Showing papers on "Nonlinear system published in 1988"

Numerical solution of boundary value problems for ordinary differential equations

Abstract: List of Examples Preface 1. Introduction. Boundary Value Problems for Ordinary Differential Equations Boundary Value Problems in Applications 2. Review of Numerical Analysis and Mathematical Background. Errors in Computation Numerical Linear Algebra Nonlinear Equations Polynomial Interpolation Piecewise Polynomials, or Splines Numerical Quadrature Initial Value Ordinary Differential Equations Differential Operators and Their Discretizations 3. Theory of Ordinary Differential Equations. Existence and Uniqueness Results Green's Functions Stability of Initial Value Problems Conditioning of Boundary Value Problems 4. Initial Value Methods. Introduction. Shooting Superposition and Reduced Superposition Multiple Shooting for Linear Problems Marching Techniques for Multiple Shooting The Riccati Method Nonlinear Problems 5. Finite Difference Methods. Introduction Consistency, Stability, and Convergence Higher-Order One-Step Schemes Collocation Theory Acceleration Techniques Higher-Order ODEs Finite Element Methods 6. Decoupling. Decomposition of Vectors Decoupling of the ODE Decoupling of One-Step Recursions Practical Aspects of Consistency Closure and Its Implications 7. Solving Linear Equations. General Staircase Matrices and Condensation Algorithms for the Separated BC Case Stability for Block Methods Decomposition in the Nonseparated BC Case Solution in More General Cases 8. Solving Nonlinear Equations. Improving the Local Convergence of Newton's Method Reducing the Cost of the Newton Iteration Finding a Good Initial Guess Further Remarks on Discrete Nonlinear BVPS 9. Mesh Selection. Introduction Direct Methods A Mesh Strategy for Collocation Transformation Methods General Considerations 10. Singular Perturbations. Analytical Approaches Numerical Approaches Difference Methods Initial Value Methods 11. Special Topics. Reformulation of Problems in 'Standard' Form Generalized ODEs and Differential Algebraic Equations Eigenvalue Problems BVPs with Singularities Infinite Intervals Path Following, Singular Points and Bifurcation Highly Oscillatory Solutions Functional Differential Equations Method of Lines for PDEs Multipoint Problems On Code Design and Comparison Appendix A. A Multiple Shooting Code Appendix B. A Collocation Code References Bibliography Index.

Navier-Stokes equations

Abstract: Both an original contribution and a lucid introduction to mathematical aspects of fluid mechanics, Navier-Stokes Equations provides a compact and self-contained course on these classical, nonlinear, partial differential equations, which are used to describe and analyze fluid dynamics and the flow of gases.

Exact controllability, stabilization and perturbations for distributed systems

Abstract: Exact controllability is studied for distributed systems, of hyperbolic type or for Petrowsky systems (like plate equations).The control is a boundary control or a local distributed control. Exact controllability consists in trying to drive the system to rest in a given finite time. The solution of the problems depends on the function spaces where the initial data are taken, and also depends on the function space where the control can be chosen.A systematic method (named HUM, for Hilbert Uniqueness Method) is introduced. As the terminology indicates, it is based on Uniqueness results (classical or new) and on Hilbert spaces constructed (in infinitely many ways) by using Uniqueness. A number of applications are indicated.Having a general method for exact controllability implies having a general method for stabilization. This leads to new (and complicated ...) nonlinear Riccati’s type PDEs, to be compared with direct methods (when available).In the last part of the paper, we consider how all this behaves fo...

Nonlinear-optical loop mirror

Abstract: A nonlinear device for ultrafast processing is proposed. This device is based on the nonlinear propagation in a waveguide loop formed by connecting the output ports of a conventional coupler. The device is shown to have potentially useful characteristics for unequal coupling ratios and has the ability to operate on entire pulses when soliton effects are included.

Discrete self-focusing in nonlinear arrays of coupled waveguides.

Abstract: We show that a nonlinear array of coupled waveguides can exhibit discrete self-focusing that in the continuum approximation obeys the so-called nonlinear Schrodinger equation. This process has much in common with the biophysical model of Davydov.

Optimal infinite-horizon feedback laws for a general class of constrained discrete-time systems: stability and moving-horizon approximations

Abstract: Stability results are given for a class of feedback systems arising from the regulation of time-varying discrete-time systems using optimal infinite-horizon and moving-horizon feedback laws. The class is characterized by joint constraints on the state and the control, a general nonlinear cost function and nonlinear equations of motion possessing two special properties. It is shown that weak conditions on the cost function and the constraints are sufficient to guarantee uniform asymptotic stability of both the optimal infinite-horizon and moving-horizon feedback systems. The infinite-horizon cost associated with the moving-horizon feedback law approaches the optimal infinite-horizon cost as the moving horizon is extended.

