Pair Of Imaginary Eigenvalues Research Articles

The 0:1 resonance bifurcation associated with the supercritical Hamiltonian pitchfork bifurcation

We consider the non-semisimple 0:1 resonance (i.e., the unperturbed equilibrium has two purely imaginary eigenvalues ( and ) and a non-semisimple double-zero one) Hamiltonian bifurcation with one distinguished parameter, which corresponds to the supercritical Hamiltonian pitchfork bifurcation. Based on BCKV singularity theory established by Broer et al. (Z. Angew. Math. Phys. 44:389-432, 1993), this bifurcation essentially triggered by the reversible universal unfolding with respect to BCKV-restricted morphisms of the planar non-semisimple singularity (the is regarded as distinguished parameter with respect to the external parameter λ). We first give the plane bifurcation diagram of the integrable Hamiltonian on each level of integral in detail, which is related to the usual supercritical Hamiltonian pitchfork bifurcation. Then, we use the -symmetry generated by the additional pair of imaginary eigenvalues to reconstruct the above plane bifurcation phenomenon caused by the zero eigenvalue pair into the case with two degrees of freedom. Finally, we prove the persistence of typical bifurcation scenarios (e.g., 2-dimensional invariant tori and the symmetric homoclinic orbit) under the small Hamiltonian perturbations, as proposed by Broer et al. (Z Angew Math Phys 44: 389-432, 1993). An example system (the coupled Duffing oscillator) with strong linear coupling and weak local nonlinearity is given for this bifurcation.

Fear and Group Defense Effect of a Holling Type IV Predator-Prey System Intraspecific Competition

Field and experimental data on aquatic ecosystem species show the effect of fear on changing prey demographics. The fear effect has an impact on aquatic ecosystems, such as species migration to settled areas. In this paper, the type of research described is a literature study. The cost effect assigned to the reproductive system of the prey population and the predation function response are given as Holling Type IV for research purposes to model the fear effect. Some research novelties, the equilibrium points are all shown in the population dynamics system model with an analysis of positive equilibrium. Positive and biologically realistic equilibrium points were analyzed using the Routh-Hurwitz criterion which is mathematically a local asymptotically stable. A pair of imaginary eigenvalues with a negative real part can increase population growth. An equilibrium region showing equilibrium for several parameters such as extinction, no predators, and two populations coexisting in a sustainable manner. The correlation and fluctuation of fear and fear cost were investigated to obtain a better model. The results of the numerical simulations show that the prey population becomes more daring to fight or fighting power with significant prey growth rates or high predator mortality rates. Doi: 10.28991/ESJ-2023-07-02-06 Full Text: PDF

Impact of fear effect on the growth of prey in a predator-prey interaction model

Several field data and experiments on a terrestrial vertebrates exhibited that the fear of predators would cause a substantial variability of prey demography. Fear for predator population enhances the survival probability of prey population, and it can greatly reduce the reproduction of prey population. Based on the experimental evidence, we proposed and analyzed a prey-predator system introducing the cost of fear into prey reproduction with Holling type-II functional response. We investigate all the biologically feasible equilibrium points, and their stability is analyzed in terms of the model parameters. Our mathematical analysis exhibits that for strong anti-predator responses can stabilize the prey-predator interactions by ignoring the existence of periodic behaviors. Our model system undergoes Hopf bifurcation by considering the birth rate r0 as a bifurcation parameter. For larger prey birth rate, we investigate the transition to a stable coexisting equilibrium state, with oscillatory approach to this equilibrium state, indicating that the greatest characteristic eigenvalues are actually a pair of imaginary eigenvalues with real part negative, which is increasing for r0. We obtained the conditions for the occurrence of Hopf bifurcation and conditions governing the direction of Hopf bifurcation, which imply that the prey birth rate will not only influence the occurrence of Hopf bifurcation but also alter the direction of Hopf bifurcation. We identify the parameter regions associated with the extinct equilibria, predator-free equilibria and coexisting equilibria with respect to prey birth rate, predator mortality rates. Fear can stabilize the predator-prey system at an interior steady state, where all the species can exists together, or it can create the oscillatory coexistence of all the populations. We performed some numerical simulations to investigate the relationship between the effects of fear and other biologically related parameters (including growth/decay rate of prey/predator), which exhibit the impact that fear can have in prey-predator system. Our numerical illustrations also demonstrate that the prey become less sensitive to perceive the risk of predation with increasing prey growth rate or increasing predators decay rate.

