Invariant Manifold Approach Research Articles

The invariant manifold approach applied to global long-time dynamics of FitzHugh-Nagumo systems

We consider the FitzHugh-Nagumo system equipped with boundary condition of Dirichlet type on some two/three-dimensional domains. This system describes the signal transmission across the axonal membrane in neurophysiology. It is a semilinear parabolic PDE for the voltage variable coupled with a first-order ODE of space-time type for the recovery variable. We prove that there exists a finite-dimensional global manifold in the case of the fast recovery variable. Since the manifold is uniformly attracting, it gives geometric insight into the global long-time dynamics of the solutions. The proof is based on an abstract invariant manifold theorem for dynamical systems on a Banach space.

Order reduction of nonlinear quasi-periodic systems subjected to external excitations

This paper presents order reduction techniques for nonlinear quasi-periodic systems subjected to external excitations. The order reduction techniques presented here are based on the Lyapunov–Perron (L–P) Transformation. For a class of non-resonant quasi-periodic systems, the L-P Transformation can convert a linear quasi-periodic system into a linear time-invariant one. This Linear Time-Invariant (LTI) system retains the dynamics of the original quasi-periodic system. Once this LTI system is obtained, the tools and techniques available for analysis of LTI systems can be used, and the results could be obtained for the original quasi-periodic system via the L–P Transformation. This approach is similar to the Lyapunov–Floquet (L–F) transformation to convert a linear time-periodic system into an LTI system and perform analysis and control.Order reduction is a systematic way of constructing dynamical system models with relatively smaller states that accurately retain large-scale systems’ essential dynamics. This work presents reduced-order modeling techniques for nonlinear quasi-periodic systems subjected to external excitations. The methods proposed here use the L–P Transformation that makes the linear part of transformed equations time-invariant. In this work, two order reduction techniques are suggested. The first method is simply an application of the well-known Guyan like reduction method to nonlinear systems. The second technique is based on the concept of an invariant manifold for quasi-periodic systems.The ‘quasi-periodic invariant manifold’ based technique yields ‘reducibility conditions’. These conditions (referred to in the perturbation literature as resonance conditions) help us understand the system’s various types of resonant interactions. These resonances indicate energy interactions between the system states, nonlinearity, and external excitation. To retain the essential dynamical characteristics, one must preserve all these ‘resonant’ states in the reduced-order model. Thus, if the ‘reducibility conditions’ are satisfied then only, a nonlinear order reduction based on the quasi-periodic invariant manifold approach is possible. It is found that the invariant manifold approach yields good results. These methodologies are general and can be used for parametric study, sensitivity analysis, and controller design.

Invariant manifold approach for quantifying the dynamics of highly inertial particles in steady and time-periodic incompressible flows.

The dynamics of finite-sized particles with large inertia are investigated in steady and time-dependent flows through the numerical solution of the invariance equation, describing the spatiotemporal evolution of the slow/inertial manifold representing the effective particle velocity field. This approach allows for an accurate reconstruction of the effective particle divergence field, controlling clustering/dispersion features of particles with large inertia for which a perturbative approach is either inaccurate or not even convergent. The effect of inertia on heavy and light particles is quantified in terms of the rate of contraction/expansion of volume elements along a particle trajectory and of the maximum Lyapunov exponent for systems exhibiting chaotic orbits, such as the time-periodic sine-flow on the 2D torus and the time-dependent 2D cavity flow.

Dynamic scaling‐based adaptive control without scaling factor: With application to Euler–Lagrange systems

AbstractA modular dynamic scaling‐based immersion and invariance (I&I) adaptive control framework for a class of nonlinear system with parametric uncertainties is presented in this paper. The framework is based on an invariant manifold approach which allows for predefined target dynamics to be assigned to the closed‐loop systems. The integrability obstacle typically inherent to I&I methodology is overcome by the matrix reconstruction and dynamic scaling technique. The prominent feature is that this methodology can be implemented without scaling factor, and hence the introduction of the scaling factor is just to prove the additive disturbance brought by the matrix reconstruction can be eliminated by constant feedback gains. Moreover, the bounded robustness against three different types of disturbance is verified. As an application, Euler‐Lagrange systems with unknown inertia parameters are applied to illustrate the effectiveness of the proposed method.

