Search

We consider analytic maps and vector fields defined in R 2 × T d , having a d -dimensional invariant torus T. The map (resp. vector field) restricted to T defines a rotation of Diophantine frequency vector ω ∈ R d , and its derivative restricted to transversal directions to T does not diagonalize. In this context, we give conditions on the coefficients of the nonlinear terms of the map (resp. vector field) under which T possesses stable and unstable invariant manifolds, and we show that such invariant manifolds are analyitic away from the invariant torus. We also provide effective algorithms to compute approximations of parameterizations of the invariant manifolds, and a posteriori theorems that can be used to validate numerical computations. Moreover, we present some applications of the results. [ABSTRACT FROM AUTHOR]

The use of ideal memristors in a continuous-time (CT) nonlinear circuit is known to greatly enrich the dynamic behavior with respect to the memristorless counterpart, which is a crucial property for applications in future analog electronic circuits. This can be explained via the flux–charge analysis method (FCAM), according to which CT circuits with ideal memristors have for structural reasons first integrals (or invariants of motion, or conserved quantities) and their state space can be foliated in infinitely many invariant manifolds where they can display different dynamics. The paper introduces a new discretization scheme for the memristor which, differently from those adopted in the literature, guarantees that the first integrals of the CT memristor circuits are preserved exactly in the discretization, and that this is true for any step size. This new scheme makes it possible to extend FCAM to discrete-time (DT) memristor circuits and rigorously show the existence of invariant manifolds and infinitely many coexisting attractors (extreme multistability). Moreover, the paper addresses standard bifurcations varying the discretization step size and also bifurcations without parameters, i.e. bifurcations due to varying the initial conditions for fixed step size and circuit parameters. The method is illustrated by analyzing the dynamics and flip bifurcations with and without parameters in a DT memristor–capacitor circuit and the Poincaré–Andronov–Hopf bifurcation in a DT Murali–Lakshmanan–Chua circuit with a memristor. Simulations are also provided to illustrate bifurcations in a higher-order DT memristor Chua's circuit. The results in the paper show that DT memristor circuits obtained with the proposed discretization scheme are able to display even richer dynamics and bifurcations than their CT counterparts, due to the coexistence of infinitely many attractors and the possibility to use the discretization step as a parameter without destroying the foliation in invariant manifolds. [ABSTRACT FROM AUTHOR]

Here we prove that the dynamics on any two center-manifolds of a fixed point of any C k , 1 dynamical system of finite dimension with k ≥ 1 are C k -conjugate to each other. For pedagogical purpose, we also extend Perron's method for differential equations to diffeomorphisms to construct the theory of invariant manifolds and invariant foliations at fixed points of dynamical systems of finite dimensions. [ABSTRACT FROM AUTHOR]

• The low-thrust transfer problem between the L2 northern and southern halo orbits in the Earth-Moon CRTBP is solved. • A flowchart for the calculation of the transfer trajectories between halo orbits based on the time-, energy-, and fuel-optimal criteria is presented. • The transfer configurations are constructed using two different initializations: the multi-revolution structures and the invariant manifold theory. • A strategy to quickly search for the location of the transfer endpoints on the target orbits is suggested to obtain the best transfer solution. This paper presents a study of the low-thrust transfer problem between L2 northern and southern halo orbits using the Pontryagin minimum principle (PMP) in the Earth-Moon circular restricted three-body problem (CRTBP). The northern and southern halo families exist in the vicinity of the Moon, and have the same orbital period distribution, which can be used for space missions such as lunar navigation constellations, lunar relay communications and lunar space stations. Traditional low-thrust transfer trajectories are multi-revolution structures, which are modeled in this problem as multiple rotations around the libration point. The characteristics of the transfer are affected by the number of revolutions. Moreover, we propose a novel approach using invariant manifolds associated with periodic orbits in the CRTBP to construct L2 north–south halo orbit transfers. This approach results in superior transfer performance, which involves shorter time of flight, lower fuel consumption and reduced engine thrust requirements. A complete set of low-thrust time-, energy- and fuel-optimal transfer algorithms using moving point strategies and continuation methods is proposed. The computational procedures and analysis presented in this paper provide a reliable reference for the design of related orbital transfer missions. [ABSTRACT FROM AUTHOR]

This paper explores how to identify a reduced order model (ROM) from a physical system. A ROM captures an invariant subset of the observed dynamics. We find that there are four ways a physical system can be related to a mathematical model: invariant foliations, invariant manifolds, autoencoders and equation-free models. Identification of invariant manifolds and equation-free models require closed-loop manipulation of the system. Invariant foliations and autoencoders can also use off-line data. Only invariant foliations and invariant manifolds can identify ROMs, and the rest identify complete models. Therefore, the common case of identifying a ROM from existing data can only be achieved using invariant foliations. Finding an invariant foliation requires approximating high-dimensional functions. For function approximation, we use polynomials with compressed tensor coefficients, whose complexity increases linearly with increasing dimensions. An invariant manifold can also be found as the fixed leaf of a foliation. This only requires us to resolve the foliation in a small neighbourhood of the invariant manifold, which greatly simplifies the process. Combining an invariant foliation with the corresponding invariant manifold provides an accurate ROM. We analyse the ROM in case of a focus type equilibrium, typical in mechanical systems. The nonlinear coordinate system defined by the invariant foliation or the invariant manifold distorts instantaneous frequencies and damping ratios, which we correct. Through examples we illustrate the calculation of invariant foliations and manifolds and at the same time show that Koopman eigenfunctions and autoencoders fail to capture accurate ROMs under the same conditions. [ABSTRACT FROM AUTHOR]

