Bifurcation Normal Form Research Articles

Bifurcations and Early-Warning Signs for SPDEs with Spatial Heterogeneity

AbstractBistability is a key property of many systems arising in the nonlinear sciences. For example, it appears in many partial differential equations (PDEs). For scalar bistable reaction-diffusions PDEs, the bistable case even has taken on different names within communities such as Allee, Allen-Cahn, Chafee-Infante, Nagumo, Ginzburg-Landau,$$\Phi ^4$$Φ4, Schlögl, Stommel, just to name a few structurally similar bistable model names. One key mechanism, how bistability arises under parameter variation is a pitchfork bifurcation. In particular, taking the pitchfork bifurcation normal form for reaction-diffusion PDEs is yet another variant within the family of PDEs mentioned above. More generally, the study of this PDE class considering steady states and stability, related to bifurcations due to a parameter is well-understood for the deterministic case. For the stochastic PDE (SPDE) case, the situation is less well-understood and has been studied recently. In this paper we generalize and unify several recent results for SPDE bifurcations. Our generalisation is motivated directly by applications as we introduce in the equation a spatially heterogeneous term and relax the assumptions on the covariance operator that defines the noise. For this spatially heterogeneous SPDE, we prove a finite-time Lyapunov exponent bifurcation result. Furthermore, we extend the theory of early warning signs in our context and we explain, the role of universal exponents between covariance operator warning signs and the lack of finite-time Lyapunov uniformity. Our results are accompanied and cross-validated by numerical simulations.

Equivalence of MTS and CMR methods associated with the normal form of Hopf bifurcation for delayed reaction–diffusion equations

In this paper, we prove the equivalence of the multiple time scales (MTS) method and the center manifold reduction (CMR) method for deriving the normal form of Hopf bifurcation for delayed reaction–diffusion equations. First, we generalize the MTS method associated with functional differential equations to delayed reaction–diffusion equation for solving the normal form of Hopf bifurcation. This provides a user-friendly approach of analyzing the stability of time-periodic solutions, particularly, for the engineering researchers who are not familiar with CMR method. Second, we outline the CMR method associated with the computation of normal form of Hopf bifurcation for delayed reaction–diffusion equations. Third, we prove that the two methods can derive equivalent up to the third order normal form of Hopf bifurcation. Finally, different types of examples are presented, such as reaction–diffusion equations with Neumann boundary condition and Dirichlet boundary conditions, single delay and two delays, spatially homogeneous and spatially inhomogeneous periodic solutions. We obtain the equivalent Hopf bifurcation normal forms results derived by using the CMR and MTS methods for above examples, respectively.

Nonlinear stochastic modelling with Langevin regression.

Many physical systems characterized by nonlinear multiscale interactions can be modelled by treating unresolved degrees of freedom as random fluctuations. However, even when the microscopic governing equations and qualitative macroscopic behaviour are known, it is often difficult to derive a stochastic model that is consistent with observations. This is especially true for systems such as turbulence where the perturbations do not behave like Gaussian white noise, introducing non-Markovian behaviour to the dynamics. We address these challenges with a framework for identifying interpretable stochastic nonlinear dynamics from experimental data, using forward and adjoint Fokker–Planck equations to enforce statistical consistency. If the form of the Langevin equation is unknown, a simple sparsifying procedure can provide an appropriate functional form. We demonstrate that this method can learn stochastic models in two artificial examples: recovering a nonlinear Langevin equation forced by coloured noise and approximating the second-order dynamics of a particle in a double-well potential with the corresponding first-order bifurcation normal form. Finally, we apply Langevin regression to experimental measurements of a turbulent bluff body wake and show that the statistical behaviour of the centre of pressure can be described by the dynamics of the corresponding laminar flow driven by nonlinear state-dependent noise.

