Numerical Bifurcation Research Articles

Numerical Methods for Delay Differential Equations with Threshold State-Dependent Delay

Threshold delays arise naturally in a wide variety of dynamical systems. Standard routines for performing numerical analysis of delay differential equations (DDEs) typically handle discrete constant or state-dependent delays, but are not directly applicable to distributed delay problems. We show how to reformulate the threshold delay problem as a discrete delay DDE, with the delay as an extra variable, by differentiating the threshold condition. This allows initial value problems to be solved in Matlab using ddesd. We also describe two implementations of the threshold delay DDE in DDE-BIFTOOL to perform numerical bifurcation analysis. One approach enables us to determine steady state bifurcations via a correction of characteristic values from a modified but related problem with discrete delays. The second approach involves introducing dummy constant delays which are used to discretize the integral in the threshold condition. This allows the threshold delay to be solved for directly within DDE-BIFTOOL, and enables the full functionality of DDE-BIFTOOL, so we can compute periodic orbits as well as steady states, along with their stability and bifurcations. We illustrate these techniques on a gene regulation model where the delay, driven by a transport process, is implicitly defined by a threshold condition. The speed of the transport process depends on the state of the system making the delay both state-dependent and distributed.

On a mathematical model of tumor-immune system interactions with an oncolytic virus therapy

<p style='text-indent:20px;'>We investigate a mathematical model of tumor–immune system interactions with oncolytic virus therapy (OVT). Susceptible tumor cells may become infected by viruses that are engineered specifically to kill cancer cells but not healthy cells. Once the infected cancer cells are destroyed by oncolysis, they release new infectious virus particles to help kill surrounding tumor cells. The immune system constructed includes innate and adaptive immunities while the adaptive immunity is further separated into anti-viral or anti-tumor immune cells. The model is first analyzed by studying boundary equilibria and their stability. Numerical bifurcation analysis is performed to investigate the outcomes of the oncolytic virus therapy. The model has a unique tumor remission equilibrium, which is unlikely to be stable based on the parameter values given in the literature. Multiple stable positive equilibria with tumor sizes close to the carrying capacity coexist in the system if the tumor is less antigenic. However, as the viral infection rate increases, the OVT becomes more effective in the sense that the tumor can be dormant for a longer period of time even when the tumor is weakly antigenic.</p>

Emergence of quasiperiodic regimes in a neutral delay model of flute-like instruments: Influence of the detuning between resonance frequencies

<p style='text-indent:20px;'>Musical instruments display a wealth of dynamics, from equilibria (where no sound is produced) to a wide diversity of periodic and non-periodic sound regimes. We focus here on two types of flute-like instruments, namely a recorder and a pre-hispanic Chilean flute. A recent experimental study showed that they both produce quasiperiodic sound regimes which are avoided or played on purpose depending on the instrument. We investigate the generic model of sound production in flute-like musical instruments, a system of neutral delay-differential equations. Using time-domain simulations, we show that it produces stable quasiperiodic oscillations in good agreement with experimental observations. A numerical bifurcation analysis is performed, where both the delay time (related to a control parameter) and the detuning between the resonance frequencies of the instrument – a key parameter for instrument makers – are considered as bifurcation parameters. This demonstrates that the large detuning that is characteristic of prehispanic Chilean flutes plays a crucial role in the emergence of stable quasiperiodic oscillations.</p>

From A.Tondl's Dutch contacts to Neimark-Sacker-bifurcation

We present a description of the many contacts of A. Tondl with Dutch scientists involving nonlinear dynamics models for mechanics. One of the topics is Neimark-Sacker bifurcation that leads to the presence of families of quasi-periodic solutions that are geometrically organised and visualised in tori. A new model in the spirit of A. Tondl, containing interaction of self-excited and parametrically excited oscillators is analysed to find this bifurcation and quasi-periodic solutions. The analysis using averaging in combination with numerical bifurcation tools Matcont and Auto produces a picture of rich dynamical phenomena with several surprises among which a special quasi-periodic solution produced by the averaged equation.