Invariant manifolds are important for describing the dynamics of systems of ODEs. In this article we classify invariant algebraic manifolds in the model of double convection given by a three-dimensional quadratic differential system. We find the explicit solutions to this system whose trajectories lie on invariant manifolds. [ABSTRACT FROM AUTHOR]

The orbital motion of a spacecraft with a solar sail in the vicinity of the Sun–Earth collinear libration point is studied. The study is carried out within the Hill approximation of the circular restricted three-body problem. The collinear libration point is unstable, but instability can be a positive factor in the tasks of maneuvering in interplanetary space. This is because a small change in velocity near an unstable libration point can lead to a significant change in orbital motion. This property can be used in the development of projects to counteract the comet–asteroid hazard. A scheme for constructing the trajectories of interception of a spacecraft with a potentially hazardous object is proposed. One of the stages of this scheme, an optimal escape from the vicinity of the libration point using light pressure forces, is considered. The results of numerical simulations are presented. [ABSTRACT FROM AUTHOR]

We study two rather similar evolutionary partial differential equations. One of them was obtained by Sivashinsky and the other by Kuramoto. The Kuramoto version was taken as the basic version of the equation that became known as the Kuramoto–Sivashinsky equation. We supplement each version of the Kuramoto–Sivashinsky equation with natural boundary conditions and, for the proposed boundary-value problems, study local bifurcations arising near a homogeneous equilibrium when they change stability. The analysis is based on the methods of the theory of dynamical systems with an infinite-dimensional phase space, namely, the methods of integral manifolds and normal forms. For all boundary-value problems, asymptotic formulas are obtained for solutions that form integral manifolds. We also point out boundary conditions under which the dynamics of solutions of the corresponding boundary-value problems of the two versions of the Kuramoto–Sivashinsky equation are significantly different. [ABSTRACT FROM AUTHOR]

This article concerns the global dynamics of a continuous planar piecewise linear differential system with three zones. We give global phase portraits in the Poincaré disc and classify bifurcation diagrams under certain parametric conditions, when the dynamics of central linear zone is anti-saddle. Rich dynamical behaviors are demonstrated, from which we observe homoclinic loops appearing in three linear zones and limit cycles occurring in three linear zones which surround a node or node-focus. [ABSTRACT FROM AUTHOR]

We prove a flow box theorem for smooth 2-dimensional slow-fast vector fields, providing a simple normal form that is obtained by smooth coordinate changes, without having to change the time. We introduce a notion of 2d slow-fast diffeomorphism, define the log-determinant integral and prove a normal form theorem similar to the flow box theorem for slow-fast vector fields. The normal form is used in proving the relevance of the log-determinant integral. [ABSTRACT FROM AUTHOR]

On an arbitrary Banach space, assuming that a linear nonautonomous difference equation xm+1 = Amxm admits a very general type of trichotomy, we establish conditions for the existence of global Lipschitz invariant center manifolds of the perturbed equation xm+1 = Amxm + fm(xm). Our results not only improve results already existing in the literature, but also include new cases. [ABSTRACT FROM AUTHOR]

We consider a Z 2 -equivariant flow in R 4 with an integral of motion and a hyperbolic equilibrium with a transverse homoclinic orbit Γ. We provide criteria for the existence of stable and unstable invariant manifolds of Γ. We prove that if these manifolds intersect transversely, creating a so-called super-homoclinic, then in any neighborhood of this super-homoclinic there exist infinitely many multi-pulse homoclinic loops. An application to a system of coupled nonlinear Schrödinger equations is considered. [ABSTRACT FROM AUTHOR]

The controlled motion of a solar sail is considered, with the reflectivity being a control parameter. The equations of controlled orbital motion of the solar sail in the vicinity of a Sun–Earth collinear libration point are investigated. The model of the circular restricted three-body problem of the Sun–Earth system is used to describe the controlled orbital motion of the solar sail. The laws of change in reflectivity to keep the solar sail in the vicinity of the libration point by means of light pressure forces are constructed, and an estimate of the controllability region is given. The control laws constructed for a linearized model of the equations of orbital motion are extended to the nonlinear case. The proposed technique for constructing the control laws enhances the potential in solving the problems of motion stabilization in the vicinity of a collinear libration point, where the light pressure force can have a significant efficiency. Numerical simulations of controlled motion are presented. [ABSTRACT FROM AUTHOR]

We consider the periodic boundary value problem for two variants of a weakly dissipative complex Ginzburg–Landau equation. In the first case, we study a variant of such an equation that contains the cubic and quintic nonlinear terms. We study the problem of local bifurcations of traveling periodic waves under stability changes. We show that a countable set of two-dimensional invariant tori arises as a result of such bifurcations. Both types of bifurcations are possible in the considered formulation of the problem, soft (postcritical) and hard (subcritical) ones, depending on the choice of the coefficients in the equation. We obtain asymptotic formulas for the solutions forming the invariant tori. We also study the periodic boundary value problem for the equation that is called the nonlocal Ginzburg–Landau equation in physics. We show that the boundary value problem in the considered variant has an infinite-dimensional global attractor. We present the solutions forming such an attractor. [ABSTRACT FROM AUTHOR]

