Search

In this paper, the interesting dynamics of chaos induced by the effect of the variation of internal average heat during operation in the DC motor control by the full bridge drive are analyzed. By using simple powerful tools of analyzing nonlinear dynamical systems like phase portraits, time traces and frequency spectrum in the MATLAB-SIMULINK environment, we showed that under certain conditions, the PMDC motor develops different behaviors as periodic limit cycles, and chaotic attractors, when the motor drive different form of external load torque and the windings resistance variation. This paper presents the first studies on the variation of the average heat of the motor and the amplitude of the triangular load torque to produce the strange phenomena like chaos as far as our knowledge go. A chaos control of the unstable regime is proposed to stabilize the PMDC motor in a desire regime. This contribution is very important in industry because some unexplained dynamical behaviors of the DC motor driven by a full bridge now can be avoided. [ABSTRACT FROM AUTHOR]

This paper aims to investigate the effect of motor suspension parameters, specifically damping and stiffness, on the bifurcations of a bogie system. A motor bogie model with a nonlinear smooth equivalent conicity function is established. The study includes qualitative analyses of the stability and the Hopf bifurcation of the equilibrium, with the running speed as the single parameter. Furthermore, the generalised Hopf bifurcation and Hopf-Hopf bifurcation of the equilibrium, based on two motor suspension parameters, are also analysed. The paper delves into the codimension-1 and codimension-2 bifurcations of the limit cycles generated by the Hopf bifurcation, including Neimark–Sacker bifurcation, fold bifurcation, cusp bifurcation, 1:3 resonance, and 1:4 resonance. Analytical investigations reveal that the cusp bifurcation alters the type of fold bifurcation (subcritical or supercritical), and the fold bifurcation influences the stability and bifurcation direction of the limit cycle. Additionally, resonance occurs between the motor and the bogie frame. Since the subcritical (supercritical) Neimark–Sacker bifurcation produces an unstable (a stable) torus, the resonance points associated with the subcritical Neimark–Sacker bifurcation will lead to the instability of the motor bogie. The bifurcation analysis on the motor suspension parameters in this paper offers a theoretical reference for enhancing the stability of the motor bogie. [ABSTRACT FROM AUTHOR]

This paper introduces a new concept of a zero-platform delta robot with three key parameters affecting the shape and size of the workspace. This concept is applied to directly bring the torus configuration into the links of the robot and shows its usefulness in configuring and generating the workspace conveniently. Analyzing the workspace of parallel robots, such as delta robots, requires extensive computation due to the constraints between the links, typically requiring complex equations or numerical methods. This paper proposes a new method for quickly estimating the shape and size of the workspace using a cross-section diagram based on a geometrical analysis of the zero-platform delta robot. The shape and size of the workspace can be rapidly estimated because the intersection of three cross-section diagrams needs only the torus's 2D operation. Comparing the workspace between the cross-section diagram and the 3D CAD software, this paper shows that the cross-section diagram can analyze the shape and size of the workspace quickly and give a more geometrical understanding of the workspace. [ABSTRACT FROM AUTHOR]

Disjoint paths are defined as paths between the source and destination nodes where the intermediate nodes in any two paths are disjoint. They are helpful in fault-tolerance routing and securing message distribution in the network. Several research papers were proposed to solve the problem of finding disjoint paths for a variety of interconnection networks such as Hypercube, Generalized Hypercube, Mesh, Torus, Gaussian, Eisenstein–Jacobi, and many other topologies. In this research, we have developed a general algorithm that constructs maximal node-to-node disjoint paths for symmetric networks where all paths are shortest. The algorithm presented in this paper outperforms other algorithms in finding not only the disjoint paths but shortest and maximal disjoint paths with a complexity of O (n 2) . In addition, we have simulated the proposed algorithm on different networks. The solution of unsolved problem in Cube-Connected-Cycles is given in the simulation results. [ABSTRACT FROM AUTHOR]

