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In the mid 1980s, Lichtenstein, Pnueli, and Zuck proved a classical theorem stating that every formula of Past LTL (the extension of Linear Temporal Logic (LTL) with past operators) is equivalent to a formula of the form \(\bigwedge _{i=1}^n {\mathbf {G}}{\mathbf {F}}\varphi _i \vee {\mathbf {F}}{\mathbf {G}}\psi _i\) , where φi and ψi contain only past operators. Some years later, Chang, Manna, and Pnueli built on this result to derive a similar normal form for LTL. Both normalization procedures have a non-elementary worst-case blow-up, and follow an involved path from formulas to counter-free automata to star-free regular expressions and back to formulas. We improve on both points. We present direct and purely syntactic normalization procedures for LTL, yielding a normal form very similar to the one by Chang, Manna, and Pnueli, that exhibit only a single exponential blow-up. As an application, we derive a simple algorithm to translate LTL into deterministic Rabin automata. The algorithm normalizes the formula, translates it into a special very weak alternating automaton, and applies a simple determinization procedure, valid only for these special automata. [ABSTRACT FROM AUTHOR]

In this paper, we provide the normal form for the Hopf bifurcation of a class of the reaction‐diffusion equation with memory‐based diffusion and nonlocal effect, where the delay is present in the differential term, similar to the chemotaxis model with time delay. The eigenvalue problems and the decomposition of the phase space are discussed in detail. Through a series of variable transformations, we obtain the third‐order truncated normal form of the model constrained on the central manifold and its equivalent equation in polar coordinates. Then, with the help of the dynamic analysis for the finite dimensional equations, the key parameters for determining the direction and stability of the Hopf bifurcation are given. These theoretical results are applied to the Bazykin's model, the stability, Turing bifurcation and Hopf bifurcation of the equilibrium are demonstrated through both theoretical and numerical methods. [ABSTRACT FROM AUTHOR]

The cognitive abilities of animals, such as memory, have a significant impact on their movement in space. In this paper, we consider a radio-dependent model with memory-based diffusion under the conditions of Neumann boundary. The stability of a positive equilibrium and the existence of the Turing–Hopf bifurcation induced by memory diffusion and memory delay are carried out in details. Notably, our findings indicate that with a relatively short average memory period, the large memory diffusion can stabilize an otherwise unstable equilibrium. In addition, the third-order truncated normal form for the Turing–Hopf bifurcation restricted to the central manifold is derived, which can reveal the generation of some steady-state and time-periodic solutions with spatial heterogeneity. The coefficients within the normal form are systematically determined through matrix operations, and these results can also be applied to other models with memory diffusion. Ultimately, leveraging the theoretical findings, we elucidated the intricate spatial-temporal dynamics and their associated parameter scopes caused by Turing–Hopf bifurcation through numerical simulations. [ABSTRACT FROM AUTHOR]

This paper investigates a discrete conformable fractional order bacteria population model in a microcosm. When the parameters of this discrete model satisfy some conditions, its positive fixed point has a double eigenvalue - 1 with geometric multiplicity 1, corresponding to a 1:2 strong resonance. Applying the normal form theory and the approximation theory by a flow, we firstly transform this discrete model into a two-dimensional ordinary differential system. Then we use the bifurcation theory to analyze the qualitative properties of this ordinary differential system near the degenerate equilibrium. Therefore, the qualitative properties of the corresponding fixed point are obtained, which show that this discrete model undergoes the period-doubling bifurcation, the Neimark-Sacker bifurcation, the homoclinic bifurcation and the double closed orbit bifurcation. We also add a hybrid controller to this model such that the fixed point is locally asymptotically stable. Finally, numerical simulations are given to verify the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we introduce prey-taxis to the diffusive Bazykin system and study the codimension-two Turing–Turing bifurcation of this modified system. For the local system, i.e. without diffusion terms and the prey-taxis term, we investigate the stability of the unique positive equilibrium. Then, for the diffusive Bazykin system with prey-taxis, we examine the stability of the unique positive constant steady state, Turing instability, and Turing bifurcation. Moreover, by choosing the random diffusion coefficients as bifurcation parameters, we investigate the Turing–Turing bifurcation of the diffusive Bazykin system with prey-taxis. Furthermore, to demonstrate and classify the spatiotemporal dynamics near the Turing–Turing bifurcation point, we derive an algorithm for calculating the third-order truncated normal form of this bifurcation using the center manifold theorem and normal form theory. Finally, to verify the theoretical analysis results and demonstrate the effectiveness of the derived algorithm, we perform several numerical simulations using matlab software. We also investigate the influence of the prey-taxis coefficient on the Turing instability of the diffusive Bazykin system with prey-taxis. [ABSTRACT FROM AUTHOR]