This paper is concerned with equivariant normal forms of semilinear functional differential equations (FDEs) in general Banach spaces. The analysis is based on the theory previously developed for autonomous delay differential equations and on the existence of invariant manifolds. We show that in the neighborhood of trivial solutions, variables can be chosen so that the form of the reduced vector field relies not only on the information of the linearized system at the critical point but also on the inherent symmetry. We observe that the normal forms give critical information about dynamical properties, such as generic local branching spatiotemporal patterns of equilibria and periodic solutions. As an important application of equivariant normal forms, we not only establish equivariant Hopf bifurcation theorem for semilinear FDEs in general Banach spaces, but also in a natural way derive criteria for the existence, stability, and bifurcation direction of branches of bifurcating periodic solutions. We employ these general results to obtain the existence of infinite many small-amplitude wave solutions for a delayed Ginzburg-Landau equation on a two-dimensional disk with the homogeneous Dirichlet boundary condition. [ABSTRACT FROM AUTHOR]

This paper discerns the invariant manifold of a class of ill-posed stochastic evolution equations driven by a nonlinear multiplicative noise. To be more precise, we establish the existence of mean-square random unstable invariant manifold and only mean-square stable invariant set. Due to the lack of the Hille-Yosida condition, we construct a modified variation of constants formula by the resolvent operator. With the price of imposing an unusual condition involving a non-decreasing map, we set up the Lyapunov-Perron method and derive the required estimates. We also emphasize that the Lyapunov-Perron map in the forward time loses the invariant due to the adaptedness, we alternatively establish the existence of mean-square random stable sets. [ABSTRACT FROM AUTHOR]

We study existence and multiplicity of positive ground states for the scalar curvature equation Δ u + K (| x |) u n + 2 n - 2 = 0 , x ∈ R n , n > 2 , when the function K : R + → R + is bounded above and below by two positive constants, i.e. 0 < K ̲ ≤ K (r) ≤ K ¯ for every r > 0 , it is decreasing in (0 , R) and increasing in (R , + ∞) for a certain R > 0 . We recall that in this case ground states have to be radial, so the problem is reduced to an ODE and, then, to a dynamical system via Fowler transformation. We provide a smallness non perturbative (i.e. computable) condition on the ratio K ¯ / K ̲ which guarantees the existence of a large number of ground states with fast decay, i.e. such that u (| x |) ∼ | x | 2 - n as | x | → + ∞ , which are of bubble-tower type. We emphasize that if K(r) has a unique critical point and it is a maximum the radial ground state with fast decay, if it exists, is unique. [ABSTRACT FROM AUTHOR]

We analyze a singularly Kuramoto-Sivashinsky perturbed Camassa-Holm equation with methods of the geometric singular perturbation theory. Especially, we study the persistence of smooth and peaked solitons. Whether a solitary wave of the original Camassa-Holm equation is smooth or peaked depends on whether there is linear dispersion, i.e. whether 2 k = 0. If 2 k > 0 , then a unique smooth solitary wave persists with selected wave speed under singular Kuramoto-Sivashinsky perturbation just as what happens in the KS-KdV equation. On the other hand, we show that if there is no linear dispersion, i.e. 2 k = 0 , then any observable peaked soliton fails to persist. This case is non-typical since the related slow manifold blows up and the classical geometric singular perturbation theory is not available. [ABSTRACT FROM AUTHOR]

The aim of this paper is to construct invariant manifolds for a coupled system, consisting of a parabolic equation and a second-order ordinary differential equation, set on T3\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb {T}^3$$\end{document} and subject to periodic boundary conditions. Notably, the “spectral gap condition" does not hold for the system under consideration, leading to the use of the spatial averaging principle, together with the graph transform method. This approach facilitates the construction of the relevant invariant manifold, characterized by attributes such as Lipschitz continuity, local invariance, infinite dimensionality, and exponential tracking, thus mirroring the properties traditionally associated with a classical global manifold. [ABSTRACT FROM AUTHOR]

The controlled motion of a spacecraft with a solar sail in interplanetary space near the collinear libration points and of the Sun–Earth system is investigated. The equations of the circular restricted three-body problem and their modifications are used to describe the orbital motion. We present a technique for constructing the orbital motion control laws and provide an estimate of the controllability region when solving the problem of stabilization or long-term spacecraft keeping in the vicinity of and . This technique is based on the solution of the control problem for a linearized model of the equations of orbital motion and the transfer of results to the nonlinear case. Such a technique for the construction of control laws extends the possibilities both in solving the problems of motion stabilization and in solving the problems of maneuvering in the vicinity of a collinear libration point. We estimate the controllability region for a spacecraft with a solar sail and present the trajectories of possible spacecraft motion that are provided by the solar sail orientation when flying from the vicinity of a collinear libration point. It is in these regions of space that the light pressure force can have a significant efficiency due to the relatively small magnitude of the gravitational force. As an example, we consider a model of a spacecraft with a solar sail whose sail area-to-mass ratio corresponds to the spacecraft from the implemented IKAROS project and is less than that of the CubeSat spacecraft from the implemented LightSail-2 project by an order of magnitude. [ABSTRACT FROM AUTHOR]

