states by computing their stability and bifurcation structure as a function of the main control parameter of the system. Furthermore, we show that by increasing the dispersion of fourth order, bright localized states can be also stabilized.

Analysis of the response of damped and parametrically driven, strongly anharmonic Klein-Gordon chain - Part 2: Beat-waves

This paper continues a series of publications devoted to the analysis of strongly nonlinear wave phenomena, emerging in genuinely anharmonic, Klein-Gordon cyclic chains, subjected to a traveling-wave parametric excitation. In this study, we analyze a special type of nonlinear waves, which will be further referred to as beat-waves. These are spatially extended, and strongly modulated waves manifested by the extreme amplitude modulation, which can be of stationary or nonstationary type. Stationary beat-waves of Klein-Gordon chains propagate with constant group velocity, that depends on the beat-wave amplitude. In the vicinity of 2:1 parametric resonance, we derive the parametrically driven, discrete p-Schrödinger (PD-DpS) model, which depicts the slow modulation of the response of a parametric oscillator chain. Special coordinates' transformation applied on the PD-DpS model allows the exact reduction of extremely modulated beat-waves to an invariant manifold of this slow-flow system. The system dynamics reduced to the beat-wave invariant manifold is depicted by three collective coordinates. In the undamped case, the analysis of the slow-flow system dynamics on the invariant manifold reveals a special family of nonlinear beat-waves characterized by an extreme amplitude modulation. Among these beat-wave solutions lying on the manifold, our analysis identifies some special solutions manifested by the solitary-pulse like, temporal evolution of the beat-wave amplitude which can either decay to zero or to a constant nonzero value. We derive the exact analytical solutions for these special types of waves as well as obtain an explicit analytical approximation for the beat-waves of other types. In the damped case, analysis of the slow-flow system dynamics on the invariant manifold fully recovers the mechanism of system synchronization on the special, stationary beat-wave solutions characterized by the harmonically modulated envelope. Importantly, the derived reduced order model accurately captures the transient evolution stage as well as the steady-state stage of the collective beat-wave response of PD-DpS model, initialized on the beat-wave invariant manifold. A detailed bifurcation analysis of the stationary undamped as well as the stationary damped beat-waves recovers their entire bifurcation structure and enables to fully characterize the zones of system parameters guaranteeing the existence of this special type of non-linear wave solutions. The results of analytical study of various types of beat wave solutions are in excellent agreement with the results of numerical simulation of both PD-DpS as well as the original, parametrically driven, genuinely an-harmonic, discrete Klein-Gordon model.

A Synopsis of the Noninvertible, Two-Dimensional, Border-Collision Normal Form with Applications to Power Converters

The border-collision normal form is a canonical form for two-dimensional, continuous maps comprised of two affine pieces. In this paper, we provide a guide to the dynamics of this family of maps in the noninvertible case where the two pieces fold onto the same half-plane. Most significantly we identify parameter regimes for the occurrence of key bifurcation structures, such as period-incrementing, period-adding, and robust chaos. We characterize the simplest and most dominant bifurcations and illustrate various dynamical possibilities such as invariant circles, two-dimensional attractors, and several cases of coexisting attractors. We then apply the results to a classic model of a boost converter for adjusting the voltage of direct current. It is known that for one combination of circuit parameters the model exhibits a border-collision bifurcation that mimics supercritical period-doubling and is noninvertible due to the switching mechanism of the converter. We find that over a wide range of parameter values, even though the dynamics created in border-collision bifurcations is in general extremely diverse, the bifurcation in the model can only mimic period-doubling, although it can be subcritical.

Complete Bifurcation Analysis of the Vilnius Chaotic Oscillator

The paper is dedicated to the numerical and experimental study of nonlinear oscillations exhibited by the Vilnius chaotic generator. The motivation for the work is defined by the need for a comprehensive analysis of the dynamics of the oscillators being embedded into chaotic communication systems. These generators should provide low-power operation while ensuring the robustness of the chaotic oscillations, insusceptible to parameter variations and noise. The work focuses on the investigation of the dependence of nonlinear dynamics of the Vilnius oscillator on the operating voltage and component parameter changes. The paper shows that the application of the Method of Complete Bifurcation Groups reveals the complex smooth and non-smooth bifurcation structures, forming regions of robust chaotic oscillations. The novel tool—mode transition graph—is presented, allowing the comparison of experimental and numerical results. The paper demonstrates the applicability of the Vilnius oscillator for the generation of robust chaos, and highlights the need for further investigation of the inherent trade-off between energy efficiency and robustness of the obtained oscillations.

Kink–antikink interaction forces and bound states in a nonlinear Schrödinger model with quadratic and quartic dispersion

In the present work we explore the competition of quadratic and quartic dispersion in producing kink-like solitary waves in a model of the nonlinear Schrödinger type bearing cubic nonlinearity. We present the first 6 families of multikink solutions and explore their bifurcations as the strength of the quadratic dispersion is varied. We reveal a rich bifurcation structure for the system, connecting two-kink states with states involving 4-, as well as 6-kinks. The stability of all of these states is explored. For each family, we discuss a “lower branch” adhering to the energy landscape of the 2-kink states. We also, however, study in detail the “upper branches” bearing higher numbers of kinks. In addition to computing the stationary states and analyzing their stability within the partial differential equation model, we develop an effective particle ordinary differential equation theory that is shown to be surprisingly efficient in capturing the kink equilibria and normal (as well as unstable) modes. Finally, the results of the bifurcation analysis are corroborated by means of direct numerical simulations involving the excitation of the states in a targeted way in order to explore their instability-induced dynamics.