Nonlinear Network Dynamics with Consensus\u2013Dissensus Bifurcation

We study a nonlinear dynamical system on networks inspired by the pitchfork bifurcation normal form. The system has several interesting interpretations: as an interconnection of several pitchfork systems, a gradient dynamical system and the dominating behaviour of a general class of nonlinear dynamical systems. The equilibrium behaviour of the system exhibits a global bifurcation with respect to the system parameter, with a transition from a single constant stationary state to a large range of possible stationary states. Our main result classifies the stability of (a subset of) these stationary states in terms of the effective resistances of the underlying graph; this classification clearly discerns the influence of the specific topology in which the local pitchfork systems are interconnected. We further describe exact solutions for graphs with external equitable partitions and characterize the basins of attraction on tree graphs. Our technical analysis is supplemented by a study of the system on a number of prototypical networks: tree graphs, complete graphs and barbell graphs. We describe a number of qualitative properties of the dynamics on these networks, with promising modelling consequences.

An Approach to State Signal Shaping by Limit Cycle Model Predictive Control

A novel nonlinear model predictive control approach for state signal shaping is proposed. The control strategy introduces a residual shape cost kernel based on the dynamics of circular limit cycles from a supercritical Neimark-Sacker bifurcation normal form. This allows the controller to impose a fundamental harmonic state signal shape with a specific frequency and amplitude. An application example for harmonic compensation in distribution grids integrated with renewable energies is presented. The controller is tasked with the calculation of the reference current for an active power filter used for load compensation. The results achieved are successful, reducing the harmonic distortion to satisfactory levels while ensuring the correct frequency and amplitude.

Nonlinear stability analysis of motor bogies for high-speed trains

The traction motor flexibly suspended on the bogie frame is conducive to the lateral dynamic performance of high-speed trains. At present, most of the researches about the influences of suspension parameters of the traction motor on the stability of the bogie are limited to the linear system. In this paper, according to the suspension mode of the traction motor of a certain type of high-speed train in China, the dynamic equations of the motorized bogie with eight degrees of freedom are derived, and the nonlinear stability of the bogie system is analyzed. The bifurcation diagrams of the bogie system with different motor suspension parameters are obtained by using the continuation algorithm, and the linear and nonlinear critical speeds of the system are studied. The study shows that the suspension stiffness, damping coefficient, and the mass of the motor significantly affect the critical speeds of the bogie system. Then the mechanisms of the influence of suspension parameters on the linear and nonlinear critical speeds of the bogie are analyzed by the root locus method and Hopf bifurcation normal form theory, respectively.

Optimal phase control of biological oscillators using augmented phase reduction

We develop a novel optimal control algorithm to change the phase of an oscillator using a minimum energy input, which also minimizes the oscillator's transversal distance to the uncontrolled periodic orbit. Our algorithm uses a two-dimensional reduction technique based on both isochrons and isostables. We develop a novel method to eliminate cardiac alternans by connecting our control algorithm with the underlying physiological problem. We also describe how the devised algorithm can be used for spike timing control which can potentially help with motor symptoms of essential and parkinsonian tremor, and aid in treating jet lag. To demonstrate the advantages of this algorithm, we compare it with a previously proposed optimal control algorithm based on standard phase reduction for the Hopf bifurcation normal form, and models for cardiac pacemaker cells, thalamic neurons, and circadian gene regulation cycle in the suprachiasmatic nucleus. We show that our control algorithm is effective even when a large phase change is required or when the nontrivial Floquet multiplier is close to unity; in such cases, the previously proposed control algorithm fails.

Quasi-periodic intermittency in oscillating cylinder flow

Fluid dynamics induced by periodically forced flow around a cylinder is analysed computationally for the case when the forcing frequency is much lower than the von Kármán vortex shedding frequency corresponding to the constant flow velocity condition. By using the Koopman mode decomposition approach, we find a new normal form equation that extends the classical Hopf bifurcation normal form by a time-dependent term for Reynolds numbers close to the Hopf bifurcation value. The normal form describes the dynamics of an observable and features a forcing (control) term that multiplies the state, and is thus a parametric – i.e. not an additive – forcing effect. We find that the dynamics of the flow in this regime is characterized by alternating instances of quiescent and strong oscillatory behaviour and that this pattern persists indefinitely. Furthermore, the spectrum of the associated Koopman operator is shown to possess quasi-periodic features. We establish the theoretical underpinnings of this phenomenon – that we name quasi-periodic intermittency – using the new normal form model and show that the dynamics is caused by the tendency of the flow to oscillate between the unstable fixed point and the stable limit cycle of the unforced flow. The quasi-periodic intermittency phenomenon is also characterized by positive finite-time Lyapunov exponents that, over a long period of time, asymptotically approach zero.