We assign some kind of invariant manifolds to a given integrable PDE (its discrete or semidiscrete variant). First, we linearize the equation around its arbitrary solution u. Then we construct a differential (respectively, difference) equation compatible with the linearized equation for any choice of u. This equation defines a surface called a generalized invariant manifold. In a sense, the manifold generalizes the symmetry, which is also a solution to the linearized equation. In this paper, we concentrate on continuous and discrete models of hyperbolic type. It is known that such kinds of equations have two hierarchies of symmetries, corresponding to the characteristic directions. We have shown that a properly chosen generalized invariant manifold allows one to construct recursion operators that generate these symmetries. It is surprising that both recursion operators are related to different parametrizations of the same invariant manifold. Therefore, knowing one of the recursion operators for the hyperbolic type integrable equation (having no pseudo-constants) we can immediately find the second one. [ABSTRACT FROM AUTHOR]

We consider the Cahn–Hilliard equation in the case where its solution depends on two spatial variables, with homogeneous Dirichlet and Neumann boundary conditions, and also periodic boundary conditions. For these three boundary value problems, we study the problem of local bifurcations arising when changing stability by spatially homogeneous equilibrium states. We show that the nature of bifurcations that lead to spatially inhomogeneous solutions is strongly related to the choice of boundary conditions. In the case of homogeneous Dirichlet boundary conditions, spatially inhomogeneous equilibrium states occur in a neighborhood of a homogeneous equilibrium state, depending on both spatial variables. An alternative scenario is realized in analyzing the Neumann problem and the periodic boundary value problem. In these, as a result of bifurcations, invariant manifolds formed by spatially inhomogeneous solutions occur. The dimension of these manifolds ranges from 1 to 3. In analyzing three boundary value problems, we use methods of infinite-dimensional dynamical system theory and asymptotic methods. Using the integral manifold method together with the techniques of normal form theory allows us to analyze the stability of bifurcating invariant manifolds and also to derive asymptotic formulas for spatially inhomogeneous solutions forming these manifolds. [ABSTRACT FROM AUTHOR]

In this paper, a robust output voltage-tracking algorithm is proposed for dc/dc boost power conversion systems based on a variable invariant manifold. A systematic multivariable approach, considering not only nonlinearity but also the parametric uncertainties, is used for deriving the control law. The proposed method has two features. First, a variable cut-off frequency algorithm is constructed to automatically adjust the invariant manifold, improving the transient output voltage-tracking performance. Second, nonlinear disturbance observers are introduced to enable the control law to exponentially recover the desirable tracking performance without any offset errors in the variable invariant manifold. The performance of the proposed technique is experimentally confirmed with a 3-kW dc/dc boost power conversion system. [ABSTRACT FROM AUTHOR]

We consider an autonomous system admitting an invariant manifold M. The following questions are discussed: (i) what are the conditions ensuring exponential stability of the invariant manifold? (ii) does every motion attracting byMtend to some motion onM(i.e. have an asymptotic phase)? (iii) what is the geometrical structure of the set formed by orbits approaching a given orbit? We get an answer to (i) in terms of Lyapunov functions omitting the assumption that the normal bundle of M is trivial. An affirmative answer to (ii) is obtained for invariant manifold M with partially hyperbolic structure of tangent bundle. In this case, the existence of asymptotic phase is obtained under new conditions involving contraction rates of the linearized flow in normal and tangential to M directions. To answer the question (iii), we show that a neighborhood of M has a structure of invariant foliation each leaf of which corresponds to motions with common asymptotic phase. In contrast to theory of cascades, our technique exploits the classical Lyapunov-Perron method of integral equations. [ABSTRACT FROM AUTHOR]

This article studies the flocking behavior of self-organized agents in two species. First, referring to the work of Olfati-Saber and the classical Cucker-Smale model, we establish a discrete system describing the flocking dynamic of the agents in two species. Second, by using the LaSalle's invariance principle, we show that the system with global interaction will achieve unconditional time-asymptotic flocking, and the system with local interaction has a time-asymptotic flocking under certain assumptions. Moreover, we investigate the local asymptotic stability of a class of flocking solutions. Finally, some numerical simulations and qualitative results are presented. [ABSTRACT FROM AUTHOR]

Gradient descent-based optimization methods underpin the parameter training of neural networks, and hence comprise a significant component in the impressive test results found in a number of applications. Introducing stochasticity is key to their success in practical problems, and there is some understanding of the role of stochastic gradient descent in this context. Momentum modifications of gradient descent such as Polyak's Heavy Ball method (HB) and Nesterov's method of accelerated gradients (NAG), are also widely adopted. In this work our focus is on understanding the role of momentum in the training of neural networks, concentrating on the common situation in which the momentum contribution is fixed at each step of the algorithm. To expose the ideas simply we work in the deterministic setting. Our approach is to derive continuous time approximations of the discrete algorithms; these continuous time approximations provide insights into the mechanisms at play within the discrete algorithms. We prove three such approximations. Firstly we show that standard implementations of xed momentum methods approximate a time-rescaled gradient descent flow, asymptotically as the learning rate shrinks to zero; this result does not distinguish momentum methods from pure gradient descent, in the limit of vanishing learning rate. We then proceed to prove two results aimed at understanding the observed practical advantages of xed momentum methods over gradient descent, when implemented in the non-asymptotic regime with xed small, but non-zero, learning rate. We achieve this by proving approximations to continuous time limits in which the small but xed learning rate appears as a parameter; this is known as the method of modified equations in the numerical analysis literature, recently rediscovered as the high resolution ODE approximation in the machine learning context. In our second result we show that the momentum method is approximated by a continuous time gradient flow, with an additional momentum-dependent second order time-derivative correction, proportional to the learning rate; this may be used to explain the stabilizing eect of momentum algorithms in their transient phase. Furthermore in a third result we show that the momentum methods admit an exponentially attractive invariant manifold on which the dynamics reduces, approximately, to a gradient flow with respect to a modified loss function, equal to the original loss function plus a small perturbation proportional to the learning rate; this small correction provides convexification of the loss function and encodes additional robustness present in momentum methods, beyond the transient phase. [ABSTRACT FROM AUTHOR]

