Unstable Steady State Research Articles

Turing instabilities for three interacting species

In this paper, I prove necessary and sufficient conditions for the existence of Turing instabilities in a general system with three interacting species. Turing instabilities describe situations when a stable steady state of a reaction system (ordinary differential equation) becomes an unstable homogeneous steady state of the corresponding reaction–diffusion system (partial differential equation). Similarly to a well-known inequality condition for Turing instabilities in a system with two species, I find a set of inequality conditions for a system with three species. Furthermore, I distinguish conditions for the Turing instability when spatial perturbations grow steadily and the Turing–Hopf instability when spatial perturbations grow and oscillate in time simultaneously.

Adaptive delayed feedback control for stabilizing unstable steady states.

Delayed feedback control is a commonly used control method for stabilizing unstable periodic orbits and unstable steady states. The present paper proposes an adaptive tuning delay time rule for delayed feedback control focused on stabilizing unstable steady states. The rule is designed to slowly vary the delay time, increasing the difference between the past and current states of dynamical systems, which induces the delay time to automatically fall into the stability region. We numerically confirm that the tuning rule works well for the Stuart-Landau oscillator, FitzHugh-Nagumo model, and Lorenz system.

Monostable pulsating traveling waves in discrete periodic media with delay

The purpose of this article is to investigate various qualitative properties of pulsating traveling waves for a delayed lattice dynamical system with global interaction in periodic habitat. Under some appropriate assumptions, we first establish a Harnack type of inequality for its wave profile system. Then, we show that all wave profiles remain strictly increasing in the propagation process. Based on the Harnack’s inequality and monotonicity result, the other effort of the article is devoted to the exponential decay of the non-critical pulsating traveling waves towards the unstable steady state. Finally, we obtain the uniqueness of all non-critical pulsating traveling waves.

Time-delay Feedback Control of Fractional Chaotic Rössler Oscillator

In this research, we explore the utilization of the time-delayed feedback method to stabilize unstable steady states and aperiodic orbits within the chaotic fractional-order Rössler Oscillator. Employing the time-delay feedback control algorithm, we identify specific parameter ranges enabling the successful stabilization of unstable equilibria, considering variations in both feedback gain and time delay. Unlike previous research works where Caputo and Riemann-Liouville characterization of fractional derivatives are used, we are using Grünwald-Letnikov (GL) characterization because of its simplicity and ease of implementation and demonstrating the stability using the analytical and numerical analysis and plots of eigenvalues. Additionally, our analysis highlight the effectiveness of a sinusoidally modulated time delay in the control law, significantly expanding the stability region of steady states beyond the capabilities of the traditional time-delayed feedback scheme with a constant delay. Furthermore, the analysis of eigenvalues before and after applying the control strategy offers tangible insights into the system's stability dynamics.

Traveling fronts for Fisher–KPP lattice equations in almost-periodic media

This paper investigates the existence of almost-periodic traveling fronts for Fisher–KPP lattice equations in one-dimensional almost-periodic media. Using the Lyapunov exponent of the linearized operator near the unstable steady state, we give sufficient conditions for the existence of a minimal speed of traveling fronts. Furthermore, it is shown that almost-periodic traveling fronts share the same recurrence property as the structure of the media. As applications, we give some typical examples which have minimal speed, and the proof of this depends on a dynamical system approach to the almost-periodic Schrödinger operator.

Global existence, blow-up and dynamical behavior in a nonlocal parabolic problem with variational structure

In this paper, we study an initial boundary value problem for a nonlocal parabolic equation that includes a diffusion term and convex–concave nonlinearities. By establishing the Lq-estimate and analyzing its energy, we prove the existence of global solutions and obtain some blow-up conditions. Given the variational structure of the problem, we utilize the Mountain-pass theorem to demonstrate the existence of nontrivial steady-state solutions. Additionally, we establish the dynamical behavior of global solutions whose trajectory is relatively compact in H01(Ω), and which uniformly converge to some non-zero steady state after a long time, owing to the energy functional satisfying the P.S. condition. Finally, we derive an unstable steady states sequence by using another minimax theorem.