Mathematical modelling of tumor-immune system by considering the regulatory T cells role

The immune system has an important role in protecting the body from tumors. However, several clinical studies have shown that not all immune cells in the body work positively against tumors, such as regulatory T cells which are known to modulate the function of effector cells and inhibit their cytotoxic activity. The purpose of this paper is to analyze the mathematical model of tumor-immune system dynamics by considering the regulatory T cells role. Based on the model analysis results obtained eight equilibrium points, where two equilibrium points are unstable, namely the equilibrium point of normal, tumor, effector cells extinction and the equilibrium point of normal, tumor cells extinction, then four equilibrium points are conditionally asymptotically stable, namely the equilibrium point of normal and effector cells extinction, tumor and effector cells extinction, tumor cells extinction, and effector cells extinction, and two equilibrium points are thought to tend to be asymptotically stable when the existence conditions are satisfied, namely the equilibrium point of normal cells extinction and coexistence. The numerical simulation results show that regulatory T cells play an important role in inhibiting effector cells and promoting tumor cell growth. Furthermore, a numerical bifurcation analysis is performed which shows the presence of saddle-node bifurcation and bistable behavior in the system.

A mathematical analysis of an activator-inhibitor Rho GTPase model

<p style='text-indent:20px;'>Recent experimental observations reveal that local cellular contraction pulses emerge via a combination of fast positive and slow negative feedbacks based on a signal network composed of Rho, GEF and Myosin interactions [<xref ref-type="bibr" rid="b22">22</xref>]. As an examplary, we propose to study a plausible, hypothetical temporal model that mirrors general principles of fast positive and slow negative feedback, a hallmark for activator-inhibitor models. The methodology involves (ⅰ) a qualitative analysis to unravel system switching between different states (stable, excitable, oscillatory and bistable) through model parameter variations; (ⅱ) a numerical bifurcation analysis using the positive feedback mediator concentration as a bifurcation parameter, (ⅲ) a sensitivity analysis to quantify the effect of parameter uncertainty on the model output for different dynamic regimes of the model system; and (ⅳ) numerical simulations of the model system for model predictions. Our methodological approach supports the role of mathematical and computational models in unravelling mechanisms for molecular and developmental processes and provides tools for analysis of temporal models of this nature.</p>

Origin of jumping oscillons in an excitable reaction-diffusion system.

Oscillons, i.e., immobile spatially localized but temporally oscillating structures, are the subject of intense study since their discovery in Faraday wave experiments. However, oscillons can also disappear and reappear at a shifted spatial location, becoming jumping oscillons (JOs). We explain here the origin of this behavior in a three-variable reaction-diffusion system via numerical continuation and bifurcation theory, and show that JOs are created via a modulational instability of excitable traveling pulses (TPs). We also reveal the presence of bound states of JOs and TPs and patches of such states (including jumping periodic patterns) and determine their stability. This rich multiplicity of spatiotemporal states lends itself to information and storage handling.

Multiple Time Delays Induced Dynamics of p53 Gene Regulatory Network

p53 dynamics plays an important role in determining cell arrest or apoptosis upon DNA damage response. In this paper, based on a p53 gene regulatory network composed of its core regulator ATM, Mdm2 and Wip1, the effect of multiple time delays in transcription and translation of Mdm2 and Wip1 gene expression on p53 dynamics are investigated through theoretical and numerical analyses. The stability of the positive equilibrium point and the existence of Hopf bifurcation are demonstrated through analyzing the associated characteristic equation of the corresponding linearized system in five cases. Detailed numerical simulations and bifurcation analyses are performed to support the theoretical results. The results show that with the increase of a time delay, the positive equilibrium point becomes unstable, and the p53 dynamics presents an oscillating state. These results reveal that time delay has a significant impact on p53 dynamics and may provide a useful insight into developing anti-cancer therapy.

