Construct Bifurcation Diagrams Research Articles

Bifurcation analysis of the motion of a cylinder and a point vortex in an ideal fluid

We consider an integrable Hamiltonian system describing the motion of a circular cylinder and a vortex filament in an ideal fluid. We construct bifurcation diagrams and bifurcation complexes for the case in which the integral manifold is compact and for various topological structures of the symplectic leaf. The types of motions corresponding to the bifurcation curves and their stability are discussed.

Chaplygin’s ball with a rotor: Non-degeneracy of singular points

A problem of a dynamically balanced asymmetric ball with a rotor rolling over a rough horizontal plane is considered. Earlier, A. Y. Moskvin constructed bifurcation diagrams of the momentum map and bifurcation complexes in order to study the dynamics of the system and to obtain singular solutions. A natural development of this research is a fine Liouville analysis of the system. The first step in this direction is presented in the paper, namely, we verify the non-degeneracy of singularities and describe the Liouville foliation in a neighborhood of singular points of the momentum map.

Non-local kinetic and macroscopic models for self-organised animal aggregations

The last two decades have seen a surge in kinetic and macroscopic models derived to investigate the multi-scale aspects of self-organised biological aggregations. Because the individual-level details incorporated into the kinetic models (e.g., individual speeds and turning rates) make them somewhat difficult to investigate, one is interested in transforming these models into simpler macroscopic models, by using various scaling techniques that are imposed by the biological assumptions of the models. However, not many studies investigate how the dynamics of the initial models are preserved via these scalings. Here, we consider two scaling approaches (parabolic and grazing collision limits) that can be used to reduce a class of non-local 1D and 2D models for biological aggregations to simpler models existent in the literature. Then, we investigate how some of the spatio-temporal patterns exhibited by the original kinetic models are preserved via these scalings. To this end, we focus on the parabolic scaling for non-local 1D models and apply asymptotic preserving numerical methods, which allow us to analyse changes in the patterns as the scaling coefficient $\epsilon$ is varied from $\epsilon=1$ (for 1D transport models) to $\epsilon=0$ (for 1D parabolic models). We show that some patterns (describing stationary aggregations) are preserved in the limit $\epsilon\to 0$, while other patterns (describing moving aggregations) are lost. To understand the loss of these patterns, we construct bifurcation diagrams.

Effect of delay mismatch in Pyragas feedback control.

Pyragas time-delayed feedback is a control scheme designed to stabilize unstable periodic orbits, which occur naturally in many nonlinear dynamical systems. It has been successfully implemented in a number of applications, including lasers and chemical systems. The control scheme targets a specific unstable periodic orbit by adding a feedback term with a delay chosen as the period of the unstable periodic orbit. However, in an experimental or industrial environment, obtaining the exact period or setting the delay equal to the exact period of the target periodic orbit may be difficult. This could be due to a number of factors, such as incomplete information on the system or the delay being set by inaccurate equipment. In this paper, we evaluate the effect of Pyragas control on the prototypical generic subcritical Hopf normal form when the delay is close to but not equal to the period of the target periodic orbit. Specifically, we consider two cases: first, a constant, and second, a linear approximation of the period. We compare these two cases to the case where the delay is set exactly to the target period, which serves as the benchmark case. For this comparison, we construct bifurcation diagrams and determine any regions where a stable periodic orbit close to the target is stabilized by the control scheme. In this way, we find that at least a linear approximation of the period is required for successful stabilization by Pyragas control.

Chaotic Wave Solutions in a Nonlocal Model for Media With Vibrating Inclusions

The present paper is devoted to the investigation of wave solutions of a mathematical model of a complex medium that takes into account the nonlocal interaction of structural elements of the main medium and the vibration dynamics of an additional continuum (inclusions). By the methods of qualitative and numerical analysis, it is shown that an autonomous dynamic system with four-dimensional phase space has periodic, multiperiodic, quasiperiodic, and chaotic solutions. We construct bifurcation diagrams of the development of vibrations depending on the parameters of nonlocality of the dynamic equation of state of the medium

Modelling Effects of Rapid Evolution on Persistence and Stability in Structured Predator-Prey Systems