This paper presents a novel derivation of the direct parametrisation method for invariant manifolds able to build simulation-free reduced-order models for nonlinear piezoelectric structures, with a particular emphasis on applications to Micro Electro Mechanical Systems. The constitutive model adopted accounts for the hysteretic and electrostrictive response of the piezoelectric material by resorting to the Landau-Devonshire theory of ferroelectrics. Results are validated with full-order simulations operated with a harmonic balance finite element method to highlight the reliability of the proposed reduction procedure. Numerical results show a remarkable gain in terms of computing time as a result of the dimensionality reduction process over low dimensional invariant sets. Results are also compared with experimental data to highlight the remarkable benefits of the proposed model order reduction technique. • Model order reduction for structures with geometric and piezoelectric nonlinearity. • Direct computation of the invariant manifold from finite element problem. • Efficient technique accounting for electrostrictive response of the piezoelectric. • High accuracy of the results compared to full-order simulation. • Application to micro-electro mechanical systems with complex geometry. [ABSTRACT FROM AUTHOR]

We investigate the existence of a two-dimensional invariant manifold that attracts all nonzero orbits in 3 species Lotka-Volterra systems with identical linear growth rates. This manifold, which we call the balance simplex, is the common boundary of the basin of repulsion of the origin and the basin of repulsion of infinity. The balance simplex is linked to ecological models where there is 'growth when rare' and competition for finite resources. By including alternative food sources for predators we cater for predator-prey type models. In the case that the model is competitive, the balance simplex coincides with the carrying simplex which is an unordered manifold (no two points may be ordered componentwise), but for non-competitive models the balance simplex need not be unordered. The balance simplex of our models contains all limit sets and is the graph of a piecewise analytic function over the unit probability simplex. [ABSTRACT FROM AUTHOR]

We consider self-pulsing in lasers with a gain section and an absorber section via a mechanism known as Q -switching, as described mathematically by the Yamada ordinary differential equation model for the gain, the absorber and the laser intensity. More specifically, we are interested in the case that gain and absorber decay on different time-scales. We present an overall bifurcation structure by showing how the two-parameter bifurcation diagram in the plane of pump strength versus decay rate of the gain changes with the ratio between the two decay rates. In total, there are ten cases BI to BX of qualitatively different two-parameter bifurcation diagrams, which we present with an explanation of the transitions between them. Moroever, we show for each of the associated eleven cases of structurally stable phase portraits (in open regions of the parameter space) a three-dimensional representation of the organization of phase space by the two-dimensional manifolds of saddle equilibria and saddle periodic orbits. The overall bifurcation structure provides a comprehensive picture of the observable dynamics, including multistability and excitability, which we expect to be of relevance for experimental work on Q -switching lasers with different kinds of saturable absorbers. [ABSTRACT FROM AUTHOR]

The goal of this paper is to apply the method of Lagrangian descriptors to reveal the phase space mechanism by which a Caldera-type potential energy surface (PES) exhibits the dynamical matching phenomenon. Using this technique, we can easily establish that the nonexistence of dynamical matching is a consequence of heteroclinic connections between the unstable manifolds of the unstable periodic orbits (UPOs) of the upper index-1 saddles (entrance channels to the Caldera) and the stable manifolds of the family of UPOs of the central minimum of the Caldera, resulting in the temporary trapping of trajectories. Moreover, dynamical matching will occur when there is no heteroclinic connection, which allows trajectories to enter and exit the Caldera without interacting with the shallow region of the central minimum. Knowledge of this phase space mechanism is relevant because it allows us to effectively predict the existence, and nonexistence, of dynamical matching. In this work, we explore a stretched Caldera potential by means of Lagrangian descriptors, allowing us to accurately compute the critical value for the stretching parameter for which dynamical matching behavior occurs in the system. This approach is shown to provide a tremendous advantage for exploring this mechanism in comparison to other methods from nonlinear dynamics that use phase space dividing surfaces. [ABSTRACT FROM AUTHOR]