Reduced order models for geometrically nonlinear structures: Assessment of implicit condensation in comparison with invariant manifold approach

A comparison between two methods to derive reduced-order models (ROM) for geometrically nonlinear structures is proposed. The implicit condensation and expansion (ICE) method relies on a series of applied static loadings. From this set, a stress manifold is constructed for building the ROM. On the other hand, nonlinear normal modes rely on invariant manifold theory in order to keep the key property of invariance for the reduced subspaces. When the model coefficients are fully known, the ICE method reduces to a static condensation. However, in the framework of finite element discretization, getting all these coefficients is generally too computationally expensive. The stress manifold is shown to tend to the invariant manifold only when a slow/fast decomposition between master and slave coordinates can be assumed. Another key problem in using the ICE method is related to the fitting procedure when a large number of modes need to be taken into account. A simplified procedure, relying on normal form theory and identification of only resonant monomial terms in the nonlinear stiffness, is proposed and contrasted with the current method. All the findings are illustrated on beams and plates examples.

Global detection of detached periodic solution branches of friction-damped mechanical systems

Two novel methods to determine detached periodic solution branches of low-dimensional and large-scale friction-damped mechanical systems have been developed. The approach for low-dimensional systems is an extension of the global terrain method and consists of three steps: First, the non-smooth elastic Coulomb slider is temporarily modified to pose a $$C^2$$ continuous energy surface in the frequency domain. Second, the global terrain method is extended to rescale the search space consecutively along multiple eigendirections of the Hessian at the solution points, facilitating the determination of disconnected solutions. Finally, found solutions are used as the starting points for a homotopy strategy to detect the solutions of the original non-smooth problem. The method for large-scale systems based on an invariant manifold approach consists of three steps: First, the system’s dimension is decreased following a nonlinear modal reduction leading to a two-dimensional surrogate problem. Second, various subspaces for direct extrema, direct bifurcation, constant frequency, and constant amplitude solution detection are defined. Finally, a deterministic line search, a stochastic global solution method, and a newly developed deterministic line deflation are applied to the problem. The proposed methods are applied to low-dimensional and large-scale friction-damped mechanical systems simultaneously subjected to external and self-excitation as well as a low-dimensional mechanical system with shape-memory alloy nonlinearity subjected to external excitation. Their ability to find all solutions is discussed and compared with existing solution procedures such as bifurcation tracking, deflation, homotopy, and the global terrain method. Depending on the chosen subspace for the global search, the proposed methods are capable of providing additional detached solution branches (isola).

Extended complexification method to study nonlinear passive control

The present research work aims to design a passive vibration control based on nonlinear energy pumping. An extended asymptotic approach is introduced based on the invariant manifold approach for the case of 1:1 resonance. It consists in introducing an extended form of Manevitch’s complex variables, taking into consideration higher harmonics, enabling the detection of the invariant manifold of the system at fast timescale. At the slow timescale, equilibrium points and singularities are identified analytically in order to predict periodic regimes and strongly modulated responses. The example of a passive shunt loudspeaker using a nonlinear absorber is studied. Unlike classical investigations, the first and third harmonics are taken into consideration. It is demonstrated that the presence of the third harmonic improves the approximations of the results. Different cases are considered, where the obtained analytical results are in good agreement with those obtained via direct numerical integration of the principal system of equations.

Dynamical system analysis of a Dirac-Born-Infeld model: a center manifold perspective

In this paper we present the cosmological dynamics of a perfect fluid and the Dark Energy (DE) component of the Universe, where our model of the dark energy is the string-theoritic Dirac-Born-Infeld (DBI) model. We assume that the potential of the scalar field and the warp factor of the warped throat region of the compact space in the extra dimension for the DBI model are both exponential in nature. In the background of spatially flat Friedman-Robertson-Walker-Lemaitre Universe, the Einstein field equations for the DBI dark energy reduce to a system of autonomous dynamical system. We then perform a dynamical system analysis for this system. Our analysis is motivated by the invariant manifold approach of the mathematical dynamics. In this method, it is possible to reach a definite conclusion even when the critical points of a dynamical system are non-hyperbolic in nature. Since we find the complete set of critical points for this system, the center manifold analysis ensures that our investigation of this model leaves no stone unturned. We find some interesting results such as that for some critical points there are situations where scaling solutions exist. Finally we present various topologically different phase planes and stability diagrams and discuss the corresponding cosmological scenario.