On the dynamics of finite-strain rods undergoing large motions a geometrically exact approach

Abstract: The dynamics of a fully nonlinear rod model, capable of undergoing finite bending, shearing, and extension, is considered in detail. Unlike traditional nonlinear structural dynamics formulations, due to the effect of finite rotations the deformation map takes values in r 3 × SO(3), which is a differentiable manifold and not a linear space. An implicit time stepping algorithm that furnishes a canonical extension of the classical Newmark algorithm to the rotation group (SO(3)) is developed. In addition to second-order accuracy, the proposed algorithm reduces exactly to the plane formulation. Moreover, the exact linearization of the algorithm and associated configuration update is obtained in closed form, leading to a configuration-dependent nonsymmetric tangent inertia matrix. As a result, quadratic rate of convergence is attained in a Newton-Raphson iterative solution strategy. The generality of the proposed formulation is demonstrated through several numerical examples that include finite vibration, centrifugal stiffening of a fast rotating beam, dynamic instability and snap-through, and large overall motions of a free-free flexible beam. Complete details on implementation are given in three appendices.

Receding horizon control of nonlinear systems

Abstract: The receding horizon control strategy provides a relatively simple method for determining feedback control for linear or nonlinear systems; the method is especially useful for the control of slow nonlinear systems, such as chemical batch processes, where it is possible to solve, sequentially, open-loop, fixed-horizon, optimal control problems online. The authors show that receding horizon control stabilizes a wide class of nonlinear systems, extending the results of their previous paper, which require the finite-horizon value function to be continuously differentiable. >

Stable adaptive observers for nonlinear time-varying systems

Abstract: An adaptive observer/identifier for single input/single output observable nonlinear systems that can be transformed to a certain observable canonical form is described. Sufficient conditions for stability of this observer are provided. These conditions are in terms of the structure of the system and canonical form, the boundedness of the parameter variations, and the sufficient richness of some signals. The scope of the canonical form and the use of the observer/identifier is motivated by the presentation of applications to time-invariant bilinear systems, nonlinear systems in phase-variable form a biotechnological process, and a robot manipulator. In each case, the specific stability conditions are presented. >

Nonlinear gyrokinetic equations for tokamak microturbulence

Abstract: A nonlinear electrostatic gyrokinetic Vlasov equation as well as a Poisson equation have been derived in a form suitable for particle simulation studies of tokamak microturbulence and associated anomalous transport This work differs from the existing nonlinear gyrokinetic theories in toroidal geometry, since the present equations conserve energy while retaining the crucial linear and nonlinear polarization physics In the derivation, the action‐variational Lie perturbation method is utilized in order to preserve the Hamiltonian structure of the original Vlasov–Poisson system Emphasis is placed on the dominant physics of the collective fluctuations in toroidal geometry, rather than on details of particle orbits

Nonlinear Vibration Models for Extremes and Fatigue

Abstract: Hermite moment models of nonlinear random vibration are formulated. These models use response moments (skewness, kurtosis, etc.) to form non‐Gaussian contributions, made orthogonal through a Hermite series. First‐yield and fatigue failure rates are predicted from these moments, which are often simpler to estimate (from either a time. history or analytical model). Both hardening and softening nonlinear models are developed. These are shown to be more flexible than the conventional Charlier and Edgeworth series, with the ability to reflect wider ranges of nonlinear behavior. Analytical moment‐based estimates of spectral densities, crossing rates, probability distributions of the response and its extremes, and fatigue damage rates are formed. These are found to compare well with exact results for various nonlinear models, including nonlinear oscillator responses and quasi‐static responses to Morison wave loads.

Polymers on disordered trees, spin glasses, and traveling waves

Abstract: We show that the problem of a directed polymer on a tree with disorder can be reduced to the study of nonlinear equations of reaction-diffusion type. These equations admit traveling wave solutions that move at all possible speeds above a certain minimal speed. The speed of the wavefront is the free energy of the polymer problem and the minimal speed corresponds to a phase transition to a glassy phase similar to the spin-glass phase. Several properties of the polymer problem can be extracted from the correspondence with the traveling wave: probability distribution of the free energy, overlaps, etc.