The Swift–Hohenberg equation (SHE) is a partial differential equation that explains how patterns emerge from a spatially homogeneous state. It has been widely used in the theory of pattern formation. Following a recent study by Bramburger and Holzer (SIAM J Math Anal 55(3):2150–2185, 2023), we consider discrete SHE on deterministic and random graphs. The two families of the discrete models share the same continuum limit in the form of a nonlocal SHE on a circle. The analysis of the continuous system, parallel to the analysis of the classical SHE, shows bifurcations of spatially periodic solutions at critical values of the control parameters. However, the proximity of the discrete models to the continuum limit does not guarantee that the same bifurcations take place in the discrete setting in general, because some of the symmetries of the continuous model do not survive discretization. We use the center manifold reduction and normal forms to obtain precise information about the number and stability of solutions bifurcating from the homogeneous state in the discrete models on deterministic and sparse random graphs. Moreover, we present detailed numerical results for the discrete SHE on the nearest-neighbor and small-world graphs. [ABSTRACT FROM AUTHOR]

In this paper, we introduce an extension of the fellow traveler property which allows fellow travelers to be at distance bounded from above by a function f (n) growing slower than any linear function. We study normal forms satisfying this extended fellow traveler property and certain geometric constraints that naturally generalize two fundamental properties of an automatic normal form — the regularity of its language and the bounded length difference property. We show examples of such normal forms and prove some nonexistence theorems. [ABSTRACT FROM AUTHOR]

In the reaction–diffusion systems, the Laplacian is usually a self-adjoint operator. Incorporating advection terms generally destroys this and yields a quite complicated decomposition of phase space in the Center Manifold Reduction (CMR) method. Considering that the Multiple Time Scales (MTS) method can be directly applied to bifurcation analysis and normal form derivation based on algebraic operations, and the computational process is relatively universal, we apply the MTS method to analyze the Hopf bifurcation problem of reaction–diffusion–advection systems with time delay. First, we introduce the MTS method for the system with a specific boundary condition, which is used to derive the normal form of Hopf bifurcation. Calculating the normal form using the MTS method can be summarized as determining the bifurcation parameters, conducting Taylor expansion based on the MTS idea, and eliminating secular terms. Then, the key coefficients in the normal form are explicitly given, which are also compared with the results obtained by the CMR method, and both methods lead to the same bifurcation results. Finally, we use the MTS method to calculate the normal form of Hopf bifurcation for a predator–prey model with mixed boundary conditions. We find that the spatially nonhomogeneous periodic solutions appear near the equilibrium in the system when the time delay exceeds the critical value, and the theoretical results are illustrated by numerical simulations. [ABSTRACT FROM AUTHOR]

We present a technique for the direct optimization of conservative backbone curves in nonlinear mechanical systems. The periodic orbits on the conservative backbone are computed analytically using the reduced dynamics of the corresponding Lyapunov subcenter manifold (LSM). In this manner, we avoid expensive full-system simulations and numerical continuation to approximate the nonlinear response. Our method aims at tailoring the shape of the backbone curve using a gradient-based optimization with respect to the system's parameters. To this end, we formulate the optimization problem by imposing constraints on the frequency-amplitude relation. Sensitivities are computed analytically by differentiating the backbone expression and the corresponding LSM. At each iteration, only the reduced-order model construction and sensitivity computation are performed, making our approach robust and efficient. [ABSTRACT FROM AUTHOR]

Gröbner-Shirshov bases theory for strict monoidal categories is established and some applications are given [ABSTRACT FROM AUTHOR]

In Straubing (Theor. Comput. Sci. 13:137–150, 1981), Straubing introduced an n-ary the Schützenberger product on finite monoids. In this paper, we examine this generalization from another point of view and obtain an alternative n-generalized monoid construction of Schützenberger product to facilitate the intelligibility of Straubing's structure. Then, we find a generator and relator set for this product. Additionally, we compute a complete rewriting system and thus we obtain normal form structure of elements of this monoid construction which could be useful for Automata Theory. This gives us the solvability of the word problem for this product. [ABSTRACT FROM AUTHOR]