China's Chang'E-4 (CE-4) probe has started its exploration on the far side of the Moon thanks to the relay satellite---Queqiao---which is around the Earth-Moon collinear libration point L 2 and provides continuous communication links between the lander and ground stations. The triangular libration points of the Earth-Moon system, L 4 and L 5 , have long been considered as potential locations for future space applications. As a possible extension of Queqiao or other future missions around L 2 , the work presented here contributes to constructing transfers from the collinear libration point ( L 2 ) to the triangular libration points. Taking the L 4 point as an example, two types of two-maneuver transfers from a planar Lyapunov orbit around L 2 to the periodic orbit around L 4 are investigated in the planar Earth-Moon system and compared with each other, including the type-I transfer utilizing invariant manifolds along with powered lunar gravity assist and the type-II transfer using the Jacobi integral. Our studies indicate that type-I transfer saves more Δ v cost than type-II transfer. For the cases studied by us, the minimum Δ v cost is about 0.1636 km/s for type-I transfer, with a corresponding transfer time of 169.84 days. For type-II transfer and the cases studied by us, the corresponding values are 0.4745 km/s and 137.19 days. At the end of the paper, the type-I transfer is generalized (with some modifications) to the ephemeris model and some example transfer trajectories are given. Although only the transfer from L 2 to L 4 is studied in this paper, the same orbit design strategy can be generalized to the transfer to the triangular libration point L 5 , and the transfers from the collinear libration point L 1 to triangular libration points. [ABSTRACT FROM AUTHOR]

We present a relatively plain, direct and intuitive methodology that accomplishes model order reduction on the invariant center manifold of large-dimensional nonlinear systems subjected to external periodic excitations. This methodology is applicable to parametrically excited (periodic coefficients) and constant coefficient nonlinear systems. Our approach is empowered by intuitive augmentation of system states and is further liberated from typical special strategies such as the need for a 'book-keeping' parameter, a detuning parameter and minimal excitation. Consequently, it is relatively straightforward to implement and applicable to a broad range of nonlinear systems with external periodic excitations. Two illustrative application cases are investigated, and their subsequent results validate the accuracy of our approach. The first case considers a parametrically excited system with a single external periodic frequency excitation. A mass-spring-damper system with constant coefficients and two heterogeneous external frequency excitations is considered in the second example. In both cases, the long-term steady-state behavior analysis shows exact or cogent agreement between the original and the reduced-order system. [ABSTRACT FROM AUTHOR]

We prove that a general class of nonlinear, non-autonomous ode s in Fréchet spaces are close to ode s in a specific normal form, where closeness means that solutions of the normal form ode satisfy the original ode up to a residual that vanishes up to any desired order. In this normal form, the centre, stable and unstable coordinates of the ode are clearly separated, which allows us to define invariant manifolds of such equations in a robust way. The main motivation is the case where the Fréchet space in question is a suitable function space, and the maps involved in an ode in this space are defined in terms of derivatives of the functions, so that the infinite-dimensional ode is a finite-dimensional pde. We show that our methods apply to a relevant class of nonlinear, non-autonomous pde s in this way. [ABSTRACT FROM AUTHOR]

A result due to M.W. Hirsch states that most competitive maps admit a carrying simplex, i.e., an invariant hypersurface of codimension one which attracts all nontrivial orbits. The common approach in the study of these maps is to focus on the dynamical behavior on the carrying simplex. However, this manifold is normally non-smooth. Therefore, not every tool coming from Differential Geometry can be applied. In this paper we prove that the restriction of the map to the carrying simplex in a neighborhood of an interior fixed point is topologically conjugate to the restriction of the map to its pseudo-unstable manifold by an invariant foliation. This implies that the linearization techniques are applicable for studying the local dynamics of the interior fixed points on the carrying simplex. We further construct the stable and unstable manifolds on the carrying simplex. Our results give partial responses to Hirsch's problem regarding the smoothness of the carrying simplex. We discuss some applications in classical models of population dynamics. [ABSTRACT FROM AUTHOR]

Explicit expressions in terms of Gaussian Hypergeometric functions are found for a 'balance' manifold that connects the non-zero steady states of a 2-species, non-competitive, scaled Lotka–Volterra system by the unique heteroclinic orbits. In this model, the parameters are the interspecific interaction coefficients which affects the form of the solution used. Similar to the carrying simplex of the competitive model, this balance simplex is the common boundary of the basin of repulsion of the origin and infinity, and is smooth except possibly at steady states. [ABSTRACT FROM AUTHOR]

We have simplified the complex chemical reaction by reducing it from a high dimension to a low dimension to observe the behavior of participating species near the equilibrium point and obtained their solution curves without affecting the accuracy of the whole mechanism. For this purpose, we have applied three well-established techniques: Quasi Equilibrium Manifold (QEM), Spectral Quasi Equilibrium Manifold (SQEM) and Intrinsic Low Dimension Manifold (ILDM) over a Ternkin-Boudart Mechanism. However, the invariant solution curves and their comparison for the reduced species of Ternkin-Boudart reaction mechanism have never been reported so far. Therefore, the aim of this article is to study such facts. All the above-mentioned model-reduction techniques are applied to solve the high dimensional complex problem and comparison between invariant curves of the reduced species is presented graphically and leading towards the higher dimensional manifold. The efficiency of ILDM is compared with the QEM and SQEM graphically and in tabulated form by using MATLAB. • A mathematical model for a complex reaction mechanism is designed. • Model reduction techniques (MRT) are applied to reduce the complexity without losing their essential information. • The solutions behavior obtained through different reduction techniques are compared graphically. • The surface solution (2D) is presented graphically. [ABSTRACT FROM AUTHOR]