Application of Floquet theory to formation flying in elliptic orbits via a Hamiltonian structure-preserving control

In this study, an improved Hamiltonian structure-preserving controller is applied for formation flying in elliptic orbits. The Lyapunov-Floquet theory is applied to the Tschauner-Hempel (TH) equation in order to generate its transformed form with two zero eigenvalues and two pairs of imaginary eigenvalues. Then, an improved Hamiltonian structure-preserving (HSP) control is introduced to the transformed system to eliminate the instability of double zeros. All bounded trajectories can be generated by basic solutions, which effectively reveal the following performances of the controller, such as the relationship between the frequencies of basic motions and control gains are detected and the wide selection of gains for stability. Local and global optimizations on gains are discussed based on different performances of the controller. One focuses on the sensitivity of output with respect to input of controller and the other aims at a minimal fuel consumption for the maximum likelihood. The ergodic relationship between the fuel consumption and initial condition is established for the minimal consumption. To avoid in-orbit collision, a collision-free strategy based on the artificial potential function is proposed to enhance the proposed controller, which make sure a minimum safety distance for the formation.

Orbital Normal Forms for a family of-zero Singularity

Consider a Dynamical system x'=F(x,µ) such that its linear part has a pair of imaginary eigenvalues and one zero eigenvalue (Hopf zero singularity). Recently, the simplest normal form for this singular system has been obtained by sl(2) Lie algebra theory and the decomposition of space into three invariant subspaces. The normal form of this singular system is divided into three general cases. In this paper, the obtained results will be extended to orbital normal form and one of the three aforementioned cases will be discussed. The orbital obtained normal form will be simpler than the previous simplest normal form.

Design of Type-2 Fuzzy Logic Power System Stabilizer Using Limit cycle approach

Power system is subjected to wide range of operating conditions. Heavily loaded power systems are subjected to Hopf bifurcations resulting in oscillatory instability. A power system shifts to the dynamic instability region encountered by unstable limit cycles, this result in increase of oscillatory behaviour of power system. Many of the authors suggested the theories like Poincare-Bendixson theory, bifurcation theory, eigenvalue theory and shooting methods for finding dynamic stability margin. Small signal stability focuses on eigenvalue computation at a given operating point. The onset of instability is obtained by finding Hopf bifurcations associated with a pair of imaginary eigenvalues using eigenvalue analysis. Computation of these HBs is a tedious and complex task in traditional approach. In this paper, a novel computational algorithm is proposed to identify the loading conditions resulting in these limit cycles, which are associated with a pair of imaginary eigenvalues, using genetic algorithm for single machine infinite bus (SMIB) and multi machine power system (MMPS) models. The derived loading conditions are used to systematically design a robust type-2 fuzzy logic power system stabiliser. The efficacy of the proposed controller is tested for linear and nonlinear models of power system under various operating conditions.

Low rank differential equations for Hamiltonian matrix nearness problems

For a Hamiltonian matrix with purely imaginary eigenvalues, we aim to determine the nearest Hamiltonian matrix such that some or all eigenvalues leave the imaginary axis. Conversely, for a Hamiltonian matrix with all eigenvalues lying off the imaginary axis, we look for a nearest Hamiltonian matrix that has a pair of imaginary eigenvalues. The Hamiltonian matrices can be allowed to be complex or restricted to be real. Such Hamiltonian matrix nearness problems are motivated by applications such as the analysis of passive control systems. They are closely related to the problem of determining extremal points of Hamiltonian pseudospectra. We obtain a characterization of optimal perturbations, which turn out to be of low rank and are attractive stationary points of low-rank differential equations that we derive. We use a two-level approach, where in the inner level we determine extremal points of the Hamiltonian $$\varepsilon $$ ? -pseudospectrum for a given $$\varepsilon $$ ? by following the low-rank differential equations into a stationary point, and on the outer level we optimize for $$\varepsilon $$ ? . This permits us to give fast algorithms--exhibiting quadratic convergence--for solving the considered Hamiltonian matrix nearness problems.