In this paper we explore the eco-evolutionary dynamics of a predator-prey model, where the prey population is structured according to a certain life history trait. The trait distribution within the prey population is the result of interplay between genetic inheritance and mutation, as well as selectivity in the consumption of prey by the predator. The evolutionary processes are considered to take place on the same time scale as ecological dynamics, i.e. we consider the evolution to be rapid. Previously published results show that population structuring and rapid evolution in such predator-prey system can stabilise an otherwise globally unstable dynamics even with an unlimited carrying capacity of prey. However, those findings were only based on direct numerical simulation of equations and obtained for particular parametrisations of model functions, which obviously calls into question the correctness and generality of the previous results. The main objective of the current study is to treat the model analytically and consider various parametrisations of predator selectivity and inheritance kernel. We investigate the existence of a coexistence stationary state in the model and carry out stability analysis of this state. We derive expressions for the Hopf bifurcation curve which can be used for constructing bifurcation diagrams in the parameter space without the need for a direct numerical simulation of the underlying integro-differential equations. We analytically show the possibility of stabilisation of a globally unstable predator-prey system with prey structuring. We prove that the coexistence stationary state is stable when the saturation in the predation term is low. Finally, for a class of kernels describing genetic inheritance and mutation we show that stability of the predator-prey interaction will require a selectivity of predation according to the life trait.

The role of the droplet deformations in the bouncing droplet dynamics

Droplets bouncing on a vibrated liquid bath open ways to methods of manipulating droplets, creating double emulsion, and performing pilot wave model experiments. In this work, we focus on the role of the droplet deformations in the vertical bouncing dynamics by neglecting the deformation of the surface of the bath. To be under this favorable condition, low viscous oil droplets are dropped over a highly viscous oil bath that is vibrated. These droplets bounce vertically on the surface of the bath and exhibit many periodic trajectories and resonant modes when tuning the forcing parameters, i.e., the oscillation of the bath. This complex dynamics emphasizes the interplay between elastic energy storage and energy dissipation in droplets at each bounce. We propose to model droplets using a bouncing mass-spring-damper system that mimics a deformable droplet bouncing on a non-deformable liquid bath. From the experimental measurements, we constructed bifurcation diagrams of the bouncing trajectories and challenged our bouncing spring model. The agreement between experiment and the spring model reveals that this model can be used to rationalize and predict a variety of bouncing droplets behaviors involving multi-periodicities.

Global analysis of Navier–Stokes and Boussinesq stochastic flows using dynamical orthogonality

AbstractWe provide a new framework for the study of fluid flows presenting complex uncertain behaviour. Our approach is based on the stochastic reduction and analysis of the governing equations using the dynamically orthogonal field equations. By numerically solving these equations, we evolve in a fully coupled way the mean flow and the statistical and spatial characteristics of the stochastic fluctuations. This set of equations is formulated for the general case of stochastic boundary conditions and allows for the application of projection methods that considerably reduce the computational cost. We analyse the transformation of energy from stochastic modes to mean dynamics, and vice versa, by deriving exact expressions that quantify the interaction among different components of the flow. The developed framework is illustrated through specific flows in unstable regimes. In particular, we consider the flow behind a disk and the Rayleigh–Bénard convection, for which we construct bifurcation diagrams that describe the variation of the response as well as the energy transfers for different parameters associated with the considered flows. We reveal the low dimensionality of the underlying stochastic attractor.

Dynamical-systems analysis and unstable periodic orbits in reacting flows behind symmetric bluff bodies

Dynamical systems analysis is performed for reacting flows stabilized behind four symmetric bluff bodies to determine the effects of shape on the nature of flame stability, acoustic coupling, and vortex shedding. The task requires separation of regular, repeatable aspects of the flow from experimental noise and highly irregular, nonrepeatable small-scale structures caused primarily by viscous-mediated energy cascading. The experimental systems are invariant under a reflection, and symmetric vortex shedding is observed throughout the parameter range. As the equivalence ratio-and, hence, acoustic coupling-is reduced, a symmetry-breaking transition to von Karman vortices is initiated. Combining principal-components analysis with a symmetry-based filtering, we construct bifurcation diagrams for the onset and growth of von Karman vortices. We also compute Lyapunov exponents for each flame holder to help quantify the transitions. Furthermore, we outline changes in the phase-space orbits that accompany the onset of von Karman vortex shedding and compute unstable periodic orbits (UPOs) embedded in the complex flows prior to and following the bifurcation. For each flame holder, we find a single UPO in flows without von Karman vortices and a pair of UPOs in flows with von Karman vortices. These periodic orbits organize the dynamics of the flow and can be used to reduce or control flow irregularities. By subtracting them from the overall flow, we are able to deduce the nature of irregular facets of the flows.