This paper investigates the qualitative structures of a discrete ratio-dependent predator–prey model near a degenerate fixed point whose eigenvalues are ± 1 . By the normal form theory, Picard iteration and Takens’s theorem, this model is transformed into an ordinary differential system. Then the qualitative structures of this differential system near the highly degenerate equilibrium are analyzed with the blowing-up method, which yields the ones of the discrete model near the fixed point by the conjugacy between the discrete model and the time-one mapping of the vector field. [ABSTRACT FROM AUTHOR]

In the present study, we analyze dynamics and bifurcations of a discrete-time Hopfield neural network based on two neurons and the same time delay. We determine stability and bifurcations of the system consisting flip, pitchfork and Neimark-Sacker bifurcations. The normal form coefficients for the all bifurcations are calculated using reducing to the corresponding center manifold, then these coefficients are numerically obtained using MatContM. Numerical analysis validates our analytical results and reveals more complex dynamical behaviors. [ABSTRACT FROM AUTHOR]

We prove that a transitive uniformly quasiconformal Anosov diffeomorphism with a two-dimensional stable or unstable distribution has a global product structure. As an application, we remove a topological assumption in a global rigidity theorem for uniformly quasiconformal Anosov diffeomorphisms of Kalinin-Sadovskaya [13]. [ABSTRACT FROM AUTHOR]

In this paper, a reaction–diffusion model that incorporates both discrete memory delay and distributed competition delay is introduced, we primary focus on the stability and Hopf bifurcation of its internal equilibrium point. A general methodology for analyzing the characteristic equations of this kind of model is established. Theoretical findings reveal that the stability of the system is contingent upon the relationship between the memory diffusion coefficient d 2 and the free diffusion coefficient d 1 . In addition, we also find a sudden change in the number of pure imaginary eigenvalues from 0 to infinity. In order to study the properties of the bifurcations, we transformed the model into an equivalent system without any integral term and provided a new formula for calculating the normal form of Hopf bifurcation. In the numerical simulation section, we delineate the crossing curves within the dual-delay σ - τ plane that signify the occurrence of Hopf bifurcations in the system. These curves reveal the presence of periodic solutions with varying amplitudes, along with the intriguing phenomenon of stability switching. Furthermore, the model demonstrated a distinct dynamic behavior, diverging from that observed in models with discrete competitive delays, these results suggest that the disruptive impact of distributed delays on the stability of system is significantly less pronounced than that of discrete delays. [ABSTRACT FROM AUTHOR]

In this manuscript, we consider a neutral functional differential equation. We make some assumptions and formulate conditions that ensure the existence of a zero-Hopf singularity. Using normal form theory and center manifold reduction, we derive the normal form up to third-order terms. [ABSTRACT FROM AUTHOR]

We consider two fluids in a 2-dimensional region: The lower fluid occupies an infinitely depth region, while the upper fluid occupies a region with a fixed upper boundary. We study the dynamics of the interface between the two fluids (interface problem) in the limit in which the interface has a space periodic profile, is close to horizontal, and has a "long wave profile". We use a Hamiltonian normal form approach to show that up to corrections of second order, the equations are approximated by two decoupled Benjamin-Ono equations.

In this paper, we consider a Leslie–Gower type reaction–diffusion predator–prey system with an increasing functional response. We mainly study the effect of three different types of diffusion on the stability of this system. The main results are as follows: (1) in the absence of prey diffusion, diffusion-driven instability can occur; (2) in the absence of predator diffusion, diffusion-driven instability does not occur and the non-constant stationary solution exists and is unstable; (3) in the presence of both prey diffusion and predator diffusion, the system can occur diffusion-driven instability and Turing patterns. At the same time, we also get the existence conditions of the Hopf bifurcation and the Turing–Hopf bifurcation, along with the normal form for the Turing–Hopf bifurcation. In addition, we conduct numerical simulations for all three cases to support the results of our theoretical analysis. [ABSTRACT FROM AUTHOR]

In this paper, using smooth invertible variable transformations and smooth invertible parameter changes, we construct a jerk normal form for the cusp bifurcation of a jerk system with two parameters which displays a nondegenerate fold bifurcation. [ABSTRACT FROM AUTHOR]