Shunt loudspeaker using nonlinear energy sink

The present paper aims to apply the concept of energy pumping for noise reduction at propagation and reception paths. This phenomenon consists in irreversible energy transfer from a linear primary system to a nonlinear energy sink, where the energy is finally dissipated. In this study, we turn a loudspeaker to an electroacoustic absorber by connecting at its transducers terminals a passive nonlinear shunt circuit playing the role of an absorber. The equivalent model consists of a linear structure describing the displacement of the loudspeaker, linearly coupled to a cubic nonlinear energy sink. For the case of 1:1 resonance, the Invariant manifold approach is applied for different time scales. It enables the detection of the slow invariant manifold and equilibrium and fold singularities at the fast and slow time scales respectively. This methodology provides a predictive tool allowing the design of the nonlinear energy sink for better control of the main system. The analytical and numerical results show that the nonlinear shunt circuit managed to expand the frequency band of the controlled system.

Dynamical system analysis of a three fluid cosmological model: an invariant manifold approach

The present paper considers a three-fluid cosmological model consisting of noninteracting dark matter, dark energy and baryonic matter in the background of the Friedman–Robertson–Walker–Lemaître flat spacetime. It has been assumed that the dark matter takes the form of dust whereas the dark energy is a quintessence (real) scalar field with exponential potential. It has been further assumed that the baryonic matter is a perfect fluid with barotropic equation of states. The field equations for this model takes the form of an autonomous dynamical system after some suitable changes of variables. Then a complete stability analysis is done considering all possible parameter (the adiabatic index of the baryonic matter and the parameter arising from the dark energy potential) values and for both the cases of hyperbolic and non-hyperbolic critical points. For non-hyperbolic critical points, the invariant manifold theory (center manifold approach) is applied. Finally various topologically different phase planes and vector field diagrams are produced and the cosmological interpretation of this model is presented.

Polynomial expansions of single-mode motions around equilibrium points in the circular restricted three-body problem

In this work, the single-mode motions around the collinear and triangular libration points in the circular restricted three-body problem are studied. To describe these motions, we adopt an invariant manifold approach, which states that a suitable pair of independent variables are taken as modal coordinates and the remaining state variables are expressed as polynomial series of them. Based on the invariant manifold approach, the general procedure on constructing polynomial expansions up to a certain order is outlined. Taking the Earth–Moon system as the example dynamical model, we construct the polynomial expansions up to the tenth order for the single-mode motions around collinear libration points, and up to order eight and six for the planar and vertical-periodic motions around triangular libration point, respectively. The application of the polynomial expansions constructed lies in that they can be used to determine the initial states for the single-mode motions around equilibrium points. To check the validity, the accuracy of initial states determined by the polynomial expansions is evaluated.

Nonlinear robust observer design using an invariant manifold approach

This paper presents a method to design a reduced order observer using an invariant manifold approach. The main advantages of this method are that it enables a systematic design approach, and (unlike most nonlinear observer design methods), it can be generalized over a larger class of nonlinear systems. The method uses specific mapping functions in a way that minimizes the error dynamics close to zero. Another important aspect is the robustness property which is due to the manifold attractivity: an important feature when an observer is used in a closed loop control system. A two degree-of-freedom system is used as an example. The observer design is validated using numerical simulation. Then experimental validation is carried out using hardware-in-the-loop testing. The proposed observer is then compared with a very well known nonlinear observer based on the off-line solution of the Riccati equation for systems with Lipschitz type nonlinearity. In all cases, the performance of the proposed observer is shown to be very high.

A machine learning approach to nonlinear modal analysis

Although linear modal analysis has proved itself to be the method of choice for the analysis of linear dynamic structures, its extension to nonlinear structures has proved to be a problem. A number of competing viewpoints on nonlinear modal analysis have emerged, each of which preserves a subset of the properties of the original linear theory. From the geometrical point of view, one can argue that the invariant manifold approach of Shaw and Pierre is the most natural generalisation. However, the Shaw–Pierre approach is rather demanding technically, depending as it does on the analytical construction of a mapping between spaces, which maps physical coordinates into invariant manifolds spanned by independent subsets of variables. The objective of the current paper is to demonstrate a data-based approach motivated by Shaw–Pierre method which exploits the idea of statistical independence to optimise a parametric form of the mapping. The approach can also be regarded as a generalisation of the Principal Orthogonal Decomposition (POD). A machine learning approach to inversion of the modal transformation is presented, based on the use of Gaussian processes, and this is equivalent to a nonlinear form of modal superposition. However, it is shown that issues can arise if the forward transformation is a polynomial and can thus have a multi-valued inverse. The overall approach is demonstrated using a number of case studies based on both simulated and experimental data.