Bifurcation Analysis of an Ecological System with State-Dependent Feedback Control and Periodic Forcing

By assuming a periodic variation in the intrinsic growth rate of the prey, a nonlinear ecological system with periodic forcing and state-dependent feedback control is proposed. The main purpose of the present paper is to study the dynamical behavior generated by periodic forcing and nonlinear impulse perturbations and their effects on pest control. To do this, we first investigate the existence and stability of the boundary periodic solution, and then we employ the numerical bifurcation techniques, mainly including one-dimensional and two-dimensional parameter bifurcation analyses, to reveal that the system exhibits rich and complex dynamic behaviors. Especially, period-adding bifurcation with chaos is found in the two-parameter bifurcation plane. Moreover, we find the periodic structure similar to Arnold tongues, and they are arranged according to the sequence of a Farey tree. In addition, one-dimensional bifurcation diagrams reveal the existence of order-[Formula: see text] periodic, and chaotic solutions, multiple coexisting attractors, period-doubling bifurcations, period-halving bifurcations, and so on. Finally, the effects of the initial population density of pests and natural enemies on the pulse frequency and the biological significance related to the numerical results are studied and discussed.

Delay-induced periodic oscillation for fractional-order neural networks with mixed delays

This article is mainly devoted to the investigation on the stability and Hopf bifurcation of fractional-order neural networks with mixed delays. Applying a suitable substitution of variable, a novel equivalent fractional-order neural networks concerning single delay is set up. By analyzing the corresponding characteristic equation of the involved fractional-order delayed neural networks and choosing the time delay as bifurcation parameter, we derive a new sufficient condition to guarantee the stability behavior and the appearance of Hopf bifurcation for the considered fractional-order delayed neural networks. The study reveals that the time delay is a key factor which has a vital impact on stability and Hopf bifurcation of neural networks. The obtained results of this work can be effectively applied to design neural networks. The numerical simulations and bifurcation diagrams are displayed to verify the rationality of the analytical results.

On the Thermal Dynamics of Metallic and Superconducting Wires. Bifurcations, Quench, the Destruction of Bistability and Temperature Blowup

In the present study, a numerical bifurcation analysis is carried out in order to investigate the multiplicity and the thermal runaway features of metallic and superconducting wires in a unified framework. The analysis reveals that the electrical resistance, combined with the boiling curve, are the dominant factors shaping the conditions of bistability—which result in a quenching process—and the conditions of multistability—which may lead to a temperature blowup in the wire. An interesting finding of the theoretical analysis is that, for the case of multistability, there are two ways that a thermal runaway may be triggered. One is associated with a high current value (“normal” runaway) whereas the other one is associated with a lower current value (“premature” runaway), as has been experimentally observed with certain types of superconducting magnets. Moreover, the results of the bifurcation analysis suggest that a static criterion of a warm or a cold thermal wave propagation may be established based on the limit points obtained.

Dynamics of Stoichiometric Autotroph-Mixotroph-Bacteria Interactions in the Epilimnion.

Autotrophs, mixotrophs and bacteria exhibit complex interrelationships containing multiple ecological mechanisms. A mathematical model based on ecological stoichiometry is proposed to describe the interactions among them. Some dynamic analysis and numerical simulations of this model are presented. The roles of autotrophs and mixotrophs in controlling bacterioplankton are explored to examine the experiments and hypotheses of Medina-Sánchez, Villar-Argaiz and Carrillo for La Caldera Lake. Our results show that the dual control (bottom-up control and top-down control) of bacteria by mixotrophs is a key reason for the ratio of bacterial and phytoplankton biomass in La Caldera Lake to deviate from the general tendency. The numerical bifurcation diagrams suggest that the competition between phytoplankton and bacteria for nutrients can also be an important factor for the decrease of the bacterial biomass in an oligotrophic lake.