The known publications related to the gyroscope theory consider only several geometries of the spinning rotors, like the disc, bars, rings, spheres, and others. The geometries of the spinning objects in engineering can have many designs that generate different values of inertial torques. A computing of inertial torques produced by the object's rotating masses depends on the radii of their locations. Gyroscopic devices with a spinning torus or ring like other objects should be computed for the quality of their operation. The simplified expression for the radius of the mass disposition for the ring is presented by its middle radius which does not give the correct results for inertial torques and motions of gyroscopic devices. The radius of distributed masses of a ring is bigger than its middle radius because of the differences in the external and internal semirings. A torus is a ring whose exact expression of the radius for distributed mass should be derived for the correct solution for the inertial torques generated by the rotating mass. The precise formulas for inertial torques acting on the spinning torus enable finding its optimal size for the gyroscopic devices. This paper presents the method and derivation of mathematical models for the inertial torques produced by the spinning torus and its interrelated angular velocities about the axes of rotations that manifest gyroscopic effects. [ABSTRACT FROM AUTHOR]

For the KPZ equation on a torus with a 1 + 1 spacetime white noise, it was shown in Dunlap et al. (Commun Pure Appl Math, 2023, https://doi.org/10.1002/cpa.22110) and Gu and Komorowski (Ann Inst H Poincare Prob Stat, 2021, arXiv:2104.13540v2) that the height function satisfies a central limit theorem, and the variance can be written as the expectation of an exponential functional of Brownian bridges. In this paper, we consider another physically relevant quantity, the winding number of the directed polymer on a cylinder, or equivalently, the displacement of the directed polymer endpoint in a spatially periodic random environment. It was shown in Gu and Komorowski (SIAM J Math Anal, arXiv:2207.14091) that the polymer endpoint satisfies a central limit theorem on diffusive scales. The main result of this paper is an explicit expression of the effective diffusivity, in terms of the expectation of another exponential functional of Brownian bridges. Our argument is based on a combination of tools from Malliavin calculus, homogenization, and diffusion in distribution-valued random environments. [ABSTRACT FROM AUTHOR]

Let T m be the maximal torus of a set of m × m unitary diagonal matrices. Let T be a collection of all maps that rigidly rotate every circle of latitude of the sphere with a fixed angle. T is also a maximal torus, and we shall prove in this paper that T is the topological limit inf of T m . [ABSTRACT FROM AUTHOR]

In this paper, we investigate the inviscid limit ν → 0 for time-quasi-periodic solutions of the incompressible Navier–Stokes equations on the two-dimensional torus T 2 , with a small time-quasi-periodic external force. More precisely, we construct solutions of the forced Navier–Stokes equation, bifurcating from a given time quasi-periodic solution of the incompressible Euler equations and admitting vanishing viscosity limit to the latter, uniformly for all times and independently of the size of the external perturbation. Our proof is based on the construction of an approximate solution, up to an error of order O (ν 2) and on a fixed point argument starting with this new approximate solution. A fundamental step is to prove the invertibility of the linearized Navier–Stokes operator at a quasi-periodic solution of the Euler equation, with smallness conditions and estimates which are uniform with respect to the viscosity parameter. To the best of our knowledge, this is the first positive result for the inviscid limit problem that is global and uniform in time and it is the first KAM result in the framework of the singular limit problems. [ABSTRACT FROM AUTHOR]

We define slopes in the flat torus as the set of equivalence classes of the solutions of linear equations in ℝ². The definition is equivalent to that of closed geodesics in the flat torus passing through the equivalence class of the point (0, 0). In this paper we derive formulas for counting the number of points in the intersection of multiple slopes in the flat torus. [ABSTRACT FROM AUTHOR]

