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We propose a descattering method that can be easily applied to food production lines. The system consists of several sets of linear image sensors and linear light sources slanted at different angles. The images captured by these sensors are partially clear along the direction perpendicular to the sensors. We computationally integrate these images on the frequency domain into a single clear image. The effectiveness of the proposed method is assessed by simulation and real-world experiments. The results show that our method recovers clear images. We demonstrate the applicability of the proposed method to a real production line by a prototype system.

A Hopf bifurcation in the Kuramoto-Daido model is investigated based on the generalized spectral theory and the center manifold reduction for a certain class of frequency distributions. The dynamical system of the order parameter on a four-dimensional center manifold is derived. It is shown that the dynamical system undergoes a Hopf bifurcation as the coupling strength increases, which proves the existence of a periodic two-cluster state of oscillators., arXiv admin note: substantial text overlap with arXiv:1609.04126

In his classical work on synchronization, Kuramoto derived the formula for the critical value of the coupling strength corresponding to the transition to synchrony in large ensembles of all-to-all coupled phase oscillators with randomly distributed intrinsic frequencies. We extend this result to a large class of coupled systems on convergent families of deterministic and random graphs. Specifically, we identify the critical values of the coupling strength (transition points), between which the incoherent state is linearly stable and is unstable otherwise. We show that the transition points depend on the largest positive or/and smallest negative eigenvalue(s) of the kernel operator defined by the graph limit. This reveals the precise mechanism, by which the network topology controls transition to synchrony in the Kuramoto model on graphs. To illustrate the analysis with concrete examples, we derive the transition point formula for the coupled systems on Erdős-Renyi, small-world, and \begin{document}$ k$\end{document} -nearest-neighbor families of graphs. As a result of independent interest, we provide a rigorous justification for the mean field limit for the Kuramoto model on graphs. The latter is used in the derivation of the transition point formulas. In the second part of this work [ 8 ], we study the bifurcation corresponding to the onset of synchronization in the Kuramoto model on convergent graph sequences.

In our previous work [ 3 ], we initiated a mathematical investigation of the onset of synchronization in the Kuramoto model (KM) of coupled phase oscillators on convergent graph sequences. There, we derived and rigorously justified the mean field limit for the KM on graphs. Using linear stability analysis, we identified the critical values of the coupling strength, at which the incoherent state looses stability, thus, determining the onset of synchronization in this model. In the present paper, we study the corresponding bifurcations. Specifically, we show that similar to the original KM with all-to-all coupling, the onset of synchronization in the KM on graphs is realized via a pitchfork bifurcation. The formula for the stable branch of the bifurcating equilibria involves the principal eigenvalue and the corresponding eigenfunctions of the kernel operator defined by the limit of the graph sequence used in the model. This establishes an explicit link between the network structure and the onset of synchronization in the KM on graphs. The results of this work are illustrated with the bifurcation analysis of the KM on Erdős-Renyi, small-world, as well as certain weighted graphs on a circle.

The Kuramoto model of coupled second order damped oscillators on convergent sequences of graphs is analyzed in this work. The oscillators in this model have random intrinsic frequencies and interact with each other via nonlinear coupling. The connectivity of the coupled system is assigned by a graph which may be random as well. In the thermodynamic limit the behavior of the system is captured by the Vlasov equation, a hyperbolic partial differential equation for the probability distribution of the oscillators in the phase space. We study stability of mixing, a steady state solution of the Vlasov equation, corresponding to the uniform distribution of phases. Specifically, we identify a critical value of the strength of coupling, at which the system undergoes a pitchfork bifurcation. It corresponds to the loss of stability of mixing and marks the onset of synchronization. As for the classical Kuramoto model, the presence of the continuous spectrum on the imaginary axis poses the main difficulty for the stability analysis. To overcome this problem, we use the methods from the generalized spectral theory developed for the original Kuramoto model. The analytical results are illustrated with numerical bifurcation diagrams computed for the Kuramoto model on Erd\H{o}s--R\'enyi and small-world graphs. Applications of the second-order Kuramoto model include power networks, coupled pendula, and various biological networks. The analysis in this paper provides a mathematical description of the onset of synchronization in these systems.