Parameter-space topology of models for cell polarity

Reaction–diffusion systems have been widely successful in the theoretical description of biological patterning phenomena, giving rise to numerous models based on differing mechanisms, mathematical implementations and parameter choices. However, even for models with common design features, the diversity of mathematical realizations may hinder the identification of common behavior. Here, we analyze three different reaction–diffusion models for cell polarity that feature conservation of mass, rapid cytoplasmic diffusion and bistability via a cusp bifurcation of uniform states. In all three models, the nonuniform polar states are front solutions, and growth of domains ceases through stalling of a propagating front. For these three models we find a characteristic parameter space topology, comprising a region of linear instability that loops around the cusp point and that is enclosed by a ‘comet-shaped’ region of nonuniform domain states. We propose a minimal model based on the cusp bifurcation normal form that includes essential characteristics of all cell polarity models considered. For this minimal model, we provide a complete analytical description of the parameter space topology, and find that the instability loop appears as a generic property of the cusp bifurcation. This topological analysis provides a unifying understanding of earlier mathematically distinct models and is suitable to classify future models.

Periodically Forced Hopf Bifurcation

We study a periodically forced system of ODEs near a point of Hopf bifurcation, where the forcing is pure harmonic, has small amplitude $\varepsilon$, and has forcing frequency $\omega_F$ that is near the Hopf frequency $\omega_H$. In this system we vary the forcing frequency and determine all small amplitude periodic solutions to the forced system that have frequency $\omega_F$. In other words, we determine how the number of $\frac{2\pi}{\omega_F}$-periodic solutions to the forced system changes with $\omega_F$. This problem is complicated because of the existence of three small parameters: the amplitude of the forcing $\varepsilon$, the deviation of the bifurcation parameter from the point of Hopf bifurcation $\lambda$, and the relative deviation of the forcing frequency from the Hopf frequency $\omega=\frac{\omega_F-\omega_H}{\omega_F}$. Our results are presented in terms of bifurcation diagrams of amplitude of periodic solution versus $\omega$ for fixed $\varepsilon$ and $\lambda$. We assume that the unforced system has a supercritical Hopf bifurcation at $\lambda=0$ and that the coefficient of the cubic term of the Hopf bifurcation normal form is $\gamma_R+i\gamma_I$. We find that the qualitative form of the bifurcation diagrams depends on $\gamma=\gamma_I/\gamma_R$. For example, if $\lambda<0$ (so that the equilibrium is stable and there are no small amplitude periodic solutions in the unforced system), then multiplicity of periodic solutions of the forced system occurs in the bifurcation diagrams precisely when $|\gamma|>\sqrt{3}$.

Experimental unfolding of the nonlinear dynamics of a cable-mass suspended system around a divergence-Hopf bifurcation

Systematic experimental investigation of the finite amplitude dynamics of a multiple internally resonant suspended cable-mass, subjected to anti-phase support motion at primary resonance, is accomplished. Upon getting hints from a basic system configuration assumed as reference setup about the multiple bifurcation event possibly governing transition to complex dynamics, an improved experimental apparatus is used to make it technically accessible. Results obtained by varying three control parameters, namely the frequency and amplitude of excitation and the temperature of a thermostatic chamber embedding the experimental system, allow us to characterize in-depth various occurring classes of motion in terms of time and spatial complexity, to describe peculiar and/or persistent features of transition to nonregular dynamics, and to trace them back to a canonical scenario from bifurcation theory. Variable response paths are detected via bifurcation diagrams and spectra of singular values of measurement results, and overall behaviour charts are built in the excitation parameter space. Considering the temperature as a controllable parameter shows to be fundamental for: (i) indirectly setting cable material properties to values for which the conjectured codimension 2 bifurcation becomes apparent, (ii) qualitatively referring the experimental unfolding of regular and nonregular cable dynamics to the theoretical unfolding of the divergence-Hopf bifurcation normal form, and (iii) determining system response not only in the strict neighbourhood of the organizing divergence-Hopf bifurcation but also in the ensuing postcritical regions where the dependence of material damping on temperature affects secondary bifurcations to low-dimensional homoclinic chaos.