This paper reports the research work carried out with the goal of geometrically and algebraically describing, as well as topologically classifying, the curves resulting from the intersection of a torus with families of parabolic and elliptical cylinders within a purely Euclidean framework. The parabolic cylinders under analysis have generatrices parallel to the axis of the torus, whereas the elliptical cylinders, centered at the same point as the torus, have axes either aligned with or orthogonal to the torus's axis. For the topological classification of these intersection curves, we consider their number of connected components and self-intersection points. GeoGebra, which was used to create the 3D visual geometric representations of the intersection curves, and Maple, which was used to perform the essential symbolic algebraic calculations, were critical computational tools in the development of this work. Theoretical and computational approaches are interwoven throughout this study, with the computational work serving as the foundation for exploration and providing insights that contributed to the theoretical validation of the results revealed through GeoGebra simulations. [ABSTRACT FROM AUTHOR]

We are concerned with the isentropic compressible Navier–Stokes system in the two-dimensional torus, with rough data and vacuum; the initial velocity belongs to the Sobolev space H 1 and the initial density is only bounded and nonnegative. Arbitrary regions of vacuum are admissible, and no compatibility condition is required. Under these assumptions and for large enough bulk viscosity, global solutions have been constructed in Danchin and Mucha (Commun Pure Appl Math 76:3437–3492, 2023). The main goal of the paper is to establish that these solutions converge exponentially fast to a constant state, and to specify the convergence rate in terms of the viscosity coefficients. We prove similar exponential decay results for the solutions to the inhomogeneous incompressible Navier–Stokes equations, thereby extending to the torus the recent paper (Danchin et al. in Ann Anal Non Lin IHP, 2023) where bounded domains are considered. [ABSTRACT FROM AUTHOR]

This paper is concerned with function reconstruction from samples. The sampling points used in several approaches are (1) structured points connected with fast algorithms or (2) unstructured points coming from, e.g., an initial random draw to achieve an improved information complexity. We connect both approaches and propose a subsampling of structured points in an offline step. In particular, we start with quasi-Monte Carlo (QMC) points with inherent structure and stable L_2 reconstruction properties. The subsampling procedure consists of a computationally inexpensive random step followed by a deterministic procedure to further reduce the number of points while keeping its information. In these points functions (belonging to a reproducing kernel Hilbert space of bounded functions) will be sampled and reconstructed from whilst achieving state of the art error decay. Our method is dimension-independent and is applicable as soon as we know some initial quadrature points. We apply our general findings on the d-dimensional torus to subsample rank-1 lattices, where it is known that full rank-1 lattices lose half the optimal order of convergence (expressed in terms of the size of the lattice). In contrast to that, our subsampled version regains the optimal rate since many of the lattice points are not needed. Moreover, we utilize fast and memory efficient Fourier algorithms in order to compute the approximation. Numerical experiments in several dimensions support our findings. [ABSTRACT FROM AUTHOR]

The classical Hard Lefschetz theorem (HLT), Hodge-Riemann bilinear relation theorem (HRR) and Lefschetz decomposition theorem (LD) are stated for a power of a Kähler class on a compact Kähler manifold. These theorems are not true for an arbitrary class, even if it contains a smooth strictly positive representative. Dinh-Nguyên proved the mixed HLT, HRR and LD for a product of arbitrary Kähler classes. Instead of products, they asked whether determinants of Griffiths positive k\times k matrices with (1,1)-form entries in \mathbb {C}^n satisfy these theorems in the linear case. This paper answered their question positively when k=2 and n=2,3. Moreover, assume that the matrix only has diagonalized entries, for k=2 and n\geqslant 4, the determinant satisfies HLT for bidegrees (n-2,0), (n-3,1), (1,n-3) and (0,n-2). In particular, for k=2 and n=4,5 with this extra assumption, the determinant satisfies HRR, HLT and LD. Two applications: First, a Griffiths positive 2\times 2 matrix with (1,1)-form entries, if all entries are \mathbb {C}-linear combinations of the diagonal entries, then its determinant also satisfies these theorems. Second, on a complex torus of dimension \leqslant 5, the determinant of a Griffiths positive 2\times 2 matrix with diagonalized entries satisfies these theorems. [ABSTRACT FROM AUTHOR]