Mathematical modelling, numerical and experimental analysis of one-degree-of-freedom oscillator with Duffing-type stiffness

The aim of the work is to develop and test simple mathematical models suitable for fast and realistic simulations of bifurcation dynamics of systems with a double potential well generated by a pair of repulsive magnets positioned transversely to the direction of motion, where there are motion resistances typical for rolling guides found in industry. Mathematical modelling of a harmonically forced one-degree-of-freedom oscillator with magnetically modified elasticity generating a double symmetrical minimum of potential was carried out. The system is composed of a cart moving along a linear rolling bearing with inertial excitation by means of a stepper motor with an unbalanced disc. The stiffness is realized by a pair of neodymium magnets of axes perpendicular to the direction of motion, in a position corresponding to the static equilibrium position of the oscillator, with additional mechanical linear springs, which generate a system similar to the Duffing system. Different models of the interaction force between cylindrical neodymium magnets were developed. The model parameters are estimated based on experimental data. Then the model is validated through additional experimental and numerical bifurcation analysis of the system. A very good agreement between numerical simulations and experimental data is obtained.

Numerical Bifurcation Analysis of a Film Flowing over a Patterned Surface through Enhanced Lubrication Theory

The bifurcation analysis of a film falling down an hybrid surface is conducted via the numerical solution of the governing lubrication equation. Instability phenomena, that lead to film breakage and growth of fingers, are induced by multiple contamination spots. Contact angles up to 75∘ are investigated due to the full implementation of the free surface curvature, which replaces the small slope approximation, accurate for film slope lower than 30∘. The dynamic contact angle is first verified with the Hoffman–Voinov–Tanner law in case of a stable film down an inclined plate with uniform surface wettability. Then, contamination spots, characterized by an increased value of the static contact angle, are considered in order to induce film instability and several parametric computations are run, with different film patterns observed. The effects of the flow characteristics and of the hybrid pattern geometry are investigated and the corresponding bifurcation diagram with the number of observed rivulets is built. The long term evolution of induced film instabilities shows a complex behavior: different flow regimes can be observed at the same flow characteristics under slightly different hybrid configurations. This suggest the possibility of controlling the rivulet/film transition via a proper design of the surfaces, thus opening the way for relevant practical application.

Doubling of a closed invariant curve in an impulsive Goodwin’s oscillator with delay

In the present paper, we focus on the doubling of closed invariant curves associated with quasiperiodic dynamics. We consider a 5D map derived from a hybrid model originating from systems biology and containing a continuous part with time delay and pulse-modulated feedback. Using numerical bifurcation analysis, we show that doubling bifurcation takes place on a closed 2D invariant manifold. We explain how such a configuration of the phase space can be created and highlight the role of delay.

Disconnected stationary solutions for 3D Kolmogorov flow problem: preliminary results

The extension of the classical A.N. Kolmogorov’s flow problem for the stationary 3D Navier-Stokes equations on a stretched torus for velocity vector function is considered. A spectral Fourier method with the Leray projection is used to solve the problem numerically. The resulting system of nonlinear equations is used to perform numerical bifurcation analysis. The problem is analyzed by constructing solution curves in the parameter-phase space using previously developed deflated pseudo arc-length continuation method. Disconnected solutions from the main solution branch are found. These results are preliminary and shall be generalized elsewhere.

Continuation and Bifurcation in Nonlinear PDEs \u2013 Algorithms, Applications, and Experiments

Numerical continuation and bifurcation methods can be used to explore the set of steady and time–periodic solutions of parameter dependent nonlinear ODEs or PDEs. For PDEs, a basic idea is to first convert the PDE into a system of algebraic equations or ODEs via a spatial discretization. However, the large class of possible PDE bifurcation problems makes developing a general and user–friendly software a challenge, and the often needed large number of degrees of freedom, and the typically large set of solutions, often require adapted methods. Here we review some of these methods, and illustrate the approach by application of the package pde2path to some advanced pattern formation problems, including the interaction of Hopf and Turing modes, patterns on disks, and an experimental setting of dead core pattern formation.