Abstract In this paper, a Tuned Bistable Nonlinear Energy Sink (TBNES) considering local and global potentials is implemented to decrease the vibrations of the host structural system. The structural system consists of two coupled linear symmetric oscillators and the nonlinear energy sink consists of a mass-spring-damper which is connected to the ground by means of linear and nonlinear local potentials. Based on Newton's second law, the governing equations of motion are derived. Then, using multiple scales-harmonic balance method, the response of the system for 1:1 external resonance case is extracted and the effects of various parameters on the energy absorption of the TBNES are investigated for both single mode interaction dynamics and complete full dynamics of the system. It is shown that adding potential element greatly impacts on the invariant manifolds of the system. For first mode interactions, implementing any nonlinear local spring leads to limit the stable non-zero domain of the absorber. Furthermore, the absorber performance declines by increasing the nonlinearity of the spring. So, it seems that for first mode excitation the best option is to have a local potential with minimum nonlinearity. But, for the second mode interaction, nonlinear local potential can enhance the absorber performance. In the following, the absorption capability of some famous absorbers are compared and it is obtained that adding local potentials results in greatly increase the absorption performance, especially for second mode interaction of the system. But, it strongly increases the nonlinearity of the absorber's response which makes it unpredictable. Highlights • Tuned Bistable Nonlinear Energy Sinks considering local and global potentials are investigated. • Using multiple scales method, the response of the system for 1:1 external resonance case is studied. • For first mode excitation the best option is to have a local potential with minimum nonlinearity. • Local potential results in highly increase the absorption performance for second mode absorption. [ABSTRACT FROM AUTHOR]

The optimally time-dependent (OTD) modes form a time-evolving orthonormal basis that captures directions in phase space associated with transient and persistent instabilities. In the original formulation, the OTD modes are described by a set of coupled evolution equations that need to be solved along the trajectory of the system. For many applications where real-time estimation of the OTD modes is important, such as control or filtering, this is an expensive task. Here, we examine the low-dimensional structure of the OTD modes. In particular, we consider the case of slow-fast systems, and prove that OTD modes rapidly converge to a slow manifold, for which we derive an asymptotic expansion. The result is a parametric description of the OTD modes in terms of the system state in phase space. The analytical approximation of the OTD modes allows for their offline computation, making the whole framework suitable for real-time applications. In addition, we examine the accuracy of the slow-manifold approximation for systems in which there is no explicit time-scale separation. In this case, we show numerically that the asymptotic expansion of the OTD modes is still valid for regions of the phase space where strongly transient behavior is observed, and for which there is an implicit scale separation. We also find an analogy between the OTD modes and the Gram{Schmidt vectors (also known as orthogonal or backward Lyapunov vectors), and thereby establish new properties of the former. Several examples of low-dimensional systems are provided to illustrate the analytical formulation. [ABSTRACT FROM AUTHOR]

This work concerns the dynamics of a class of stochastic dynamical systems with a multiplicative non-Gaussian Lévy noise. We first establish the existence of stable and unstable foliations for this kind of system via the Lyapunov-Perron method. Then we examine the geometric structure of the invariant foliations, and their relation with invariant manifolds. Also we illustrate our results in an example. [ABSTRACT FROM AUTHOR]

Abstract In this paper we study entire radial solutions for the quasilinear p -Laplace equation Δ p u + k (x) f (u) = 0 where k is a radial positive weight and the nonlinearity behaves e.g. as f (u) = u | u | q − 2 − u | u | Q − 2 with q < Q. In particular we focus our attention on solutions (positive and sign changing) which are infinitesimal at infinity, thus providing an extension of a previous result by Tang (2001). [ABSTRACT FROM AUTHOR]

Oscillatory rhythms are investigated in a model network where a pair of excitatory neurons interact via an inhibitory neuron. Each individual neuron is modeled as a relaxation oscillator, and a slow/fast decomposition process reduces the problem of a stable oscillatory rhythm to the problem of a stable fixed point of a map. Further reduction of the map is made by finding an invariant manifold. A detailed analysis of the reduced map reveals how different types of stable oscillatory rhythms arise, depending on the intrinsic properties of the individual neurons and the synaptic time constant. [ABSTRACT FROM AUTHOR]

The Lorenz-96 model is widely used as a test model for various applications, such as data assimilation methods. This symmetric model has the forcing F ∈ ℝ and the dimension n ∈ ℕ as parameters and is ℤ n -equivariant. In this paper, we unravel its dynamics for F < 0 using equivariant bifurcation theory. Symmetry gives rise to invariant subspaces that play an important role in this model. We exploit them in order to generalize results from a low dimension to all multiples of that dimension. We discuss symmetry for periodic orbits as well. Our analysis leads to proofs of the existence of pitchfork bifurcations for F < 0 in specific dimensions n : In all even dimensions, the equilibrium (F , ... , F) exhibits a supercritical pitchfork bifurcation. In dimensions n = 4 k , k ∈ ℕ , a second supercritical pitchfork bifurcation occurs simultaneously for both equilibria originating from the previous one. Furthermore, numerical observations reveal that in dimension n = 2 q p , where q ∈ ℕ ∪ { 0 } and p is odd, there is a finite cascade of exactly q subsequent pitchfork bifurcations, whose bifurcation values are independent of n. This structure is discussed and interpreted in light of the symmetries of the model. [ABSTRACT FROM AUTHOR]