Interactions of inhibitory neurons produce gamma oscillations (30–80 Hz) in the local field potential, which is known to be involved in functions such as cognition and attention. In this study, the modified theta model is considered to investigate the theoretical relationship between the microscopic structure of inhibitory neurons and their gamma oscillations under a wide class of distribution functions of tonic currents on individual neurons. The stability and bifurcation of gamma oscillations for the Vlasov equation of the model is investigated by the generalized spectral theory. It is shown that as a connection probability of neurons increases, a pair of generalized eigenvalues crosses the imaginary axis twice, which implies that a stable gamma oscillation exists only when the connection probability has a value within a suitable range. On the other hand, when the distribution of tonic currents on individual neurons is the Lorentzian distribution, the Vlasov equation is reduced to a finite dimensional dynamical system. The bifurcation analyses of the reduced equation exhibit equivalent results with the generalized spectral theory. It is also demonstrated that the numerical computations of neuronal population follow the analyses of the generalized spectral theory as well as the bifurcation analysis of the reduced equation.

A spectral theory of linear operators on a rigged Hilbert space is applied to Schr\"odinger operators with exponentially decaying potentials and dilation analytic potentials. The theory of rigged Hilbert spaces provides a unified approach to resonances (generalized eigenvalues) for both classes of potentials without using any spectral deformation techniques. Generalized eigenvalues for one dimensional Schr\"odinger operators (ordinary differential operators) are investigated in detail. A certain holomorphic function $\mathbb{D}(\lambda)$ is constructed so that $\mathbb{D}(\lambda) = 0$ if and only if $\lambda $ is a generalized eigenvalue. It is proved that $\mathbb{D}(\lambda)$ is equivalent to the analytic continuation of the Evans function. In particular, a new formulation of the Evans function and its analytic continuation is given.

The Red-headed Wood Pigeon, Columba janthina nitens , is an endemic and endangered subspecies of the Ogasawara Islands. This pigeon moves irregularly among island habitats. However, its range and patterns of movement, particularly between the Bonin and the Volcano Islands, which are two remote island groups approximately 150 km apart, remain unclear. In this study, we conducted a survey on the uninhabited Kita-Iwojima Island of the Volcano Islands to collect direct evidence of pigeon movement between the two island groups and to reveal their food resource availability. Pigeon food composition was also analyzed. During the study period in Kita-Iwojima, we observed two individuals banded in Chichijima in the Bonin Islands. Food composition was estimated by fecal DNA analysis and compared with a fruit census of Kita-Iwojima, which differed from fruits observed in two monitored islands of the Bonin Islands, Chichijima and Hahajima. The pigeons might move among these islands to use available food resources, reflecting limitations of food resources in a single island habitat. Fruits detected in feces of the pigeons on Kita-Iwojima were not from plants observed on the island but rather derived from plants observed on Chichijima and Hahajima, likely indicating high movement capacity of pigeons among the islands. However, the foraging habitat of the Redheaded Wood Pigeon is limited to areas of low elevation in Kita-Iwojima despite apparent food sources at higher elevations. Therefore, factors beyond food abundance, such as geographical features, might affect habitat use of pigeons on the island.

The normal form theory for polynomial vector fields is extended to those for C∞ vector fields vanishing at the origin. Explicit formulas for the C∞ normal form and the near identity transformation that brings a vector field into its normal form are obtained by means of the renormalization group method. The dynamics of a given vector field such as the existence of invariant manifolds is investigated via its normal form. The C∞ normal form theory is applied to prove the existence of infinitely many periodic orbits of two dimensional systems, which is not shown from polynomial normal forms.