This paper presents a mathematical model of the cardiovascular system (CVS) designed to simulate both normal and pathological conditions within the systemic circulation. The model introduces a novel representation of the CVS through a change of coordinates, transforming it into the "quadratic normal form". This model facilitates the implementation of a sliding mode observer (SMO), allowing for the estimation of system states and the detection of anomalies, even though the system is linearly unobservable. The primary focus is on identifying valvular heart diseases, which are significant risk factors for cardiovascular diseases. The model's validity is confirmed through simulations that replicate hemodynamic parameters, aligning with existing literature and experimental data. [ABSTRACT FROM AUTHOR]

We consider two fluids in a 2-dimensional region: The lower fluid occupies an infinitely depth region, while the upper fluid occupies a region with a fixed upper boundary. We study the dynamics of the interface between the two fluids (interface problem) in the limit in which the interface has a space periodic profile, is close to horizontal, and has a "long wave profile". We use a Hamiltonian normal form approach to show that up to corrections of second order, the equations are approximated by two decoupled Benjamin-Ono equations. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the normal form a class of partial functional differential equations, which takes account of delays in the diffusion terms as well as a wider scope of nonlinear terms. We first study the associated linear theory, mainly including the spectral properties of infinitesimal generator, formal adjoint and decomposition of phase space. Based on these results, the normal form theory for nonlinear equation is established, which can be used to study the local dynamics near the steady state for such equations. As an application, we consider the Hopf bifurcation problem of a scalar diffusive equation with delay not only involved in the diffusive term but also in reaction terms. The normal form, depending on the original coefficients, up to the third order term is calculated, which allows us to determine the direction of Hopf bifurcation and stability of bifurcated periodic solutions. The results are then applied to study the scalar population model with memory-based diffusion for modeling the movement of highly-developed animals. [ABSTRACT FROM AUTHOR]

In this paper, the stability and local bifurcations of a composite laminated circular cylindrical shell with radially pre-stretched membranes are explored by using analytical and numerical methods. On the basis of the four-dimensional averaged equation in the case of 1:1 internal resonance, three types of critical points are studied in detail. They are characterized as (1) one pair of purely imaginary eigenvalues and two negative eigenvalues; (2) a simple zero and one pair of purely imaginary eigenvalues; (3) two pairs of purely imaginary eigenvalues in nonresonant case. With the aid of normal form theory and Maple software, the steady-state solutions and the stability regions of the initial equilibrium solutions are obtained. The explicit expressions for the critical bifurcation curves leading to static bifurcation and Hopf bifurcation are also presented. The presence of Hopf bifurcation indicates that the circular cylindrical shell will flutter. The results contribute to the design of reasonable structure parameters to avoid flutter. Finally, numerical simulations are also presented to demonstrate the good agreement with the analytical predictions. [ABSTRACT FROM AUTHOR]

We continue studies of the nonlocal erosion equation that is used as a mathematical model of the formation of a spatially inhomogeneous relief on semiconductor surfaces. We show that such a relief can form as a result of local bifurcations in the case where the stability of the spatially homogeneous equilibrium state changes. We consider a periodic boundary-value problem and study its codimension- bifurcations. For solutions describing an inhomogeneous relief, we obtain asymptotic formulas and study their stability. The analysis of the mathematical problem is based on modern methods of the theory of dynamical systems with an infinite-dimensional phase space, in particular, on the method of integral manifolds and on the theory of normal forms. [ABSTRACT FROM AUTHOR]

The patchy inflammatory pattern in the gastrointestinal tract is one of the typical features of Crohn's disease, and revealing the evolutionary mechanism of Crohn's disease is helpful to further understand its pathogenesis. However, the research on the pathogenesis of Crohn's disease is not comprehensive. To this end, this paper investigates the spatiotemporal dynamics of the phagocyte-bacterium model with diffusion and explores the evolutionary mechanism of Crohn's disease from the perspective of bifurcation analysis. Theoretically, we use the normal form theory of the reaction-diffusion equation and the neutral manifold theory to derive the normal form of the system near the Turing–Hopf bifurcation. Then, the results are verified by numerical simulations, and it is found that small disturbances in the control parameters led to four different types of dynamic states in the system: uniform state, spatially non-uniform steady state, spatially uniform periodic state, and spatially non-uniform periodic state. At the same time, the effects of small alterations in intestinal permeability and phagocytes diffusion rate near the Turing–Hopf bifurcation point on pattern structure and bacterial biomass of Crohn's disease were revealed. In addition, we found that there are three different types of pattern transformation in the system. These findings will help to better understand the development process of Crohn's disease, predict its pattern transformation, and provide theoretical guidance for the research and development of new drugs. Meanwhile, our research findings provide insights into the vulnerability of the mucosal immune system in Crohn's disease. [ABSTRACT FROM AUTHOR]