In this paper, we extend the theory of planar pseudo knots to the theories of annular and toroidal pseudo knots. Pseudo knots are defined as equivalence classes under Reidemeister-like moves of knot diagrams characterized by crossings with undefined over/under information. In the theories of annular and toroidal pseudo knots, we introduce their respective lifts to the solid and the thickened torus. Then, we interlink these theories by representing annular and toroidal pseudo knots as planar O -mixed and H -mixed pseudo links. We also explore the inclusion relations between planar, annular and toroidal pseudo knots, as well as of O -mixed and H -mixed pseudo links. Finally, we extend the planar weighted resolution set to annular and toroidal pseudo knots, defining new invariants for classifying pseudo knots and links in the solid and in the thickened torus. [ABSTRACT FROM AUTHOR]

This paper deals with the study of torsional energy (total squared torsion) at infinitesimal bending of curves and knots in three dimensional Euclidean space. During bending, the curve is subject to change, and its properties are changed. The effect that deformation has on the curve is measured by variations. Here, we observe the infinitesimal bending of the second order and variations of the first and the second order that occur in this occasion. The subjects of study are curves and knots, in particular torus knots. We analyze various examples both analytically and graphically, using our own calculation and visualization software tool. [ABSTRACT FROM AUTHOR]

The paper is devoted to divergence-curl results involving a divergence free measurevalued field σ = bν, whereν is a signed Radon measure on RN and b is a nonvanishing regular vector field in RN, and a gradient measure-valued field η = ∇u on RN, N ≥ 2. On the one hand, in a nonperiodic framework we prove that for any open set Ω of R², the orthogonality condition b ∇ u = 0 in Ω implies the equality div (uσ) = 0 in Ω. The key ingredient of the proof is based on the existence of a representative in L\infty∞(Ω) of the bounded variation function u in Ω. This result allows us to extend in the setting of ODE's flows the famous Franks--Misiurewicz theorem, which claims that the Herman rotation set of any continuous two-dimensional flow on the torus T2 is a closed line segment of a line of R² passing through 0R². Moreover, this nonperiodic divergence-curl result can be applied to a finite almost periodic bounded variation function u and to a finite almost periodic measure-valued field σ =bν. On the other hand, in the periodic case with dimension N > 2, assuming thatν is absolutely continuous with respect to Lebesgue's measure on the torus TN, we prove that if the product b· ∇ u is the zero measure on TN, so is the product of the TN-means bν · ∇ u. [ABSTRACT FROM AUTHOR]

In this paper we investigate some number theoretic properties of the frequencies of the Korteweg--de Vries equation on the torus, relevant for the stability of finite gap solutions. [ABSTRACT FROM AUTHOR]

Let X be a generic quasi-symmetric representation of a connected reductive group G. The GIT quotient stack X = [ X ss (ℓ) / G ] with respect to a generic ℓ is a (stacky) crepant resolution of the affine quotient X/G, and it is derived equivalent to a noncommutative crepant resolution (=NCCR) of X/G. Halpern-Leistner and Sam showed that the derived category D b (coh X) is equivalent to certain subcategories of D b (coh [ X / G ]) , which are called magic windows. This paper studies equivalences between magic windows that correspond to wall-crossings in a hyperplane arrangement in terms of NCCRs. We show that those equivalences coincide with derived equivalences between NCCRs induced by tilting modules, and that those tilting modules are obtained by certain operations of modules, which is called exchanges of modules. When G is a torus, it turns out that the exchanges are nothing but iterated Iyama–Wemyss mutations. Although we mainly discuss resolutions of affine varieties, our theorems also yield a result for projective Calabi-Yau varieties. Using techniques from the theory of noncommutative matrix factorizations, we show that Iyama–Wemyss mutations induce a group action of the fundamental group π 1 (P 1 \ { 0 , 1 , ∞ }) on the derived category of a Calabi-Yau complete intersection in a weighted projective space. [ABSTRACT FROM AUTHOR]