Simple high-accuracy resolution program for convective modelling of discontinuities

Abstract: For steady multidimensional convection, the Quadratic Upstream Interpolation for Convective Kinematics (QUICK) scheme has several attractive properties. However, for highly convective simulation of step profiles, QUICK produces unphysical overshoots and a few oscillations, and this may cause serious problems in nonlinear flows. Fortunately, it is possible to modify the convective flux by writing the normalized convected control-volume face value as a function of the normalized adjacent upstream node value, developing criteria for monotonic resolution without sacrificing formal accuracy. This results in a nonlinear functional relationship between the normalized variables, whereas standard methods are all linear in this sense. The resulting Simple High Accuracy Resolution Program (SHARP) can be applied to steady multidimensional flows containing thin shear or mixing layers, shock waves, and other frontal phenomena. This represents a significant advance in modeling highly convective flows of engineering and geophysical importance. SHARP is based on an explicit, conservative, control-volume flux formation, equally applicable to one, two, or three dimensional elliptic, parabolic, hyperbolic, or mixed-flow regimes. Results are given for the bench-mark purely convective first-order results and the nonmonotonic predictions of second- and third-order upwinding.

Existence of Semiclassical Bound States of Nonlinear Schrödinger Equations with Potentials of the Class (V)a

Abstract: (1988). Existence of Semiclassical Bound States of Nonlinear Schrodinger Equations with Potentials of the Class (V)a. Communications in Partial Differential Equations: Vol. 13, No. 12, pp. 1499-1519.

Nonlinear Stochastic Systems Theory and Applications to Physics

Abstract: I: A Summary of the Decomposition Method.- 1: The Decomposition Method.- 1.1 Introduction.- 1.2 Summary of the Decomposition Method.- 1.3 Generation of the An Polynomials.- 1.4 The An for Differential Nonlinear Operators.- 1.5 Convenient Computational Forms for the An Polynomials.- 1.6 Calculation of the An Polynomials for Composite Nonlinearities.- 1.7 New Generating Schemes - the Accelerated Polynomials.- 1.8 Convergence of the An Polynomials.- 1.9 Euler's Transformation.- 1.9.1 Solution of a Differential Equation by Decomposition.- 1.9.2 Application of Euler Transform to Decomposition Solution.- 1.9.3 Numerical Comparison.- 1.9.4 Solution of Linearized Equation.- 1.10 On the Validity of the Decomposition Solution.- 2: Effects of Nonlinearity and Linearization.- 2.1 Introduction.- 2.2 Effects on Simple Systems.- 2.3 Effects on SOlution for the General Case.- 3: Research on Initial and Boundary Conditions for Differential and Partial Differential Equations.- II: Applications to the Equations of Physics.- 4: The Burger's Equation.- 5: Heat Flow and Diffusion.- 5.1 One-Dimensional Case.- 5.2 Two-Dimensional Case.- 5.3 Three-Dimensional Case.- 5.4 Some Examples.- 5.5 Heat Conduction in an Inhomogeneous Rod.- 5.6 Nonlinear Heat Conduction.- 5.7 Heat Conduction Equation with Discontinuous Coefficients.- 5.8 Nonlinear Boundary Conditions.- 5.9 Comparisons.- 5.10 Uncoupled Equations with Coupled Conditions.- 6: Nonlinear Oscillations in Physical Systems.- 6.1 Oscillatory Motion.- 6.2 Pendulum Problem.- 6.3 The Duffing and Van der Pol Oscillators.- 7: The KdV Equation.- 8: The Benjamin-Ono Equation.- 9: The Sine-Gordon Equation.- 10: The Nonlinear Schrodinger Equation and the Generalized Schrodinger Equation.- 10.1 Nonlinear Schrodinger Equation.- 10.2 Generalized Schrodinger Equation.- 10.3 Schrodinger's Equation with a Quartic Potential.- 11: Nonlinear Plasmas.- 12: The Tricomi Problem.- 13: The Initial-Value Problem for the Wave Equation.- ChaDter 14: Nonlinear Dispersive or Dissipative Waves.- 14.1 Wave Propagation in Nonlinear Media.- 14.2 Dissipative Wave Equations.- 15: The Nonlinear Klein-Gordon Equation.- 16: Analysis of Model Equations of Gas Dynamics.- 17: A New Approach to the Efinger Model for a Nonlinear Quantum Theory for Gravitating Particles.- 18: The Navier-Stokes Equations.- Epilogue.