In this work, the Hopf and Bogdanov–Takens bifurcations of a delayed Bazykin predator–prey model with predator intraspecific interactions and ratio-dependent functional response are studied. Sufficient conditions for the existence of Hopf bifurcation are established. In the Bogdanov–Takens bifurcation, the dynamics near the nonhyperbolic equilibrium can be reduced to the study of the dynamics of the corresponding normal form restricted to the associated two-dimensional center manifold. Some numerical simulations, such as the distribution of eigenvalues, the bifurcation diagrams of Hopf and Bogdanov–Takens bifurcations and phase portraits, are given to illustrate the theoretical criteria. The theoretical and numerical simulation results illustrate that there is supercritical Hopf bifurcation and subcritical Bogdanov–Takens bifurcation in this model. We show that, the dynamics of prey and predator are very sensitive to parameters and delay perturbations which can play a great role in controlling and regulating the number of biological populations. [ABSTRACT FROM AUTHOR]

In this paper, we derive the algorithm for calculating the normal form of the double Hopf bifurcation that appears in a memory-based diffusion system via taking memory-based diffusion coefficient and the memory delay as the perturbation parameters. Using the obtained theoretical results, we study the dynamical classification near the double Hopf bifurcation point in a predator-prey system with Holling type II functional response. We show the existence of different kinds of stable spatially inhomogeneous periodic solutions, the transition from one kind to the other as well as the coexistence of two types of periodic solutions with different spatial profiles by varying the memory-based diffusion coefficient and the memory delay. [ABSTRACT FROM AUTHOR]

New approaches are proposed in the problem of constructing equivalent Hamiltonian systems for linear Lur'e equations (differential equations containing derivatives of even orders only). The approaches are based on the transition to the normal forms of the corresponding Hamiltonian systems with subsequent transformation of the resulting system. The proposed scheme does not require complex and cumbersome transformations of the original equation. It is shown that this problem is uniquely solvable. The appendix provides similar results for systems with two degrees of freedom. The effectiveness of the proposed formulas is illustrated by examples. [ABSTRACT FROM AUTHOR]

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This article investigates the unconditional well-posedness for the modified Kawahara equation and it proves that this is valid for initial data in H 1 / 4 (R). This result is a companion to a recent one [5] , which addressed the same problem for the classical Kawahara equation. [ABSTRACT FROM AUTHOR]

We investigate the local dynamics of a system of oscillators with a large number of elements and with diffusion-type couplings containing a large delay. We isolate critical cases in the stability problem for the zero equilibrium state and show that all of them are infinite-dimensional. Using special infinite normalization methods, we construct quasinormal forms, that is, nonlinear boundary value problems of parabolic type whose nonlocal dynamics determines the behavior of solutions of the original system in a small neighborhood of the equilibrium state. These quasinormal forms contain either two or three spatial variables, which emphasizes the complexity of dynamic properties of the original problem. [ABSTRACT FROM AUTHOR]