The Existence of Bifurcating Invariant Tori in a Spatially Extended Reaction-Diffusion-Convection System with Spatially Localized Amplification

We consider a spatially extended reaction-diffusion-convection system with a marginally stable ground state and a spatially localized amplification. We are interested in solutions bifurcating from the spatially homogeneous ground state in the case when pairs of imaginary eigenvalues simultaneously cross the imaginary axis. For this system we prove the bifurcation of a family of invariant tori which may contain quasiperiodic solutions. There is a serious difficulty in obtaining this result, because the linearization at the ground state possesses an essential spectrum up to the imaginary axis for all values of the bifurcation parameter. To construct the invariant tori, we use their invariance under the flow which manifests in a condition in PDE form. The nonlinear terms of this resulting PDE exhibit a loss of regularity. Since the linear part of this PDE is not smoothing, an adaption of the hard implicit function theorem (or Nash-Moser scheme) and energy estimates will be used to prove our result.

Stability switches and double Hopf bifurcation in a two-neural network system with multiple delays

Time delay is an inevitable factor in neural networks due to the finite propagation velocity and switching speed. Neural system may lose its stability even for very small delay. In this paper, a two-neural network system with the different types of delays involved in self- and neighbor- connection has been investigated. The local asymptotic stability of the equilibrium point is studied by analyzing the corresponding characteristic equation. It is found that the multiple delays can lead the system dynamic behavior to exhibit stability switches. The delay-dependent stability regions are illustrated in the delay-parameter plane, followed which the double Hopf bifurcation points can be obtained from the intersection points of the first and second Hopf bifurcation, i.e., the corresponding characteristic equation has two pairs of imaginary eigenvalues. Taking the delays as the bifurcation parameters, the classification and bifurcation sets are obtained in terms of the central manifold reduction and normal form method. The dynamical behavior of system may exhibit the quasi-periodic solutions due to the Neimark- Sacker bifurcation. Finally, numerical simulations are made to verify the theoretical results.

Singularities on the boundary of the stability domain near [formula omitted]-resonance

We study the linear differential equation x ˙ = L x in 1 : 1 -resonance. That is, x ∈ R 4 and L is 4 × 4 matrix with a semi-simple double pair of imaginary eigenvalues ( i β , − i β , i β , − i β ) . We wish to find all perturbations of this linear system such that the perturbed system is stable. Since linear differential equations are in one-to-one correspondence with linear maps we translate this problem to gl ( 4 , R ) . In this setting our aim is to determine the stability domain and the singularities of its boundary. The dimension of gl ( 4 , R ) is 16, therefore we first reduce the dimension as far as possible. Here we use a versal unfolding of L, i.e. a transverse section of the orbit of L under the adjoint action of Gl ( 4 , R ) . Repeating a similar procedure in the versal unfolding we are able to reduce the dimension to 4. A 3-sphere in this 4-dimensional space contains all information about the neighborhood of L in gl ( 4 , R ) . Considering the 3-sphere as two 3-discs glued smoothly along their common boundary we find that the boundary of the stability domain is contained in two right conoids, one in each 3-disc. The singularities of this surface are transverse self-intersections, Whitney umbrellas and an intersection of self-intersections where the surface has a self-tangency. A Whitney stratification of the 3-sphere such that the eigenvalue configurations of corresponding matrices are constant on strata allows us to describe the neighborhood of L and in particular identify the stability domain.