We provide the structure of regular/singular fast/slow decay radially symmetric solutions for a class of superlinear elliptic equations with an indefinite weight. In particular we are interested in the case where such a weight is positive in a ball and negative outside, or in the reversed situation. We extend the approach to elliptic equations in presence of Hardy potentials, i.e. to Δu+h(|x|)|x|2u+f(u,|x|)=0where h is not necessarily constant. By the use of Fowler transformation we study the corresponding dynamical systems, presenting the construction of invariant manifolds when the global existence of solutions is not ensured. [ABSTRACT FROM AUTHOR]

This paper is concerned with the Camassa–Holm equation, which is a model for shallow water waves. We first establish the existence of solitary wave solutions for the equation without delay. And then we prove the existence of solitary wave solutions for the equation with a special local delay convolution kernel and a special nonlocal delay convolution kernel by using the method of dynamical system, especially the geometric singular perturbation theory and invariant manifold theory. According to the relationship between solitary wave and homoclinic orbit, the Camassa–Holm equation is transformed into the ordinary differential equations with fast variables by using the variable substitution. It is proved that the equation with disturbance also possesses homoclinic orbit, and there exists solitary wave solution of the delayed Camassa–Holm equation. [ABSTRACT FROM AUTHOR]

In this study, the generalized Nosé–Hoover oscillator is analyzed rigorously. We show that all trajectories not belonging to the two one-dimensional invariant manifolds (one is a straight line and the other is a unit circle) must traverse a two-dimensional unit disk infinitely many times transversally along the same direction in the state space. Consequently, all of the trajectories not located in the straight line invariant manifold have at least one ω -limiting point. [ABSTRACT FROM AUTHOR]

We suggest an algorithm for seeking recursion operators for nonlinear integrable equations. We find that the recursion operator R can be represented as a ratio of the form R = L1−1 L2, where the linear differential operators L1 and L2 are chosen such that the ordinary differential equation (L2 −λL1)U = 0 is consistent with the linearization of the given nonlinear integrable equation for any value of the parameter λ ∈ C. To construct the operator L1, we use the concept of an invariant manifold, which is a generalization of a symmetry. To seek L2, we then take an auxiliary linear equation related to the linearized equation by a Darboux transformation. It is remarkable that the equation L1U˜ = L2U defines a B¨acklund transformation mapping a solution U of the linearized equation to another solution U˜ of the same equation. We discuss the connection of the invariant manifold with the Lax pairs and the Dubrovin equations. [ABSTRACT FROM AUTHOR]

As a novel approach to control the relative motion of a spacecraft formation, electromagnetic formation flight (EMFF) has some prominent advantages, such as no propellant consumption and no plume contamination. The relative motion actuated by inter-craft electromagnetic force/torque is characterized by equilibrium states which could be utilized to accomplish particular missions, such as close formation, interferometer, etc. However, to maneuver among and maintain these equilibrium configurations, the control capability would be insufficient with only electromagnetic actuation and the inertial thrust is required, so an optimal method which exploits the invariant manifold theory is investigated in this paper. Firstly, the nonlinear translational dynamic model of two-craft electromagnetic formations is derived, and then relative equilibrium configurations and their stabilities are analytically studied. Secondly, the manifolds displaying characteristics of equilibria for the unstable equilibrium configurations are analyzed for the first time. Based on these manifolds, a novel and generalized method is proposed to formulate and parameterize optimal reconfigurations from one electromagnetic equilibrium configuration to another with the electromagnetic control inputs from the initial value to the target value. Specifically, the system's uncontrolled and discontinuous flows along manifolds compose majority of the transfer trajectories, and then optimal controls are applied to a portion of the discontinuous trajectories to differentially correct them to match continuity. By defining the control histories as a finite set of time independent variables, the optimal control problem is converted to a parameter optimization problem and solved by Particle Swarm Optimization. The method can attain the shape-changing ability and consume as little inertial thrust as possible, in exchange for added electromagnetic control effort. Lastly, numerical simulations are presented to verify the feasibility and optimality of the designed method. [ABSTRACT FROM AUTHOR]

Currently the number of reduction methods used in practice in climate applications is vast and tends to be difficult to access for researchers searching for an overview of the area. In this work, we review a range of reduction methods that have been, or may be useful for connecting models of the Earth's climate system of differing complexity. We particularly focus on methods where rigorous reduction is possible. We aim to highlight the main mathematical ideas of each reduction method and also provide several benchmark examples for the reduction from climate modelling. In particular, our goal is to provide a broad overview of available methods. • We present a brief but broad overview of reduction methods applicable to climate dynamics. • Each method is illustrated using a model example and/or a discussion of the ideas behind the method. • Mathematical underpinnings of the methods are explained and compared. [ABSTRACT FROM AUTHOR]

In this paper, we consider the cooperative system xn+1=axn+byn21+yn2,yn+1=cxn21+xn2+dyn,n=0,1,…, where all parameters a,b,c,d are positive numbers and the initial conditions x0,y0 are nonnegative numbers. We describe the global dynamics of this system in a number of cases. An interesting feature of this system is that it exhibits a coexistence of locally stable equilibrium and locally stable periodic solutions as well as the Allee effect. [ABSTRACT FROM AUTHOR]