The first, second and fourth Painleve equations are studied by means of dynamical systems theory and three dimensional weighted projective spaces C P 3 ( p , q , r , s ) with suitable weights ( p , q , r , s ) determined by the Newton diagrams of the equations or the versal deformations of vector fields. Singular normal forms of the equations, a simple proof of the Painleve property and symplectic atlases of the spaces of initial conditions are given with the aid of the orbifold structure of C P 3 ( p , q , r , s ) . In particular, for the first Painleve equation, a well known Painleve's transformation is geometrically derived, which proves to be the Darboux coordinates of a certain algebraic surface with a holomorphic symplectic form. The affine Weyl group, Dynkin diagram and the Boutroux coordinates are also studied from a view point of the weighted projective space.

A Minimal multi-target helicon sputtering tool is developed to deposit a multi-layer metallic film for electrodes and interconnection wires in the Minimal Fab System.

Formal series solutions and the Kovalevskaya exponents of a quasi-homogeneous polynomial system of differential equations are studied by means of a weighted projective space and dynamical systems theory. A necessary and sufficient condition for the series solution to be a convergent Laurent series is given, which improve the well known Painlev\'{e} test. In particular, if a given system has the Painlev\'{e} property, an algorithm to construct Okamoto's space of initial conditions is given. The space of initial conditions is obtained by weighted blow-ups of the weighted projective space, where the weights for the blow-ups are determined by the Kovalevskaya exponents. The results are applied to the first Painlev\'{e} hierarchy ($2m$-th order first Painlev\'{e} equation).

The water source for Ogasawara village in Tokyo Prefecture has high concentrations of organic matter, color, and high trihalomethane formation potential. Therefore, the fungal breakdown of plants has produced leaf mold, which in turn has caused surface runoff to contain various humicsubstances. At the Ogiura water purification plant, a flocculant is added by a high injection ratio of 200-400 mg/L for removal of humicsubstances. This research focuses on humic substances in raw water in order to develop a means of separating humic acid with pH adjustment. The result shows that humic acid of raw water for water supply and purification treatment could be reduced and that the amount of chemical used for coagulation treatment could furthermore be reduced by optimal pH and PAC injection ratio under laboratory conditions. The results should serve as useful information for improving the purification treatment in areas that use raw water containing high levels of humic substances for their water supplies.

To utilize a gap mode in surface enhanced Raman scattering, we elucidated the interaction between adsorbed species and Ag nanoparticles (AgNPs). Various thiol molecules such as normal alkanethiols, thiols with a phenyl, cyclohexane or naphthalene ring on Ag films immobilized AgNPs through van der Waals force, and electrostatic interaction. Immobilized AgNPs provided enormous Raman enhancement by a factor of 107–1010 for thiol molecules at a nanogap, in consistent with that anticipated by finite difference time domain calculations. Only alkanethiols with a tert-methyl group and those with a carboxylic group did not immobilize any AgNPs probably owing to steric hindrance. A gap mode is relevant for a variety of metals even with large damping like Pt and Fe, indicating a crucial role of electric multipoles in AgNPs generated by a localized surface plasmon and induced mirror images in metal substrates for markedly enhanced electric field at a nanogap.

The Kuramoto model is a system of ordinary differential equations for describing synchronization phenomena defined as a coupled phase oscillators. In this paper, a bifurcation structure of the infinite dimensional Kuramoto model is investigated. For a certain non-selfadjoint linear operator, which defines a linear part of the Kuramoto model, the spectral theory on a space of generalized functions is developed with the aid of a rigged Hilbert space to avoid a continuous spectrum on the imaginary axis. Although the linear operator has an unbounded continuous spectrum on a Hilbert space, it is shown that it admits a spectral decomposition consisting of a countable number of eigenfunctions on a space of generalized functions. The semigroup generated by the linear operator is calculated by using the spectral decomposition to prove the linear stability of a steady state of the system. The center manifold theory is also developed on a space of generalized functions. It is proved that there exists a finite dimensional center manifold on a space of generalized functions, while a center manifold on a Hilbert space is of infinite dimensional because of the continuous spectrum on the imaginary axis. The results are applied to the stability and bifurcation theory of the Kuramoto model to obtain a bifurcation diagram conjectured by Kuramoto. If the coupling strength $K$ between oscillators is smaller than some threshold $K_c$, the de-synchronous state proves to be asymptotically stable, and if $K$ exceeds $K_c$, a nontrivial solution, which corresponds to the synchronization, bifurcates from the de-synchronous state., Comment: It will be published in Ergodic Theory and Dynamical Systems