Benjamin–Ono periodic bifurcating water waves in presence of an essential spectrum

The mathematical study of travelling waves in the potential flow of two superposed layers of perfect fluid can be set as an ill-posed evolutionary problem, in which the horizontal unbounded space variable plays the role of “time”. In this paper we consider two problems for which the bottom layer of fluid is infinitely deep: for the first problem, the upper layer is bounded by a rigid top and there is no surface tension at the interface; for the second problem, there is a free surface with a large enough surface tension. In both problems, the linearized operator Lɛ (where ɛ is a combination of the physical parameters) around 0 possesses an essential spectrum filling the entire real line, with in addition a simple eigenvalue in 0. Moreover, for ɛ<0, there is a pair of imaginary eigenvalues which meet in 0 when ɛ=0 and which disappear in the essential spectrum for ɛ>0. For ɛ>0 small enough, we prove in this paper the existence of a two parameter family of periodic travelling waves (corresponding to periodic solutions of the dynamical system). These solutions are obtained in showing that the full system can be seen as a perturbation of the Benjamin–Ono equation. The periods of these solutions run on an interval (T0,∞) possibly except a discrete set of isolated points.

DYNAMICAL COMPLEXITY NEAR NON-CONTROLLABLE 3D PIECEWISE LINEAR LUR'E SYSTEMS

A very useful approach to understand the complex behavior of a system is to consider its degenerate situations. Accordingly the structure of periodic orbits for some degenerate symmetric piecewise linear three-dimensional systems with three zones, whose linear parts share a pair of imaginary eigenvalues is analyzed. Under this hypothesis, the system is noncontrollable and its study can be reduced to the analysis of a periodic one-dimensional equation that can have up to five periodic orbits.

Water waves as a spatial dynamical system; infinite depth case

We review the mathematical results on traveling waves in one or several superposed layers of potential flow, subject to gravity, with or without surface and interfacial tension, where the bottom layer is infinitely deep. The problem is formulated as a "spatial dynamical system," and it is shown that the linearized operator of the resulting reversible system has an essential spectrum filling the real line. We consider three cases where bifurcation occurs. (i) The first case is when, in moving a parameter, two pairs of imaginary eigenvalues merge into one pair of double eigenvalues, and then split into four symmetric complex conjugate eigenvalues. (ii) The second case is when one pair of imaginary eigenvalues meet in 0, and disappear; (iii) the third case is when the phenomenon described in (ii) is superposed to the presence of another pair of imaginary eigenvalues sitting at finite distance from 0. We give a physical example for each case and more specially study the solitary waves and generalized solitary waves, emphasizing the differences, in the methods and in the results, between these cases and the finite depth case.

Homoclinic solutions of reversible systems possessing an essential spectrum

In this Note we consider bifurcations of a class of infinite dimensional reversible dynamical systems. These systems possess a family of equilibrium solutions near the origin. We also assume that the linearized operator at the origin L ɛ has an essential spectrum filling the entire real line, in addition to a simple eigenvalue at 0. Moreover, for parameter values ɛ < 0 there is a pair of imaginary eigenvalues which meet in 0 for ɛ = 0 , and which disappear for ɛ > 0 . We give assumptions on L ɛ and on the non-linear term which describe this situation. These assumptions are sufficient to prove the existence of a family of solutions homoclinic to the equilibrium solutions near the origin. The result of this Note applies when we look for solitary waves in superposed layers of perfect fluids, the bottom one being infinitely deep. To cite this article: M. Barrandon, C. R. Acad. Sci. Paris, Ser. I 339 (2004).

Reversible Equivariant Hopf Bifurcation

In this paper we study codimension-one Hopf bifurcation from symmetric equilibrium points in reversible equivariant vector fields. Such bifurcations are characterized by a doubly degenerate pair of purely imaginary eigenvalues of the linearization of the vector field at the equilibrium point. The eigenvalue movements near such a degeneracy typically follow one of three scenarios: splitting (from two pairs of imaginary eigenvalues to a quadruplet on the complex plane), passing (on the imaginary axis), or crossing (a quadruplet crossing the imaginary axis). We give a complete description of the behaviour of reversible periodic orbits in the vicinity of such a bifurcation point. For non-reversible periodic solutions, in the case of Hopf bifurcation with crossing eigenvalues, we obtain a generalization of the equivariant Hopf Theorem.