This paper is devoted to the descriptions of automorphisms and local automorphisms on complex solvable Lie algebras of maximal rank. First, it is established that any automorphism on a solvable Lie algebra of maximal rank can be represented as a product (composition) of inner, diagonal and graph automorphisms. We apply the description of automorphism to the specification of automorphisms on solvable Lie algebras of maximal rank with abelian nilradical, and to the description of automorphisms of standard Borel subalgebras of complex simple Lie algebras. Based on the representation of an automorphism, it is proved that all local automorphisms on a solvable Lie algebra of maximal rank are global automorphisms. We also present two examples of solvable Lie algebras which are not of maximal rank and have different behaviours of local automorphisms. Namely, the first algebra does not admit pure local automorphisms, while the second algebra admits a local automorphism which is not an automorphism. [ABSTRACT FROM AUTHOR]

In this paper, we present two different ways for computing the Kauffman bracket skein module of S 1 × S 2 , KBSM S 1 × S 2 , via braids. We first extend the universal Kauffman bracket type invariant V for knots and links in the Solid Torus ST, which is obtained via a unique Markov trace constructed on the generalized Temperley–Lieb algebra of type B, to an invariant for knots and links in S 1 × S 2 . We do that by imposing on V relations coming from the braid band moves. These moves reflect isotopy in S 1 × S 2 and they are similar to the second Kirby move. We obtain an infinite system of equations, a solution of which is equivalent to computing KBSM S 1 × S 2 . We show that KBSM S 1 × S 2 is not torsion free and that its free part is generated by the unknot (or the empty knot). We then present a diagrammatic method for computing KBSM S 1 × S 2 via braids. Using this diagrammatic method, we also obtain a closed formula for the torsion part of KBSM S 1 × S 2 . [ABSTRACT FROM AUTHOR]

Copyright of Journal of Chongqing University of Technology (Natural Science) is the property of Chongqing University of Technology and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

This paper introduces the fractional Fourier series on the fractional torus and proceeds to investigate several fundamental aspects. Our study includes topics such as fractional convolution, fractional approximation, fractional Fourier inversion, and the Poisson summation formula. We also explore the relationship between the decay of its fractional Fourier coefficients and the smoothness of a function. Additionally, we establish the convergence of the fractional Féjer means and Bochner–Riesz means. Finally, we demonstrate the practical applications of the fractional Fourier series, particularly in the context of solving fractional partial differential equations with periodic boundary conditions, and showcase the utility of approximation methods on the fractional torus for recovering non-stationary signals. [ABSTRACT FROM AUTHOR]

In this paper, we define novel topological invariants of doubly periodic tangles (DP tangles). DP tangles are embeddings of curves in the thickened plane with translational symmetries in two independent directions. We first organize the components of a DP tangle into different interlinked compounds, which are invariants of a DP tangle. The notion of an interlinked compound leads to the classification of DP tangles according to their directional type. We then prove that the directional type is an invariant of DP tangles using the concept of axis-motif, which can be viewed as the blueprint of a DP tangle. [ABSTRACT FROM AUTHOR]

Amelotte and Briggs [arXiv:2304.08476] constructed a family of cohomology operations induced by a torus action. We use these cohomology operations to describe higher differentials of the bicomplex corresponding to the double cohomology of the moment angle complex, as expected by Limonchenko et al. [arXiv:2112.07004]. The present paper can also be viewed as a continuation of the author's paper in [Topology Appl. 326, 1–7 (2023)]. [ABSTRACT FROM AUTHOR]

The aim of this paper is to realize the techniques of picture-valued invariants and invariants valued in free groups for long knots in the full torus. Such knots and links are of particular interest because of their relation to Legendrian knots, knotoids, 3 -manifolds and many other objects. Invariants constructed in the paper are powerful and easy to compare. This paper is a sequel of [V. O. Manturov, A free-group valued invariant of free knots, preprint (2020), arXiv:2012.15571v2]. Long knots naturally appear in the study of classical knots [T. Fiedler, More 1 -cocycles for classical knots, preprint (2020), arXiv:2004.04624; A. Mortier, A Kontsevich integral of order 1 , preprint (2018), arXiv:1810.05747]. [ABSTRACT FROM AUTHOR]