The computation of the normal form as well as its unfolding is a key step to understand the topological structure of a bifurcation. Though a lot of results have been obtained, it still remains unsolved for higher co-dimensional bifurcations. The main purpose of this paper is devoted to the computation of a codimension-3 zero-Hopf–Hopf bifurcation, at which a zero as well as two pairs of pure imaginary eigenvalues can be found from the matrix evaluated at the equilibrium point. Different distributions of eigenvalues are considered, which may behave in a non-semisimple form for 1:1 internal resonance. Based on the combination of center manifold and normal form theory, all the coefficients of normal forms and nonlinear transformations are derived explicitly in terms of parameters of the original vector field, which are obtained via a recursive procedure. Accordingly, a user friendly computer program using a symbolic computation language Maple is developed to compute the coefficients up to an arbitrary order for a specific vector field with zero-Hopf–Hopf bifurcation. Furthermore, universal unfolding parameters are derived in terms of the perturbation of physical parameters, which can be employed to investigate the local behaviors in the neighborhood of the bifurcation point. Here, we emphasize that though different norm forms based on different choices may exist, their topological structures are the same, corresponding to qualitatively equivalent dynamics. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the spatiotemporal dynamics in a diffusive two-species system with taxis term and general functional response, which means the directional movement of one species upward or downward the other one. The stability of positive equilibrium and the existences of Turing bifurcation, Turing–Hopf bifurcation and Turing–Turing bifurcation are investigated. An algorithm for calculating the normal form of the Turing–Hopf bifurcation induced by the taxis term and another parameter is derived. Furthermore, we apply our theoretical results to a cooperative Lotka–Volterra system and a predator–prey system with prey-taxis. For the cooperative system, stable equilibrium becomes unstable by taxis-driven Turing instability, which is impossible for the cooperative system without taxis. For a predator–prey system with prey-taxis, the dynamical classification near the Turing–Hopf bifurcation point is clearly described. Near the Turing–Hopf point, there are spatially inhomogeneous steady-state solution, spatially homogeneous/nonhomogeneous periodic solution and pattern transitions from one spatiotemporal state to another one. [ABSTRACT FROM AUTHOR]

This paper is concerned with a reaction-diffusion population model with nonlocal delayed effect and zero-Dirichlet boundary condition. Under the condition when the delayed feedback control is dominant, the normal form for spatially nonhomogeneous Hopf bifurcation from the sufficiently small positive equilibrium is computed by means of the normal form method and the center manifold theorem for partial functional differential equations. It is revealed that Hopf bifurcations appearing at the small positive equilibrium are forward and all bifurcating periodic solutions are locally orbitally asymptotically stable on the center manifold. To verify the validity of the obtained theoretical results, numerical simulations are also provided. [ABSTRACT FROM AUTHOR]

The local dynamics of a system of oscillators with a large number of elements and with diffusive- and advective-type couplings containing a large delay are studied. Critical cases in the problem of the stability of the zero equilibrium state are singled out, and it is shown that all of them have infinite dimensions. Applying special methods of infinite normalization, we construct quasinormal forms, namely, nonlinear boundary value problems of the parabolic type, whose nonlocal dynamics determine the behavior of the solutions of the initial system in a small neighborhood of the equilibrium state. These quasinormal forms contain either two or three spatial variables, which emphasizes the complexity of the dynamical properties of the original problem. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the periodic pattern formations with spatial multi-peaks in a classic diffusive Holling-Tanner predator-prey model with nonlocal intraspecific prey competition. The main innovation is that a spatial dependently kernel is considered in the nonlocal effect, which mathematically complicates the linear stability analysis. We first generate the existences of Hopf, Turing, Turing-Hopf and double-Hopf bifurcations, and determine the stability of the positive equilibrium. It turns out that the stable parameter region for the positive equilibrium decreases with α increasing, which implies that the parameter region of pattern formation for such kernel is smaller than the spatial average case. For double-Hopf bifurcation, we calculate the normal form up to the third-order term restricted on the center manifold, which is expressed by the original parameters of the system. Via analyzing the equivalent amplitude equations, the system exhibits stable spatially nonhomogeneous periodic patterns, the bistability of such periodic solutions, as well as unstable spatially nonhomogeneous quasi-periodic solutions, all of them possess multiple spatial peaks. Interestingly, some possible strange attractors are found numerically near the double-Hopf singularity. Biologically, the emerging spatio-temporal patterns imply that such nonlocal intraspecific competition can promote the coexistence of the prey and predator species in the form of more complex periodic states. [ABSTRACT FROM AUTHOR]

We prove a normal form theorem for principal Hamiltonian actions on Poisson manifolds around the zero locus of the moment map. The local model is the generalization to Poisson geometry of the classical minimal coupling construction from symplectic geometry of Sternberg and Weinstein. Further, we show that the result implies that the quotient Poisson manifold is linearizable, and we show how to extend the normal form to other values of the moment map. [ABSTRACT FROM AUTHOR]

In this article, we obtain new results on the unconditional well-posedness for a pair of periodic nonlinear dispersive equations using an abstract framework introduced by Kishimoto. This framework is based on a normal form reductions argument coupled with a number of crucial multilinear estimates.