Invariant manifolds of periodic orbits for piecewise linear three-dimensional systems

This paper analyses the existence of invariant manifolds of periodic orbits for a specific piecewise linear three‐dimensional system with two zones, whose linear parts share a pair of imaginary eigenvalues. This degenerate situation is obtained from the lack of controllability. The analysis proceeds by its reduction to a periodic one‐dimensional equation for which some results of the Ambrosetti–Prodi type are given.

Hopf Bifurcation and Exchange of Stability in Diffusive Media

We consider solutions bifurcating from a spatially homogeneous equilibrium under the assumption that the associated linearization possesses a continuous spectrum up to the imaginary axis, for all values of the bifurcation parameter, and that a pair of imaginary eigenvalues crosses the imaginary axis. For a reaction-diffusion-convection system we investigate the nonlinear stability of the trivial solution with respect to spatially localized perturbations, prove the occurrence of a Hopf bifurcation and the nonlinear stability of the bifurcating time-periodic solutions, again with respect to spatially localized perturbations.

On Lyapunov stability and normal forms of nonlinear systems with a nonsemisimple critical mode. I. Zero eigenvalue

This work evolved from an endeavor to derive stability criteria and Poincare normal forms for nonlinear systems associated with a nonsemisimple zero (in Part I) or a pair of imaginary eigenvalues (in Part II). The stability criteria are given in terms of the noninteracting vector restoring and restraining forces, which are motivated from the Lienard equation for nonlinear mass-damper-spring system models. Lyapunov functions are constructed explicitly to fulfill the La Salle invariant principle for local or global stability assertion. It turned out that the Lyapunov functions thus constructed apply to a wide variety of linear stability scenarios. By introducing the notions of restoring and restraining forces, how the Lyapunov functions, the stability criteria and the system dynamics interplay are also exhibited. Two distinct classes of nonlinearities which we referred to as being arithmetical and transcendental, emerged. In some sense, such systems carry nonlinear lags coexisting with the linear lead. In particular, a characteristic of the nonlinear dynamics, a staircase structure, is discovered. Further extension is also made to incorporate nondestabilizing perturbation, which bears important bifurcational implications.

On Lyapunov stability and normal forms of nonlinear systems with a nonsemisimple critical mode. II. Imaginary eigenvalues pair

For pt. I see ibid., vol. 47, no. 6 (June 2000). This paper studies nonlinear systems associated with or linked to a nonsemisimple (NSS) pair of imaginary eigenvalues. Stability criteria and Poincare normal forms (PNF's) are derived via explicit construction of Lyapunov functions. It is a counterpart and continuation of Part I, which considered the case with an NSS zero eigenvalue. While originating from stability analysis for a nonsemisimple critical mode, the development extends to coexisting (semi-) simple critical modes and to stable modes, as in Part I. The present NSS imaginary pairs (NSSIP) case is shown to possess certain characteristics common to NSS zero (NSSZ), such as the nonlinear lag and the staircase structure, but retains its own identities as well. These include the less surprising, the submodular variable-pair circles which emerge from the units of the potential generator and the more remarkable stricter specification in the dynamics, which is possibly attributable to the inherited conjugacy. An elementary partial differential equation traversing the development shows potential for nontrivial generalization.

Arnold tongues and mode–jumping in the supercritical post–buckling of an archetypal elastic structure

The buckling and post–buckling of a classical simply supported elastic strut on a stiffening elastic foundation is treated as a nonlinear dynamical system. For supercritical loads this fourth–order system has a pair of imaginary eigenvalues, indicating the possibility of either a double or a quasi–periodic response. Finite lengths impose constraints that restrict such responses to the rational positions on the real line of the ratio of competing wavelengths: for irrational positions the response locks onto a single periodic wave. The extent of the mode–locking is described in parameter space by the geometry of Arnold tongues. Critical positions on the tongues are identified which mark conditions for mode–jumping from one wavelength to another.