Nonlinear vibration modes of an offshore articulated tower

Compliant structures, due to their low stiffness and hostile environmental loads, may present large displacements and rotations, leading to significant nonlinear effects. This work applies the theory of nonlinear normal modes to investigate the nonlinear vibrations of a discrete two-degree-of-freedom conceptual model of an offshore compliant articulated tower. The effects of buoyancy, added mass, ocean currents and waves are considered in the analysis. The elastic restoring forces are modeled, based on Augusti's model, using two orthogonal rotational springs. The invariant-manifold approach is then applied to the equations of motion, and the resulting equations are solved through an asymptotic expansion. The derived nonlinear normal modes are then used to reduce the problem to a single degree-of-freedom nonlinear oscillator in each mode. The stability of the solution is investigated through Floquet theory and Mathieu charts. Multiplicity of stable and unstable modes is detected using Poincaré sections. Similar and non-similar modes are also identified. The results of the reduced order model are compared to the numerical solutions of the original equations of motion. The favorable comparisons between both solutions confirm that nonlinear normal modes are a good alternative for the nonlinear analysis of an articulated tower and similar offshore structures.

Multi-objective transfer to libration-point orbits via the mixed low-thrust and invariant-manifold approach

The multi-objective optimization of transfer trajectories from an orbit near Earth to a periodic libration-point orbit in the Sun–Earth system using the mixed low-thrust and invariant-manifold approach is investigated in this paper. A two-objective optimization model is proposed based on the mixed low-thrust and invariant-manifold approach. The circular restricted three-body model (CRTBP) is utilized to represent the motion of a spacecraft in the gravitational field of the Sun and Earth. The transfer trajectory is broken down into several segments; both low-thrust propulsion and stable manifolds are utilized based on the CRTBP in different segments. The fuel cost, which is generated only by the low-thrust trajectory for transferring the spacecraft from an orbit near Earth to a stable manifold, is minimized. The total flight time, which includes the time during which the spacecraft is controlled by the low-thrust trajectory and the time during which the spacecraft is moving on the stable manifold, is also minimized. Using the nondominated sorting genetic algorithm for the resulting multi-objective optimization problem, highly promising Pareto-optimal solutions for the transfer of the spacecraft are found. Via numerical simulations, it is shown that tradeoffs between time of flight and fuel cost can be quickly evaluated using this approach. Furthermore, for the same time of flight, transfer trajectories based on the mixed-transfer method can save a larger amount of fuel than the low-thrust method alone.

A control Liapunov function approach to generalized and regularized descent methods for zero finding1

This paper revisits a class of recently proposed so-called invariant manifold methods for zero finding of ill-posed problems, showing that they can be profitably viewed as homotopy methods, in which the homotopy parameter is interpreted as a learning parameter. Moreover, it is shown that the choice of this learning parameter can be made in a natural manner from a control Liapunov function approach CLF. From this viewpoint, maintaining manifold invariance is equivalent to ensuring that the CLF satisfies a certain ordinary differential equation, involving the learning parameter, that allows an estimate of rate of convergence. In order to illustrate this approach, algorithms recently proposed using the invariant manifold approach, are rederived, via CLFs, in a unified manner. Adaptive regularization parameters for solving linear algebraic ill-posed problems were also proposed. This paper also shows that the discretizations of the ODEs to solve the zero finding problem, as well as the different adaptive choices of the regularization parameter, yield iterative methods for linear systems, which are also derived using the Liapunov optimizing control LOC method.

The Effect of Slow Flow Dynamics on the Oscillations of a Singular Damped System with an Essentially Nonlinear Attachment

We study a three degree of freedom autonomous system with damping, composed of two linear coupled oscillators with an essentially nonlinear lightweight attachment. In particular, we are interested in strongly nonlinear interactions between the linear oscillators and the essentially nonlinear attachment. First, we reduce our system to a non-autonomous second order nonlinear damped oscillator. Then, we introduce a slow-fast partition of the dynamics and average out the main frequency components in order to obtain a reduced system that is studied through the Slow Invariant Manifold (SIM) approach. Depending on the pa- rameters of the system we find different interesting nonlinear dynamical phenomena. With the help of the SIM approach we can study how the parameters of the original problem influence the asymptotic behavior of the orbits of the system. This is accomplished with the application of Tikhonov’s theorem. We classify the different cases of the dynamics according to the values of the parameters and the theoretically predicted asymptotic behavior of the orbits. Interesting phenomena are reported such as orbit capture, relaxation oscillations and complex structure of the basins of attraction.