A Novel 3-D Jerk System, Its Bifurcation Analysis, Electronic Circuit Design and a Cryptographic Application

This paper introduces a new chaotic jerk system with three cubic nonlinear terms. The stability properties of the three equilibrium points of the proposed jerk system are analyzed in detail. We show that the three equilibrium points of the new chaotic jerk system are unstable and deduce that the jerk system exhibits self-excited chaotic attractors. The bifurcation structures of the proposed jerk system are investigated numerically, showing period-doubling, periodic windows and coexisting bifurcations. An electronic circuit design of the proposed jerk system is designed using PSPICE. As an engineering application, a new image-encryption approach based on the new chaotic jerk system is presented in this research work. Experimental results demonstrate that the suggested encryption mechanism is effective with high plain-image sensitivity and the reliability of the proposed chaotic jerk system for various cryptographic purposes.

The role of exponential asymptotics and complex singularities in self-similarity, transitions, and branch merging of nonlinear dynamics

We study a prototypical example in nonlinear dynamics where transition to self-similarity in a singular limit is fundamentally changed as a parameter is varied. Here, we focus on the complicated dynamics that occur in a generalised unstable thin-film equation that yields finite-time rupture. A parameter, n, is introduced to model more general disjoining pressures. For the standard case of van der Waals intermolecular forces, n=3, it was previously established that a countably infinite number of self-similar solutions exist leading to rupture. Each solution can be indexed by a parameter, ϵ=ϵ1>ϵ2>⋯>0, and the prediction of the discrete set of solutions requires examination of terms beyond-all-orders in ϵ. However, recent numerical results have demonstrated the surprising complexity that exists for general values of n. In particular, the bifurcation structure of self-similar solutions now exhibits branch merging as n is varied. In this work, we shall present key ideas of how branch merging can be interpreted via exponential asymptotics.

Pressure Characteristics and Vortex Observation in Chiral-Symmetric Space Orthogonal Bifurcation

In aerospace engine delivery systems, “one-in-two-out” bifurcation structures are commonly used for flow distribution to downstream pipelines. There are two common “one-in-two-out” bifurcation structures in aircraft engines: the planar orthogonal bifurcation and the spatial orthogonal bifurcation. By adjusting the flow supply upstream and the cross-sectional diameter downstream, the flow distribution in the two branches can be adjusted, i.e., the “splitting ratio” changes. In this paper, a dismantling and flexible experimental system is constructed to measure the pressure signals in each channel and use non-linear dynamic analysis methods to extract pressure characteristics. The particle image velocimetry (PIV) technique combined with the fine rope tracing technique is creatively used to observe the vortex structure in the cross section of the downstream branch. The study found that for spatial orthogonal bifurcation, the pressure signal characteristics in each channel are basically the same at larger splitting ratios, regardless of the chirality. As the splitting ratio decreases, the difference in pressure signal characteristics between the two branches gradually becomes evident and becomes related to the chirality. Moreover, unlike the planar orthogonal bifurcation structure, a complete large vortex structure has not been found in the downstream branch of the spatial orthogonal bifurcation structure, regardless of changes in the splitting ratio, and it is unrelated to the chirality.

A Unique Bifurcation Structure in One-Dimensional Model of Current-Controlled Step-Up DC–DC Converter with Photovoltaic Module

A Unique Bifurcation Structure in One-Dimensional Model of Current-Controlled Step-Up DC–DC Converter with Photovoltaic Module

Biomimetic optimized vertically aligned annular fins for fast latent heat thermal energy storage

Latent heat thermal energy storage (LHTES) technology has been widely used in balancing energy demand and intermittent energy supply, as well as thermal management in buildings, but suffers from slow heat storage rates. Here, biomimetic topology-optimized vertically aligned annular fins are proposed to accelerate heat storage performance. Taking the maximum average temperature as the optimization objective function, optimized biomimetic fins can shorten the melting time by 45.9% compared with traditional fins as verified by experimental measurements. Such a fast heat storage rate is also superior to that of pitch-varying fins and diameter-varying fins by 18.4% and 23.5%, respectively. Meanwhile, the temperature uniformity is greatly improved by biomimetic fins with a reduction of average temperature difference in phase change materials (PCMs) by 75.3%. The underlying mechanism can be attributed to accelerated melting of PCMs close to the outer bottom boundary due to unique bifurcation structures of topology optimized fins. The fractal dimension of optimized fins is similar to that of leaf vein structures, demonstrating that proposed topology structures are close to Natural’s optimal solutions. This study provides novel biomimetic structures for accelerating thermal energy storage rate and promotes the application of biomimetics in LHTES techniques.

Hopf Bifurcation Analysis of Mixed Traffic and Its Implications for Connected and Autonomous Vehicles

Capturing the evolution mechanism of traffic dynamics is of great significance when alleviating traffic congestion and improving traffic efficiency. While several studies have presented stability analyses of various traffic systems, most have relied on linear analysis methods, which cannot capture the complex nonlinear dynamic behavior of heterogeneous traffic flow. To address this shortcoming, this paper derives a generic Hopf bifurcation structure that can be applied to multiclass traffic models. The proposed bifurcation structure is investigated to understand the characteristics of heterogeneous traffic flow in a connected and autonomous environment as case studies. The results, based on selected on-field calibrated traffic models, show that (1) the linear analysis results deviate significantly from the actual instability of the mixed traffic system, which illustrates the necessity of a bifurcation analysis; and (2) connected human-driven vehicles and connected autonomous vehicles have the ability to alleviate the formation and propagation of traffic oscillations. Finally, the theoretical analysis results are verified through simulation experiments.