An Organizing Center of Codimension Four in a Predator-Prey Model with Generalist Predator: From Tristability and Quadristability to Transients in a Nonlinear Environmental Change

In this paper, we take the Rosenzweig–MacArthur (RM) model with generalist predator as an example in a constant or changing environment. When the environment is fixed, we provide a more easily verifiable classification, in terms of the coefficients of the system with nilpotent linear part and general higher terms, to determine the types and codimension of nilpotent singularities in a general planar system. Second, by using the existing classification and some algebraic methods, we show that the highest codimension of a nilpotent focus is 4 and the sample RM model with generalist predator can exhibit nilpotent focus bifurcation of codimension 4. Our results indicate that generalist predation can cause not only richer bifurcations and dynamics (such as multitype tristability and quadristability, a figure-eight loop) but also the possible extirpation of prey. When the environment is changing, we study the impact of the rate and intensity of a nonlinear environmental change on dynamics. The key observations on the asymptotic and transient dynamics include (i) transient tracking on unstable steady states or oscillations, and transient-related regime shifts; (ii) slow and fast regime shifts; (iii) regulation of transient dynamics by the environmental change parameters and ; (iv) slow negative or fast positive environmental change can delay or even avoid population extirpation.

Analysis of High-Temperature Superconducting Current Leads: Multiple Solutions, Thermal Runaway, and Protection

The multiple steady states of Ag/Bi2212-composite high-Tc superconducting leads modeling current delivery to a superconducting magnet have been numerically calculated. The model is based on longitudinal conduction combined with convective heat dissipation from a helium gas stream along the conductor. Because of the nonlinearities introduced by the voltage–current relationship and the temperature-dependent material properties, up to three solutions have been identified within the range of parameters considered. Linear stability analysis reveals that two of them are stable, i.e., the superconducting and the normal branches, while the remaining one is unstable. The limit points separating the stable from the unstable steady states form the blow-up threshold, beyond which any further increase in the operating current results in a thermal runway. Interesting findings are that for low filling ratios no bounded solution exists when the length of the lead exceeds the lower limit point, while very high maximum temperatures may be encountered along the normal solution branch. The effect of various parameters such as the conduction–convection parameter, the applied current, and the reduction in coolant flow (LOFA) on the bifurcation structure and their stabilization effect on the blow-up threshold are also evaluated. Apart from the steady and unsteady operating modes, the multiplicity analysis is also used to identify the range of the design and operating variables where safe operation, with a sufficient margin from the onset of instabilities, may be established, thus facilitating the protection of the leads and the device connected to it.

Bistable traveling waves in degenerate competitive systems

This article studies the bistable traveling wave solutions for diffusion-competition systems, in which the degenerate nonlinearities appear and two different diffusion modes are admissible. The corresponding reaction system has two locally asymptotically stable steady states and two unstable steady states. Using the abstract theory established for monotone semiflows, we obtain the existence of monotone bistable traveling wave solutions. The uniqueness of wave speed is confirmed by constructing appropriate upper and lower solutions. When the initial value satisfies proper conditions, it is proved that the wave speed is the expansion speed of two functions defined by the corresponding Cauchy problem, which implies that the sign of wave speed determines the relative competitiveness in population dynamics. Numerically, it seems that the species with stronger degeneracy may have stronger competitiveness under bistable assumption and other proper conditions.

Subharmonic locking and frequency combs in frequency-swept semiconductor lasers.

We analyze a delay differential equationmodel for a swept semiconductor laser and demonstrate existence of various periodic solutions that are subharmonically locked to the sweep rate. These solutions provide optical frequency combs in spectral domain. We investigate the problem numerically and show that, due to the translational symmetry of the model, there exists a hysteresis loop formed by branches of steady states solutions, bridges of periodic solutions connecting stable and unstable steady state branches, and isolated branches of limit cycles. We discuss the role of bifurcation points and limit cycles embedded into the loop in the formation of the subharmonic dynamics.

Data-driven control of agent-based models: An Equation/Variable-free machine learning approach