Towards Accurate PWL Approximations of Parameter-Dependent Nonlinear Dynamical Systems With Equilibria and Limit Cycles

This paper deals with piecewise-linear (PWL) approximations of nonlinear dynamical systems dependent on parameters and allowing the presence of few equilibria and/or limit cycles only. A method to derive the parameters of the PWL model is proposed that is based on the minimization of functionals defined to take into account a priori some dynamical features of the systems to be approximated. The method is validated by applying it to two simple dynamical systems, i.e., the cusp bifurcation normal form and the supercritical Hopf bifurcation normal form. The robustness of the approximations is checked, with a view to circuit implementations

Structurally Stable PWL Approximation of Nonlinear Dynamical Systems Admitting Limit Cycles: An Example

In this paper, we propose a variational method to derive the coefficients of piecewise-linear (PWL) models able to accurately approximate nonlinear functions, which are vector fields of autonomous dynamical systems described by continuous-time state-space models dependent on parameters. Such dynamical systems admit limit cycles, and the supercritical Hopf bifurcation normal form is chosen as an example of a system to be approximated. The robustness of the approximations is checked, with a view to circuit implementations.

NORMAL FORM OF DOUBLE HOPF BIFURCATION IN FORCED OSCILLATORS

A general four-dimensional normal form of a double Hopf bifurcation is considered. As a particular case, the normal form of a forced (non-autonomous) non-linear oscillator having two frequencies, namely the linear natural frequency and the excitation frequency, is studied in detail. When these two frequencies form two purely imaginary sub-blocks of order two in the real Jordan block, the system constitutes a double Hopf bifurcation. In this paper, the normal form of the double Hopf bifurcation is reduced when the two frequencies are not in resonance. In order to use the method of normal form, the non-autonomous problem is transformed into an autonomous one by a generalized co-ordinate transformation. The method of undetermined coefficients is used to find the double Hopf bifurcation normal form. The coefficients of similar monomials rather than similar powers of ε are compared to get the normal form to various orders. The steady state periodic solutions and the bifurcation equations of the forced non-linear vibration system in the case of non-resonant are studied. A Mathematica program is designed to find the normal form. Three examples are given to use the Mathematica program and to compare them with the existing results.

Degenerate Hopf bifurcations via feedback system theory: Higher-order harmonic balance

In this work we present some higher-order Hopf bifurcation formulas to analyze the rich dynamic behavior of chemical systems under the failure of some hypotheses of the classical Hopf theorem. The treatment is done entirely in an equivalent formulation in the so-called “frequency domain”. This approach gives a natural interpretation about some Hopf degeneracies involving limit points on the periodic branch and multiple periodic solutions. Moreover, we calculate the expression for the second curvature coefficient, extending previous results using this formulation. This coefficient is equivalent to one of the higher-order coefficients in the Hopf bifurcation normal form used previously by other researchers.

Normal forms for chemical reaction systems via the affine transformation

Many chemical reaction systems may be described by a system of ordinary differential equations. In general, a nonlinear change of variables is required to transform a system model to a much simpler equation called the normal form, which retains all important local nonlinear features of the system. For systems with a single nonlinearity, an affine transformation that retains the original steady-state and eigenvalue structure brings the system to a much simpler form, the companion normal form, which in the cases of two- and three-dimensional systems is shown to coincide with the classical F 1 and G 1 bifurcation normal forms that are based on Jordan block structure. The theory is applied to establish the properties of a general allosterically regulated enzymatic reaction sequence and of an isothermal catalytic reaction system.