Numerical solution and bifurcation analysis of nonlinear partial differential equations with extreme learning machines

We address a new numerical method based on a class of machine learning methods, the so-called Extreme Learning Machines (ELM) with both sigmoidal and radial-basis functions, for the computation of steady-state solutions and the construction of (one-dimensional) bifurcation diagrams of nonlinear partial differential equations (PDEs). For our illustrations, we considered two benchmark problems, namely (a) the one-dimensional viscous Burgers with both homogeneous (Dirichlet) and non-homogeneous boundary conditions, and, (b) the one- and two-dimensional Liouville–Bratu–Gelfand PDEs with homogeneous Dirichlet boundary conditions. For the one-dimensional Burgers and Bratu PDEs, exact analytical solutions are available and used for comparison purposes against the numerical derived solutions. Furthermore, the numerical efficiency (in terms of numerical accuracy, size of the grid and execution times) of the proposed numerical machine-learning method is compared against central finite differences (FD) and Galerkin weighted-residuals finite-element (FEM) methods. We show that the proposed numerical machine learning method outperforms in terms of numerical accuracy both FD and FEM methods for medium to large sized grids, while provides equivalent results with the FEM for low to medium sized grids; both methods (ELM and FEM) outperform the FD scheme. Furthermore, the computational times required with the proposed machine learning scheme were comparable and in particular slightly smaller than the ones required with FEM.

Analytical and Numerical Bifurcation Analysis of Circuits Based on Nonlinear Resonators

This work presents new frequency-domain methods for the analysis and simulation of circuits based on a nonlinear resonator, with operation ranges delimited by turning points. Insightful analytical conditions fulfilled at these turning points are derived, which will enable an identification of the effect of each circuit element on their location in the solution curve. The cusp points, or co-dimension two bifurcations at which two turning points merge into one, thus delimiting the multivalued intervals, are directly calculated for the first time to our knowledge. In addition, a new numerical method, compatible with the use of commercial harmonic balance, is presented for the straightforward tracing of the multivalued solution curves, together with a new procedure to determine the locus of turning points in terms of any two analysis parameters. This relies on the use of a new mathematical condition to obtain a surface of turning points in the space defined by the two parameters and the input power. The methods have been applied to a wireless power-transfer system based on a recently proposed configuration, obtaining very good agreement with the experimental results.

The effect of characteristic times on collective modes of two quorum sensing coupled identical ring oscillators

There has long been interest in methods to achieve variability of collective modes in systems of coupled identical oscillators. Extensive multistability and a dynamically rich parameter space exist for two identical ring oscillators mean-field coupled by the quorum sensing mechanism when there is significant heterogeneity of the timescales of the ring elements’ variables. These findings have been shown in numerical and electrical circuit models based on the Repressilator, a ring oscillator using repressor genes as the inverter elements. Here we investigate the effect of reducing the degree of timescale heterogeneity. We find the 2D parameter map of dynamical regimes for the previously established heterogeneous timescale, using the coupling strength and the isolated oscillator strength for the parameters. The map shows a sea of anti-phase (AP) oscillation, with a large island of complex oscillations bounded by Neimark-Sacker torus bifurcation of the AP. The island is dominated by chaos produced by period doubling cascades of two prominent limit cycles (an unusual resonant cycle and a highly asymmetric cycle) and by torus destruction. Extensive regions of multistability exist, including between different oscillatory states, and, notably, coexistence of the steady state region and oscillatory states. Using numerical bifurcation analysis with timescale heterogeneity as a continuation parameter, we follow the evolution of the collective modes, reveal the appearance of new dynamical regimes, and determine changes in the multistability. We find that decreasing the heterogeneity causes: period-doubling cascades of the AP outside of the island, thereby creating new regimes of complexity; an increase in the dominance of chaos due to a reduction in the size of regions dominated by limit cycles; loss of stable oscillatory states inside steady state regions, converting those regions from multistable to sole dynamic steady state regions; and the appearance of in-phase oscillations and its sole dominance.