To increase the utilization of wave and other renewable energy resources, an integrated system consisting of an offshore wind turbine and a wave energy converter (WEC) could be used to harvest the potential energy. In this study, a dimensionless optimization method is developed for shape optimization of a hollow cylindrical WEC, and an optimal shape is obtained using a differential evolution (DE) algorithm. The frequency domain response characteristics of the WEC with different geometric shapes and viscous damping loads are studied. The numerical model of the wind-wave integrated system, which consists of a semisubmersible platform and the WEC, is developed and used. The dynamic responses of the integrated system with and without using the WEC optimum section are compared. The results show that the dimensionless optimization method utilized in this paper is very applicable for hollow cylindrical WECs. A smaller inner radius and larger draft increase the heave RAO amplitude of the WEC significantly. In addition, optimization of the WEC shape and power take-off (PTO) damping coefficient can significantly improve the energy capture of the integrated system, which increases by 32.03%. The research results of this paper provide guidance for achieving the optimum design of offshore wind-wave energy integrated systems and quantify the benefits of using optimum designs in the produced wave energy power. In addition, the proposed dimensionless optimization method is generic and can be widely applied to different types of WECs. [ABSTRACT FROM AUTHOR]

In this paper, we provide the mathematical foundation for an explicit and universal feature of cover time for a large class of random work processes, which was previously observed by Chupeau et al (2015 Nat. Phys. 11 844–7). Specifically, we rigorously establish that the fluctuations of the cover time, normalized by the mean first passage time, follow a Gumbel distribution, for finite-range, symmetric, irreducible random walks on a torus of dimension three or higher. The result contributes to a better understanding of cover-time behavior in random search processes, especially on the efficiency of exhaustive searches. Our approach builds upon the work of Belius (2013 Probab. Theory Relat. Fields 157 635–89) on cover times for simple random walks, leveraging a strong coupling between the random walk and random interlacements. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a discretization scheme for the Yang–Mills equations in the two-dimensional case using a framework based on discrete exterior calculus. Within this framework, we define discrete versions of the exterior covariant derivative operator and its adjoint, which capture essential geometric features similar to their continuous counterparts. Our focus is on discrete models defined on a combinatorial torus, where the discrete Yang–Mills equations are presented in the form of both a system of difference equations and a matrix form. [ABSTRACT FROM AUTHOR]

Quasiperiodic systems, related to irrational numbers, are space-filling structures without decay or translation invariance. How to accurately recover these systems, especially for low-regularity cases, presents a big challenge in numerical computation. In this paper, we propose a new algorithm, the finite points recovery (FPR) method, which is available for both continuous and low-regularity cases, to address this challenge. The FPR method first establishes a homomorphism between the lower-dimensional definition domain of quasiperiodic function and the higherdimensional torus, and then recovers the global quasiperiodic system by employing an interpolation technique with finite points in the definition domain without dimensional lifting. Furthermore, we develop accurate and efficient strategies of selecting finite points according to the arithmetic properties of irrational numbers. The corresponding mathematical theory, convergence analysis, and computational complexity analysis on choosing finite points are presented. Numerical experiments demonstrate the effectiveness and superiority of the FPR approach in recovering both continuous quasiperiodic functions and piecewise constant Fibonacci quasicrystals while existing spectral methods encounter difficulties in recovering piecewise constant quasiperiodic functions. [ABSTRACT FROM AUTHOR]

The Boltzmann equation of kinetic theory gives a statistical description of a gas of interacting particles. It is well known that the Boltzmann equation is related to the Euler and Navier–Stokes equations in the field of gas dynamics. In this paper we are concerned with the incompressible Navier–Stokes–Fourier limit of the Boltzmann equation. We prove the incompressible Navier–Stokes–Fourier limit globally in time and the time decay rate of the solution to the rescaled Boltzmann equation in a torus. For ɛ small, by using the truncated expansion and L x , v 2 – L x , v ∞ method, we prove such a limit for the general potentials γ ∈ (− 3 , 1 ] . [ABSTRACT FROM AUTHOR]