The Kuramoto model, which describes synchronization phenomena, is a system of ordinary differential equations on $N$-torus defined as coupled harmonic oscillators. The order parameter is often used to measure the degree of synchronization. In this paper, the moments systems are introduced for both of the Kuramoto model and its continuous model. It is shown that the moments systems for both systems take the same form. This fact allows one to prove that the order parameter of the $N$-dimensional Kuramoto model converges to that of the continuous model as $N\to \infty$.

A multi-Poisson structure on a Lie algebra $\mathfrak{g}$ provides a systematic way to construct completely integrable Hamiltonian systems on $\mathfrak{g}$ expressed in Lax form $\partial X_\lambda /\partial t = [X_\lambda , A_\lambda ]$ in the sense of the isospectral deformation, where $X_\lambda , A_\lambda \in \mathfrak{g}$ depend rationally on the indeterminate $\lambda $ called the spectral parameter. In this paper, a method for modifying the isospectral deformation equation to the Lax equation $\partial X_\lambda /\partial t = [X_\lambda , A_\lambda ] + \partial A_\lambda /\partial \lambda $ in the sense of the isomonodromic deformation, which exhibits the Painlev\'e property, is proposed. This method gives a few new Painlev\'e systems of dimension four.

Bryan's Shearwater (Puffinus bryani) was described in 2011 on the basis of a specimen collected on the Midway Atoll in 1963. This specimen and another recorded on Midway in the early 1990s are the sole reliable records to date. Since 1997, we have found six specimens of a remarkably small Puffinus shearwater morphologically similar to Bryan's Shearwater on the Bonin Islands, northwestern Pacific. In this study, we examined the Bonin samples genetically and confirm that they are of Bryan's Shearwater. A morphological analysis suggests that the small body size and relatively long tail are characteristics of this species. Because the most recent individual was found on an islet to the north of Chichijima Island in 2011, the species has evidently survived in the Bonin Islands, where it may breed, although the exact location remains unclear. Three of the individuals found on an islet off Chichijima Island were carcasses preyed upon by black rats (Rattus rattus). Attempts were made to eradicate rats f...

The existence of stable periodic orbits and chaotic invariant sets of singularly perturbed problems of fast–slow type having Bogdanov–Takens bifurcation points in its fast subsystem is proved by means of the geometric singular perturbation method and the blow-up method. In particular, the blow-up method is effectively used for analyzing the flow near the Bogdanov–Takens type fold point in order to show that a slow manifold near the fold point is extended along the Boutroux's tritronquee solution of the first Painleve equation in the blow-up space.