We present an Equation/Variable free machine learning (EVFML) framework for the control of the collective dynamics of complex/multiscale systems modeled via microscopic/agent-based simulators. The approach obviates the need for construction of surrogate, reduced-order models. The proposed implementation consists of three steps: (A) from high-dimensional agent-based simulations, machine learning, in particular, non-linear manifold learning (Diffusion Maps (DMs)) identifies a set of coarse-grained variables that parametrize the low-dimensional manifold on which the emergent/collective dynamics evolve. The out-of-sample extension and pre-image problems, i.e. the construction of non-linear mappings from the high-dimensional input space to the low-dimensional manifold and back, are solved by coupling DMs with the Nyström extension and Geometric Harmonics, respectively; (B) having identified the manifold and its coordinates, we exploit the Equation-free approach to perform numerical bifurcation analysis of the emergent dynamics; then (C) based on the previous steps, we design data-driven embedded wash-out controllers that drive the agent-based simulators to their intrinsic, imprecisely known, emergent open-loop unstable steady-states, thus demonstrating that the scheme is robust against numerical approximation errors and modeling uncertainty. The efficiency of the framework is illustrated by controlling emergent unstable (i) traveling waves of a deterministic agent-based model of traffic dynamics, and (ii) equilibria of a stochastic financial market model with mimesis.

Bistable traveling waves in a delayed competitive system

<p style='text-indent:20px;'>This paper is concerned with the existence, uniqueness and asymptotic stability of traveling wave solutions in a delayed competitive system with spatial diffusion. When the interspecific competition is strong, the corresponding functional differential system has two stable and two unstable steady states. Firstly, we establish the existence of bistable traveling wave solutions by the theory of monotone semiflows. With the help of squeezing technique based on comparison principle, the asymptotic stability of traveling wave solutions is proved, which further implies the uniqueness of wave speed and wave profile in the sense of phase shift.</p>

Calculation of the steady states in dynamic semiconductor laser models

We discuss numerical challenges in calculating stable and unstable steady states of widely used dynamic semiconductor laser models. Knowledge of these states is valuable when analyzing laser dynamics and different properties of the lasing states. The example simulations and analysis mainly rely on 1(time)+1(space)-dimensional traveling-wave models, where the steady state defining conditions are formulated as a system of nonlinear algebraic equations. The performed steady state calculations reveal limitations of the Lang-Kobayashi model, explain nontrivial bias threshold relations in lasers with several electrical contacts, or predict and explain transient dynamics when simulating such lasers.

Complex dynamics near extinction in a predator-prey model with ratio dependence and Holling type III functional response

In this paper, we analyze a recently proposed predator-prey model with ratio dependence and Holling type III functional response, with particular emphasis on the dynamics close to extinction. By using Briot-Bouquet transformation we transform the model into a system, where the extinction steady state is represented by up to three distinct steady states, whose existence is determined by the values of appropriate Lambert W functions. We investigate how stability of extinction and coexistence steady states is affected by the rate of predation, predator fecundity, and the parameter characterizing the strength of functional response. The results suggest that the extinction steady state can be stable for sufficiently high predation rate and for sufficiently small predator fecundity. Moreover, in certain parameter regimes, a stable extinction steady state can coexist with a stable prey-only equilibrium or with a stable coexistence equilibrium, and it is rather the initial conditions that determine whether prey and predator populations will be maintained at some steady level, or both of them will become extinct. Another possibility is for coexistence steady state to be unstable, in which case sustained periodic oscillations around it are observed. Numerical simulations are performed to illustrate the behavior for all dynamical regimes, and in each case a corresponding phase plane of the transformed system is presented to show a correspondence with stable and unstable extinction steady state.

Global stability of critical-speed pulsating fronts for degenerate monostable reactions

This paper is devoted to the global exponential stability of pulsating fronts for a kind of reaction–advection–diffusion equations with degenerate monostable nonlinearity in a periodic medium. We focus on the pulsating front with critical-speed, which is unique up to translation and decays exponentially when it approaches the unstable steady state 0. Based on the sub-super solution and the squeezing methods, we show that the solution will converge to a shift of the front exponentially in time when the general initial data decays faster than a positive exponential function at one end and has a positive lower bound at the other end.

Classical route to ergodicity and scarring phenomena in a two-component Bose-Josephson junction

We consider a Bose-Josephson junction (BJJ) formed by a binary mixture of ultracold atoms to investigate the manifestation of coherent collective dynamics on ergodicity and quantum scars, unfolding the connection between them. By tuning the inter- and intra-species interaction, we demonstrate a rich variety of Josephson dynamics and transitions between them, which plays a crucial role in controlling the overall ergodic behavior. The signature of underlying classicality is revealed from the entanglement spectrum, which also elucidates the formation of quantum scars of unstable steady states and of periodic orbits leading to athermal behavior. The degree of ergodicity across the energy band and scarring phenomena can be probed from the auto-correlation function as well from the phase fluctuation of the condensates, which has relevance in cold atom experiments.