In this paper, we establish an abstract infinite dimensional KAM theorem. As an application, we use the theorem to study the 1D wave equation system \[ \begin{cases} u_{1tt} - u_{1xx} +\sigma u_1 +u_1u_2^2 = 0 \\ u_{2tt} - u_{2xx} +\mu u_2 +u_1^2 u_2 = 0 \end{cases} \] { u 1 tt − u 1 xx + σ u 1 + u 1 u 2 2 = 0 u 2 tt − u 2 xx + μ u 2 + u 1 2 u 2 = 0 under Dirichlet boundary conditions, where $ 0 \lt \sigma \in [\sigma _1,\sigma _2]\subset [0,1], $ 0 < σ ∈ [ σ 1 , σ 2 ] ⊂ [ 0 , 1 ] , $ 0 \lt \mu \in [\mu _1,\mu _2]\subset [0,1] $ 0 < μ ∈ [ μ 1 , μ 2 ] ⊂ [ 0 , 1 ] are real parameters. By establishing a block-diagonal normal form, we obtain the existence of a Whitney smooth family of small amplitude quasi-periodic solutions corresponding to finite dimensional invariant tori of an associated infinite dimensional dynamic system. [ABSTRACT FROM AUTHOR]

This paper uses the Gevrey's smooth normalization theory to investigate the local integrability of vector fields, which have linear parts with one zero eigenvalue and the others non-resonant. First, we explore the general properties of C ∞ local integrability. Then we show the same regularity of the Gevrey's smooth local integrability as that of the vector fields under Poincaré's non-resonant condition for the case that the real parts of the eigenvalues are all positive or negative. Lastly, a sharper expression of the loss of the regularity is provided by the lowest order of the resonant terms together with the indices of Gevrey's smoothness and the diophantine condition for the case that the matrix is in the diagonal form. One of the main tools utilized here is the KAM method in the Banach space fixed with a weighted norm. [ABSTRACT FROM AUTHOR]

In this article, we obtain new results on the unconditional well-posedness for a pair of periodic nonlinear dispersive equations using an abstract framework introduced by Kishimoto. This framework is based on a normal form reductions argument coupled with a number of crucial multilinear estimates. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the mathematical analysis of a mathematical model describing the virotherapy treatment of a cancer with logistic growth and the effect of viral cycle presented by a time delay. The cancer population size is divided into uninfected and infected compartments. Depending on time delay, we prove the positivity and boundedness and the stability of equilibria. We give conditions on which the viral cycle leads to “Jeff’s phenomenon” observed in laboratory and causes oscillations in cancer size via Hopf bifurcation theory. We establish an algorithm that determines the bifurcation elements via center manifold and normal form theories. We give conditions which lead to a supercritical or subcritical bifurcation. We end with numerical simulations illustrating our theoretical results. [ABSTRACT FROM AUTHOR]

For a linear autonomous system of neutral type with commensurable delays, an algorithm is given for solving the modal controllability problem (in particular, the finite spectrum assignment problem), which provides a closed-loop system with a given characteristic quasipolynomial. A procedure for editing the finite part of the spectrum is proposed. A criterion for exponential stabilization of the system under study is constructively justified. When the criterion is met, the closed-loop system can be made exponentially stable according to the proposed spectral reduction algorithm. The obtained statements and spectrum assignment algorithms are illustrated with examples. [ABSTRACT FROM AUTHOR]

Cactus groups J n are currently attracting considerable interest from diverse mathematical communities. This work explores their relations to right-angled Coxeter groups and, in particular, twin groups T w n and Mostovoy's Gauss diagram groups D n , which are better understood. Concretely, we construct an injective group 1-cocycle from J n to D n and show that T w n (and its k-leaf generalizations) inject into J n . As a corollary, we solve the word problem for cactus groups, determine their torsion (which is only even) and center (which is trivial), and answer the same questions for pure cactus groups, P J n . In addition, we yield a 1-relator presentation of the first non-abelian pure cactus group P J 4 . Our tools come mainly from combinatorial group theory. [ABSTRACT FROM AUTHOR]

We study Nijenhuis operators, that is, .1; 1/-tensors with vanishing Nijenhuis torsion under the additional assumption that they are gl-regular, i.e., every eigenvalue has geometric multiplicity one. We prove the existence of a coordinate system in which the operator takes first or second companion form, and give a local description of such operators. We apply this local description to study singular points. In particular, we obtain normal forms of gl-regular Nijenhuis operators near singular points in dimension two and discover topological restrictions for the existence of gl-regular Nijenhuis operators on closed surfaces. [ABSTRACT FROM AUTHOR]