In his classical work, Kuramoto analytically described the onset of synchronization in all-to-all coupled networks of phase oscillators with random intrinsic frequencies. Specifically, he identified a critical value of the coupling strength, at which the incoherent state loses stability and a gradual build-up of coherence begins. Recently, Kuramoto's scenario was shown to hold for a large class of coupled systems on convergent families of deterministic and random graphs. Guided by these results, in the present work, we study several model problems illustrating the link between network topology and synchronization in coupled dynamical systems. First, we identify several families of graphs, for which the transition to synchronization in the Kuramoto model starts at the same critical value of the coupling strength and proceeds in practically the same way. These examples include Erd\H{o}s-R\'enyi random graphs, Paley graphs, complete bipartite graphs, and certain stochastic block graphs. These examples illustrate that some rather simple structural properties such as the volume of the graph may determine the onset of synchronization, while finer structural features may affect only higher order statistics of the transition to synchronization. Further, we study the transition to synchronization in the Kuramoto model on power law and small-world random graphs. The former family of graphs endows the Kuramoto model with very good synchronizability: the synchronization threshold can be made arbitrarily low by varying the parameter of the power law degree distribution. For the Kuramoto model on small-world graphs, in addition to the transition to synchronization, we identify a new bifurcation leading to stable random twisted states. The examples analyzed in this work complement the results in [Chiba, Medvedev, The mean field analysis for the Kuramoto model on graphs (parts I and II), arxiv]., Comment: 18 pages, 12 figures

The formation potentials of 24 disinfection by-products, six indicators of organic matter and the number of algae were measured in raw water once a month for one year from October 2006 to September 2007 in order to investigate the quality of water in the two water purification plants on the Ogasawara Islands. It was estimated that the main source of organic matter in raw water on the Ogasawara Islands was not autochthonous organic matter but allochthonous organic matter that flows from streams into reservoirs, as determined from the correlations between the parameters of water quality, the relationship between rainfall and water quality, UV absorbance: TOC ratio and other factors. The mean annual formation potentials of disinfection by-products in filtered raw water on Chichijima Island and Hahajima Island were as follows: total haloacetic acid: 286, 194 μg·L-1, total trihalomethane: 227, 190 μg·L-1, total haloacetonitrile: 13 μg·L-1, and chloral hydrate: 13, 9 μg·L-1, respectively. The total trihalomethane formation potentials were at the maximum levels in our country and the formation potentials of other disinfection by-products were also presumed to be their maximum levels. The formation potentials of the brominated disinfection by-products in most months were higher than those of the nonbrominated disinfection by-products. The correlations of total trihalomethane formation potential (mole concentration) with color, TOC, KMnO4 consumption and UV absorbance were lower than those of the formation potentials of total haloacetic acid, total haloacetonitrile and chloral hydrate in raw water. KMnO4 consumption and UV absorbance were effective indicators of disinfection by-product formation potential for these high correlations (P

When the natural frequencies are allocated symmetrically in the Kuramoto model there exists an invariant torus of dimension [ N / 2 ] + 1 ( N is the population size). A global phase shift invariance allows us to reduce the model to N − 1 dimensions using the phase differences, and doing so the invariant torus becomes [ N / 2 ] -dimensional. By means of perturbative calculations based on the renormalization group technique, we show that this torus is asymptotically stable at small coupling if N is odd. If N is even the torus can be stable or unstable depending on the natural frequencies, and both possibilities persist in the small coupling limit.

The renormalization group (RG) method for differential equations is one of the perturbation methods which allows one to obtain invariant manifolds of a given ordinary differential equation together with approximate solutions to it. This article investigates higher order RG equations which serve to refine an error estimate of approximate solutions obtained by the first order RG equations. It is shown that the higher order RG equation maintains the similar theorems to those provided by the first order RG equation, which are theorems on well-definedness of approximate vector fields, and on inheritance of invariant manifolds from those for the RG equation to those for the original equation, for example. Since the higher order RG equation is defined by using indefinite integrals and is not unique for the reason of the undetermined integral constants, the simplest form of RG equation is available by choosing suitable integral constants. It is shown that this simplified RG equation is sufficient to determine whether the trivial solution to time-dependent linear equations is hyperbolically stable or not, and thereby a synchronous solution of a coupled oscillators is shown to be stable.