The interaction of multiple bubbles in a Hele-Shaw channel

We study the dynamics of two air bubbles driven by the motion of a suspending viscous fluid in a Hele-Shaw channel with a small elevation along its centreline via physical experiment and numerical simulation of a depth-averaged model. For a single-bubble system we establish that, in general, the bubble propagation speed monotonically increases with bubble volume so that two bubbles of different sizes, in the absence of any hydrodynamic interactions, will either coalesce or separate in a finite time. However, our experiments indicate that the bubbles interact and that an unstable two-bubble state is responsible for the eventual dynamical outcome: coalescence or separation. These results motivate us to develop an edge-tracking routine and to calculate these weakly unstable two-bubble steady states from the governing equations. The steady states consist of pairs of ‘aligned’ bubbles that appear on the same side of the centreline with the larger bubble leading. We also discover, through time-dependent simulations and physical experiment, another class of two-bubble states that, surprisingly, are stable. In contrast to the ‘aligned’ steady states, these bubbles appear on either side of the centreline and are ‘offset’ from each other. We calculate the bifurcation structures of both classes of steady states as the flow rate and bubble volume ratio are varied. We find that they exhibit intriguing similarities to the single-bubble bifurcation structure, which has implications for the existence of $n$ -bubble steady states.

Closed-loop identification of plant and disturbance models based on data-driven generalized minimum variance regulatory control

The paper addresses a closed-loop identification method based on generalized minimum variance (GMV) evaluation. Since the proposed method uses routine operation data, it requires no extra experiment with an external excitation signal. The model parameters of the plant and the disturbance are obtained simultaneously using a single set of input–output data generated by stochastic disturbance. A new variance criterion for closed-loop identification is derived through the conversion of the GMV evaluation function that has originally been developed for data-driven regulatory control. In the conversion, the feedback invariant polynomial, which is estimated from time series analysis of the closed-output signal, plays a key role. The features of the proposed approach lead to bridge closed-loop identification with control performance assessment as well as data-driven controller parameters tuning. The paper proves that the optimization of the proposed criterion results in the unique optimal solution, which corresponds to the true plant and disturbance model parameters. In numerical examples, the proposed method is applied to datasets obtained from a continuous stirred tank reactor (CSTR), which is operated around an unstable steady state. The result illustrates the effectiveness of the proposed closed-loop identification method.

Dynamic real-time optimization for nonlinear systems with Lyapunov stabilizing MPC

The present work addresses the problem of stabilizing dynamic real-time optimization and control of nonlinear systems. Dynamic real-time optimization (DRTO) is a framework for computing an optimal operating trajectory for a plant and providing corresponding set-points for an underlying control algorithm to track. This technique can be enhanced by directly modeling the underlying control algorithm (such as an MPC) in a closed-loop DRTO (CL-DRTO). This allows the CL-DRTO to predict both the controller and plant responses to set-point changes, improving performance of the overall system. Certain systems are additionally complicated by the existence of unstable steady states, which make control of the system substantially more difficult. A number of stabilizing MPC techniques have been developed in order to handle this, and a closed-loop DRTO formulation was recently proposed to handle linear systems. This work seeks to bring the improved performance of a CL-DRTO to stabilizing control systems by formulating a CL-DRTO which utilizes and explicitly models an underlying Lyapunov stabilizing MPC to achieve stabilization for nonlinear systems. The proposed formulation is compared to the previously developed formulation to demonstrate the improved closed-loop control and performance.

Dynamical analysis for a car-following model with delayed-feedback control of both velocity and acceleration differences

In this paper, based on an existing car-following model, a novel delayed-feedback control scheme depending on both velocity and acceleration differences is proposed to stabilize the unstable steady state and suppress traffic jam. The controller of each vehicle is constructed to use the difference information i.e., velocity difference and acceleration difference between the present state and the past state, and the velocity and the acceleration of other vehicles in the platoon are not required. Using stability switching and definite integral stability methods, the stable intervals of time delays and feedback gains are obtained by calculating the number of all unstable eigen-values of the characteristic equation. Then, the unstable traffic flow can be stabilized through choosing the proper time delays and feedback gains from the corresponding stable intervals. Compared with the delayed-feedback control of velocity difference, the proposed control method expands the stable delay interval and improves the robust performance. Numerical simulations under periodic and open boundary are demonstrated to verify the theoretical results.