Thermal effects on nonlinear vibrations of an axially moving beam with an intermediate spring-mass support

The thermo-mechanical nonlinear vibrations and stability of a hinged-hinged axially moving beam, additionally sup- ported by a nonlinear spring-mass support are examined via two numerical techniques. The system is subjected to a transverse harmonic excitation force as well as a thermal loading. Hamilton's principle is employed to derive the equations of motion; it is discretized into a multi-degree-freedom system by means of the Galerkin method. The steady state resonant response of the system for both cases with and without an internal resonance between the first two modes is examined via the pseudo-arclength continuation technique. In the second method, direct time integration is employed to construct bifurcation diagrams of Poincare maps of the system.

Thermal Effects on Nonlinear Vibrations of an Axially Moving Beam with an Intermediate Spring-Mass Support

The thermo-mechanical nonlinear vibrations and stability of a hinged-hinged axially moving beam, additionally supported by a nonlinear spring-mass support are examined via two numerical techniques. The system is subjected to a transverse harmonic excitation force as well as a thermal loading. Hamilton's principle is employed to derive the equations of motion; it is discretized into a multi-degree-freedom system by means of the Galerkin method. The steady state resonant response of the system for both cases with and without an internal resonance between the first two modes is examined via the pseudo-arclength continuation technique. In the second method, direct time integration is employed to construct bifurcation diagrams of Poincaré maps of the system.

Chaotic saddles in nonlinear modulational interactions in a plasma

A nonlinear model of modulational processes in the subsonic regime involving a linearly unstable wave and two linearly damped waves with different damping rates in a plasma is studied numerically. We compute the maximum Lyapunov exponent as a function of the damping rates in a two-parameter space, and identify shrimp-shaped self-similar structures in the parameter space. By varying the damping rate of the low-frequency wave, we construct bifurcation diagrams and focus on a saddle-node bifurcation and an interior crisis associated with a periodic window. We detect chaotic saddles and their stable and unstable manifolds, and demonstrate how the connection between two chaotic saddles via coupling unstable periodic orbits can result in a crisis-induced intermittency. The relevance of this work for the understanding of modulational processes observed in plasmas and fluids is discussed.

On the use of POD-based ROMs to analyze bifurcations in some dissipative systems

This paper deals with the use of POD-based reduced order models to construct bifurcation diagrams (which requires calculating steady and time-dependent attractors) in complex bifurcation problems involving dissipative systems. The method proposed in the paper relies on the observation that POD manifolds resulting from snapshots calculated in time-dependent runs for specific values of the parameters of the problem also contain the attractors for other values of the parameters. The reason for this property is explained for a general class of dissipative systems, which includes many problems of scientific/industrial interest. The consequence is that appropriate POD manifolds can be calculated in a quite computationally efficient way. The method is illustrated considering both a simple bifurcation problem for a Fisher-like equation and a fairly complex bifurcation problem for the complex Ginzburg–Landau equation.

Thermo-mechanical nonlinear dynamics of a buckled axially moving beam

The thermo-mechanical nonlinear dynamics of a buckled axially moving beam is numerically investigated, with special consideration to the case with a three-to-one internal resonance between the first two modes. The equation of motion of the system traveling at a constant axial speed is obtained using Hamilton’s principle. A closed form solution is developed for the post-buckling configuration for the system with an axial speed beyond the first instability. The equation of motion over the buckled state is obtained for the forced system. The equation is reduced into a set of nonlinear ordinary differential equations via the Galerkin method. This set is solved using the pseudo-arclength continuation technique to examine the frequency response curves and direct-time integration to construct bifurcation diagrams of Poincare maps. The vibration characteristics of the system at points of interest in the parameter space are presented in the form of time histories, phase-plane portraits, and Poincare sections.

Equilibrium gas–liquid–solid contact angle from density-functional theory

AbstractWe investigate the equilibrium of a fluid in contact with a solid boundary through a density-functional theory. Depending on the conditions, the fluid can be in one phase, gas or liquid, or two phases, while the wall induces an external field acting on the fluid particles. We first examine the case of a liquid film in contact with the wall. We construct bifurcation diagrams for the film thickness as a function of the chemical potential. At a specific value of the chemical potential, two equally stable films, a thin one and a thick one, can coexist. As saturation is approached, the thickness of the thick film tends to infinity. This allows the construction of a liquid–gas interface that forms a well-defined contact angle with the wall.