The Black-footed Albatross Diomedea nigripes is an endangered seabird that is endemic to the North Pacific. The genetic structure of Black-footed Albatross populations on three of the Hawaiian Islands and on Izu-Torishima Island, Japan, has been studied previously, using the mitochondrial cytochrome b region. Hawaiian and Japanese breeding groups are genetically different, and the genetic diversity of the birds on Izu-Torishima is lower than that of birds in the Hawaiian Islands. We analyzed 50 Black-footed Albatrosses from the Bonin Islands, where a relatively stable population persisted throughout the twentieth century. Although albatrosses in the Bonin Islands do not differ significantly from those on Izu-Torishima in their cytochrome b region sequences, they do exhibit higher genetic diversity (as high as those from the Hawaiian colonies). A statistical parsimony network revealed two clades, one primarily in the western North Pacific colonies and the other primarily in eastern North Pacific c...

The renormalization group (RG) method for differential equations is one of the perturbation methods for obtaining solutions which approximate exact solutions for a long time interval. This article shows that, for a differential equation associated with a given vector field on a manifold, a family of approximate solutions obtained by the RG method defines a vector field which is close to the original vector field in the $C^1$ topology under appropriate assumptions. Furthermore, some topological properties of the original vector field, such as the existence of a normally hyperbolic invariant manifold and its stability, are shown to be inherited from those of the RG equation. This fact is applied to the bifurcation theory.

The Bonin Islands provide breeding habitat for many species of seabird, but detailed information on breeding sites is unavailable. Here, we describe the recent species composition of breeding seabirds on 66 islands. A total of 15 species (Diomedea immutabilis, D. nigripes, Pterodroma hypoleuca, Bulweria bulwerii, Puffinus pacificus, Oceanodroma tristrami, O. matsudairae, Phaethon rubricauda, Sula leucogaster, S. dactylatra, S. sula, Thalasseus bergii, Sterna fuscata, Anous stolidus and A. minutus) was recorded breeding in the Bonin Islands based on field and literature surveys. The sole nesting record of S. sula was on a small island near Haha-jima, where it has failed to breed since a typhoon struck the island. We did not detect Puffinus lherminieri bannermani, although it bred on Kitaiwo-jima before World War II. S. leucogaster was the most widespread species and bred on 39 islands. The second most widespread species was P. pacificus, which bred on 35 islands. There was a positive relationship between species richness and island area. The distribution of breeding sites may be affected by human settlement and introduced species such as feral goats Capra aegagrus. Introduced animals should be controlled to protect the seabird fauna on the islands.

The foraging movements of six Black-footed Albatross Diomedea nigripes (an endangered species) breeding in the Mukojima Islands (a subset of the Bonin Islands) were successfully tracked over a period of two weeks using back-mounted global positioning system (GPS) data loggers (GDBL-II) during the nest-guarding period. Ninety percent of foraging was done over relatively shallow waters within 200 km of the breeding site. The population of this species in the Bonin Islands has not decreased during last ten years, while the area of long-line fishing is likely to overlap with the albatross foraging area around the islands. The effect on population should be assessed carefully.

The renormalization group (RG) method is one of the singular perturbation methods which provides asymptotic behavior of solutions of differential equations. In this article, how to construct approximate solutions by the RG method is shown with several examples and basic theorems on the RG method, such as an error estimate and the existence of invariant manifolds are given.

To establish an efficient way to utilize a gap mode plasmon in flocculates of MNPs, under external and ATR configurations, we controlled interactio n between adsorbed species and metal nanostructures. We have successfully formed flocculates of AgNPs using electrostatic interaction between di ssociated PMBA (-COO - ), protonated PATP (-NH 3+ ) and counter ions (M n+ , X - ), as well as van der Waals force between neutral PMBAs (-COOH) and PATP (-NH 2 ) on AgNPs. Detailed adsorbed state of PMBA and PATP as well as trappe d counter ions were characterized using enormous SERS enhancement in flocculation-SERS. In a gap mode under an external geometry, most of thiol molecules on Ag films immobilized AgNPs through van der Waals force and electrosta tic interaction. They showed similar Raman enhancement of 10 8 -10 9 , in accordance with those predicted by FDTD calculations. Only thiols with tert-methyl group did not immobilize any AgNPs due to steric hindrance. In a gap mode under ATR configuration, additional enhancement was obtained by a coupling of PSP and a gap mode.