Visualizing graphs and networks in non‐Euclidean space can have benefits such as natural focus+context in hyperbolic space and the familiarity of interactions in spherical space. Despite work on these topics going back to the mid 1990s, there is no survey, or a part of a survey for this area of research. In this paper we review and categorize over 60 relevant papers and analyze them by geometry, (e.g., spherical, hyperbolic, torus), by contribution (e.g., technique, evaluation, proof, application), and by graph class (e.g., tree, planar, complex). [ABSTRACT FROM AUTHOR]

Let F be a Morse–Bott foliation on the solid torus T = S 1 × D 2 into 2-tori parallel to the boundary and one singular central circle. Gluing two copies of T by some diffeomorphism between their boundaries, one gets a lens space L p , q with a Morse–Bott foliation F p , q obtained from F on each copy of T and thus consisting of two singular circles and parallel 2-tori. In the previous paper Khokliuk and Maksymenko (J Homotopy Relat Struct 18:313–356. https://doi.org/10.1007/s40062-023-00328-z, 2024) there were computed weak homotopy types of the groups D lp (F p , q) of leaf preserving (i.e. leaving invariant each leaf) diffeomorphisms of such foliations. In the present paper it is shown that the inclusion of these groups into the corresponding group D + fol (F p , q) of foliated (i.e. sending leaves to leaves) diffeomorphisms which do not interchange singular circles are homotopy equivalences. [ABSTRACT FROM AUTHOR]

In this paper, we revisit the Melnikov's persistency problem and illustrate that the Craig--Wayne--Bourgain method can be strengthened to obtain both the existence and linear stability of the invariant tori. The proof is free from the second Melnikov's condition. [ABSTRACT FROM AUTHOR]

In this paper, we consider a special kind of geometrical nonlinear oscillator with a mass parameter admitting two different dynamical states leading to a double-valued potential energy. A cylindrical manifold is introduced to formulate the equation of motion to describe the distinguished dynamical behaviors. With the help of Hamiltonian, complex bifurcations are demonstrated with varying parameters including periodic solutions, the steady states and the blowing up phenomenon near = ± π 2 to infinity. A toroidal manifold is introduced to map the infinities into (0 , ± 2 , 0) on the torus exhibiting saddle-node-like behavior, where the uniqueness of solution is lost, for which a special "collision" parameter is introduced to define the possible motion leaving from infinities. Numerical calculation is carried out to generate bifurcation diagrams using Poincaré sections for the perturbed system to exhibit complex dynamics including the coexistence of periodic solutions, chaos from the coexisting periodic doubling and also instant chaos from the coexisting periodic solutions. The results demonstrated herein this paper provide a brand new insight into the understanding of enriched nonlinear dynamics and an essential explanation about "collision" of mechanical system with both geometrical and mass parameters. [ABSTRACT FROM AUTHOR]

In this paper, we report the bifurcations of mode-locked periodic orbits occurring in maps of three or higher dimensions. The "torus" is represented by a closed loop in discrete time, which contains stable and unstable cycles of the same periodicity, and the unstable manifolds of the saddle. We investigate two types of "doubling" of such loops: (a) two disjoint loops are created and the iterates toggle between them, and (b) the length of the closed invariant curve is doubled. Our work supports the conjecture of Gardini and Sushko, which says that the type of bifurcation depends on the sign of the third eigenvalue. We also report the situation arising out of Neimark–Sacker bifurcation of the stable and saddle cycles, which creates cyclic closed invariant curves. We show interesting types of saddle-node connection structures, which emerge for parameter values where the stable fixed point has bifurcated but the saddle has not, and vice versa. [ABSTRACT FROM AUTHOR]