Invariant slow manifold approach for exact dynamic inversion of singularly perturbed linear mechanical systems with admissible output constraints

We propose an approach for the exact dynamic inversion of singularly perturbed second-order linear systems through asymptotic expansion in a singular parameter. We show that the inversion solution, corresponding to the invariant slow manifold, can be expressed as a converging infinite series under desired output constraints composed of exponential support functions in the complex domain. We provide systematic mathematical procedures to obtain the closed-form invariant slow manifold, along with required admissible boundary conditions. Numerical examples are given to validate the proposed approach.

Reduced order modeling of nonlinear time periodic systems subjected to external periodic excitations

In this work, new methodologies for order reduction of nonlinear systems with periodic coefficients subjected to external periodic excitations are presented. The periodicity of the linear terms is assumed to be non-commensurate with the periodicity of forcing vector. The dynamical equations of motion are transformed using the Lyapunov–Floquet (L–F) transformation such that the linear parts of the resulting equations become time-invariant while the forcing and nonlinearity takes the form of quasiperiodic functions. The techniques proposed here construct a reduced order equivalent system by expressing the non-dominant states as time-varying functions of the dominant (master) states. This reduced order model preserves stability properties and is easier to analyze, simulate and control since it consists of relatively small number of states in comparison with the large scale system. Specifically, two methods are discussed to obtain the reduced order model. First approach is a straightforward application of linear method similar to the ‘Guyan reduction’. The second novel technique proposed here extends the concept of ‘ invariant manifolds’ for the forced problem to construct the fundamental solution. Order reduction approach based on this extended invariant manifold technique yields unique ‘ reducibility conditions’. If these ‘ reducibility conditions’ are satisfied only then an accurate order reduction via extended invariant manifold approach is possible. This approach not only yields accurate reduced order models using the fundamental solution but also explains the consequences of various ‘ primary’ and ‘ secondary resonances’ present in the system. One can also recover ‘ resonance conditions’ associated with the fundamental solution which could be obtained via perturbation techniques by assuming weak parametric excitation. This technique is capable of handling systems with strong parametric excitations subjected to periodic and quasi-periodic forcing. It is anticipated that these order reduction techniques will provide a useful tool in the analysis and control system design of large-scale parametrically excited nonlinear systems subjected to external periodic excitations.

Reduced-Order Modeling of Parametrically Excited Micro-Electro-Mechanical Systems (MEMS)

Reduced-order modeling is a systematic way of constructing models with smaller number of states that can capture the “essential dynamics” of the large-scale systems, accurately. In this paper, reduced-order modeling and control techniques for parametrically excited MEMS are presented. The techniques proposed here use the Lyapunov-Floquet (L-F) transformation that makes the linear part of transformed equations time invariant. In this work, three model reduction techniques for MEMS are suggested. First method is simply an application of the well-known Guyan-like reduction method to nonlinear systems. The second technique is based on singular perturbation, where the transformed system dynamics is partitioned as fast and slow dynamics and the system of differential equations is converted into a differential algebraic (DAE) system. In the third technique, the concept of invariant manifold for time-periodic systems is used. The “time periodic invariant manifold” based technique yields “reducibility conditions”. This is an important result because it helps us to understand the various types of resonances present in the system. These resonances indicate a tight coupling between the system states, and in order to retain the dynamic characteristics, one has to preserve all these “resonant” states in the reduced-order model. Thus, if the “reducibility conditions” are satisfied, only then a nonlinear order reduction based on invariant manifold approach is possible. It is found that the invariant manifold approach yields the most accurate results followed by the nonlinear projection and linear technique. These methodologies are general, free from small parameter assumptions, and can be applied to a variety of MEM systems like resonators, sensors and filters. The reduced-order models can be used for parametric study, sensitivity analysis and/or controller design. The controller design is based on the reduced-order system. Thus, first the reduced-order model of the large-scale system is constructed that captures the essential dynamics. If a controller is designed to stabilize this reduced-order system, then it guarantees that the large-scale system is controlled. The theoretical framework to design linear and nonlinear controllers is also presented.