A spectral theory of linear operators on rigged Hilbert spaces (Gelfand triplets) is developed under the assumptions that a linear operator T on a Hilbert space H is a perturbation of a selfadjoint operator, and the spectral measure of the selfadjoint operator has an analytic continuation near the real axis in some sense. It is shown that there exists a dense subspace X of H such that the resolvent ( λ − T ) − 1 ϕ of the operator T has an analytic continuation from the lower half plane to the upper half plane as an X ′ -valued holomorphic function for any ϕ ∈ X , even when T has a continuous spectrum on R, where X ′ is a dual space of X. The rigged Hilbert space consists of three spaces X ⊂ H ⊂ X ′ . A generalized eigenvalue and a generalized eigenfunction in X ′ are defined by using the analytic continuation of the resolvent as an operator from X into X ′ . Other basic tools of the usual spectral theory, such as a spectrum, resolvent, Riesz projection and semigroup are also studied in terms of a rigged Hilbert space. They prove to have the same properties as those of the usual spectral theory. The results are applied to estimate asymptotic behavior of solutions of evolution equations.

A bifurcation theory for a system of globally coupled phase oscillators is developed based on the theory of rigged Hilbert spaces. It is shown that there exists a finite-dimensional center manifold on a space of generalized functions. The dynamics on the manifold is derived for any coupling functions. When the coupling function is $\sin \theta $, a bifurcation diagram conjectured by Kuramoto is rigorously obtained. When it is not $\sin \theta $, a new type of bifurcation phenomenon is found due to the discontinuity of the projection operator to the center subspace.

It is shown that the dormancy of predators induces mixed-mode oscillations and chaos in the population dynamics of a prey-predator system under certain conditions. The mixed-mode oscillations and chaos are shown to bifurcate from a coexisting equilibrium by means of the theory of fast-slow systems. These results may help to find experimental conditions under which one can demonstrate chaotic population dynamics in a simple phytoplankton-zooplankton (-resting eggs) community in a microcosm with a short duration.

It is known that the Swift-Hohenberg equation $\partial u/\partial t = -(\partial_x^2 + 1)^2u + \varepsilon (u-u^3)$ can be reduced to the Ginzburg-Landau equation (amplitude equation) $\partial A/\partial t = 4\partial_x^2 A + \varepsilon (A-3A|A|^2)$ by means of the singular perturbation method. This means that if $\varepsilon >0$ is sufficiently small, a solution of the latter equation provides an approximate solution of the former one. In this paper, a reduction of a certain class of a system of nonlinear parabolic equations $\partial u/\partial t = \mathcal{P}u + \varepsilon f(u)$ is proposed. An amplitude equation of the system is defined and an error estimate of solutions is given. Further, it is proved under certain assumptions that if the amplitude equation has a stable steady state, then a given equation has a stable periodic solution . In particular, near the periodic solution, the error estimate of solutions holds uniformly in $t>0$.

Lie theory is applied to perturbation problems of ordinary differential equations to construct approximate solutions and invariant manifolds according to the renormalization group approach of Iwasa and Nozaki [“A method to construct asymptotic solutions invariant under the renormalization group,” Prog. Theor. Phys. 116, 605 (2006)]. It is proved that asymptotic behavior of solutions is obtained from the Lie equations even if original equations have no symmetries. Normal forms of the Lie equations are introduced to investigate the existence of invariant manifolds.

The renormalization group (RG) method for differential equations is one of the perturbation methods for obtaining approximate solutions. This article shows that the RG method is effectual for obtaining an approximate center manifold and an approximate flow on it when applied to equations having a center manifold.

Two nests of the Bull-headed Shrike Lanius bucephalus were found on Chichijima I. in 1988 and 1989. This is the first breeding record of the species from the Ogasawara Island.