Smart dampers control in a Remoissenet–Peyrard substrate potential

A model of spring-block on a moving plate with a nonlinear periodic substrate potential whose shape can be varied continuously as a function of a shape parameter is investigated. The dynamical study of the system for different values of the shape parameter involves the analysis of phase space, the construction of bifurcation diagrams, and the computation of the largest Lyapunov exponent. A smart damper associated with drag coefficient is proposed to reduce stick-slip and chaotic motions. The domain of validity of the control method is derived.

DNA stretching modeled at the base pair level: Overtwisting and shear instability in elastic linkages

Stretching experiments on single DNA molecules indicate that, counterintuitive to expectations, DNA overwinds when stretched and, at large forces, undergoes a transition into an overstretched form indicated by a plateau on the force–displacement diagrams. It is believed that these effects are the result of non-linearities in the elastic response of DNA. We use a discrete, base pair level model to simulate the behavior of short DNA molecules, taking into account the sequence dependent physical properties of DNA alongside with the coupling between the kinematical step parameters, yet retaining the quadratic form of local elastic energy function. By constructing bifurcation diagrams of equilibrium configurations and studying the dependence on base pair combinations we show that the quadratic model is capable of explaining the overtwisting as a result of coupling between modes of deformation and overstretching as a result of shear instability.

Robustness analysis and behavior discrimination in enzymatic reaction networks.

Characterizing the behavior and robustness of enzymatic networks with numerous variables and unknown parameter values is a major challenge in biology, especially when some enzymes have counter-intuitive properties or switch-like behavior between activation and inhibition. In this paper, we propose new methodological and tool-supported contributions, based on the intuitive formalism of temporal logic, to express in a rigorous manner arbitrarily complex dynamical properties. Our multi-step analysis allows efficient sampling of the parameter space in order to define feasible regions in which the model exhibits imposed or experimentally observed behaviors. In a first step, an algorithmic methodology involving sensitivity analysis is conducted to determine bifurcation thresholds for a limited number of model parameters or initial conditions. In a second step, this boundary detection is supplemented by a global robustness analysis, based on quasi-Monte Carlo approach that takes into account all model parameters. We apply this method to a well-documented enzymatic reaction network describing collagen proteolysis by matrix metalloproteinase MMP2 and membrane type 1 metalloproteinase (MT1-MMP) in the presence of tissue inhibitor of metalloproteinase TIMP2. For this model, our method provides an extended analysis and quantification of network robustness toward paradoxical TIMP2 switching activity between activation or inhibition of MMP2 production. Further implication of our approach is illustrated by demonstrating and analyzing the possible existence of oscillatory behaviors when considering an extended open configuration of the enzymatic network. Notably, we construct bifurcation diagrams that specify key parameters values controlling the co-existence of stable steady and non-steady oscillatory proteolytic dynamics.

Electrochemical oscillations and bistability during anodic dissolution of vanadium electrode in acidic media—part I. Experiment

Dynamic instabilities, current oscillations and bistability observed during anodic dissolution of both stationary and rotating disk vanadium electrode in acidic phosphoric media, were reported using both dc and ac techniques. The effect of various experimental conditions, concentration of H3PO4, temperature, disk rotation rate, and external resistance, was analyzed. Systematic studies allowed the construction of bifurcation diagrams, showing the regions of oscillations and bistability, including the complex behaviors like coexistence of both dynamic regimes. Analogous comparative measurements and analyses were performed for other media—sulfuric, nitric, perchloric, and trifluoroacetic acids—also indicating complex dynamic behaviors. These complexities arise not only due to the difficulties with the precise identification of the composition of the passive layer in every acid but are also due to relatively fast dissolution of V electrode, even in the presence of passive layer, which causes permanent drift of the system’s characteristics. Due to these experimental difficulties, theoretical modeling seems to be an appropriate method to analyze the essential nonlinear dynamic properties of the studied system.

A Truly Random Number Generator Based on a Pulse-Excited Cross-Coupled Chaotic Oscillator

A non-autonomous cross-coupled chaotic circuit is presented with its derivation mechanism. To guarantee robust chaotic behavior of the circuit against parameter variations, an ideal set of parameters is determined by constructing bifurcation diagrams. A truly random number generator (TRNG), which relies on generating non-invertible binary sequences according to regional distributions of underlying chaotic signal, is also introduced. Simulation and experimental results verifying the feasibility of the circuit are given. The proposed TRNG offers much higher and constant throughput rates, allows for offset compensation and fulfills the NIST-800-22 statistical test suite without further post-processing.