Flight dynamics of aeroelastic vehicles

Abstract: The nonlinear equations of motion for an elastic airplane are developed from first principles. Lagrange's equation and the Principle of Virtual Work are used to generate the equations of motion, and aerodynamic strip theory is then employed to obtain closed-form integral expressions for the generalized forces. The inertial coupling is minimized by appropriate choice of the body-reference axes and by making use of free vibration modes of the body. The mean axes conditions are discussed, a form that is useful for direct application is developed, and the rigid-body degrees of freedom governed by these equations are defined relative to this body-reference axis. In addition, particular attention is paid to the simplifying assumptions used during the development of the equations of motion. Since closed-form, analytic expressions are obtained for the generalized aerodynamic forces, insight can be gained into the effects of parameter variations that is not easily obtained from numerical models. An example is also presented in which the modeling method is applied to a generic elastic aircraft, and the model is used to parametrically address the effects of flexibility. The importance of residualizing elastic modes in forming an equivalent rigid model is illustrated, but as vehicle flexibility is increased, even modal residualization is shown to yield a poor model.

The maximum principle for viscosity solutions of fully nonlinear second order partial differential equations

Abstract: We prove that viscosity solutions in W1,∞ of the second order, fully nonlinear, equation F(D2u, Du, u) = 0 are unique when (i) F is degenerate elliptic and decreasing in u or (ii) F is uniformly elliptic and nonincreasing in u. We do not assume that F is convex. The method of proof involves constructing nonlinear approximation operators which map viscosity subsolutions and supersolutions onto viscosity subsolutions and supersolutions, respectively. This method is completely different from that used in Lions [8, 9] for second order problems with F convex in D 2 u and from that used by Crandall & Lions [3] and Crandall, Evans & Lions [2] for fully nonlinear first order problems.

Third order nonlinear integrated optics

Abstract: The authors review progress in the application of nonlinear optical phenomena to integrated optics. Essentially all integrated optics devices such as directional couplers, Mach-Zehnder interferometers, prism couplers, etc., can be used for all-optical signal processing when at least one of the waveguiding media exhibits an intensity-dependent refractive index. Experimental and theoretical results on the operating characteristics and figures of merit of such devices, as well as a new class of phenomena based on large nonlinear changes in waveguide properties will be summarized. Also discussed are measurement techniques for evaluating waveguide nonlinearities. Progress in the implementation of other nonlinear third order interactions, such as degenerate four-wave mixing and nonlinear spectroscopy, will also be reviewed. >

PIPs and POPs: The reduction of complex dynamical systems using principal interaction and oscillation patterns

Abstract: A general method is described for constructing simple dynamical models to approximate complex dynamical systems with many degrees of freedom. The technique can be applied to interpret sets of observed time series or numerical simulations with high-resolution models, or to relate observation and simulations. The method is based on a projection of the complete system on to a smaller number of “principal interaction patterns” (PIPs). The coefficients of the PIP expansion are assumed to be governed by a dynamic model containing a small number of adjustable parameters. The optimization of the dynamical model, which in the general case can be both nonlinear and time-dependent, is carried out simultaneously with the construction of the optimal set of interaction patterns. In the linear case the PIPs reduce to the eigenoscilations of a first-order linear vector process with stochastic forcing (principal oscillation patterns, or POPs). POPs are linearly related to the “principal prediction patterns” used in linear forecasting applications. The POP analysis can also be applied as a diagnostic tool to compress the extensive information contained in the high-dimensional cross-spectral covariance matrix representing the complete second-moment structure of the system.

Fast, accurate algorithm for numerical simulation of exponentially correlated colored noise.

Abstract: Traditionally, stochastic differential equations used in the physical sciences have involved Gaussian white noise. ' In recent times, however, white noise has been replaced by colored noise in a variety of contexts. Laser noise problems and first passage time problems have been shown to necessitate the use of colored noise instead of white noise. Even the mathematical foundations for the theory of stochastic differential equations call for colored noise if the Stratonovich perspective is adopted, as it is when physical arguments are invoked. ' In each of these contexts, many speci6c problems require numerical simulation as a component of a complete analysis. This is usually a consequence of nonlinearity and the resulting intractability in purely analytic terms. Consequently, numerical-simulation algorithms have been developed, originally for white noise, and recently for colored noise as well. The simplest type of colored noise to generate is exponentially correlated colored noise. Such noise introduces only one more parameter, the correlation time for the exponential correlation, and it is easily generated by a linear damping equation driven by white noise. Our new algorithm is for this kind of colored noise. In Sec. II we review the white-noise algorithm and the differential version of the exponentially correlated, colored-noise algorithm. In Sec. III we present the integral version of the colored-noise algorithm and demonstrate its superior properties.

Simulation of nonlinear viscous fingering in miscible displacement

Abstract: The nonlinear behavior of viscous fingering in miscible displacements is studied. A Fourier spectral method is used as the basic scheme for numerical simulation. In its simplest formulation, the problem can be reduced to two algebraic equations for flow quantities and a first‐order ordinary differential equation in time for the concentration. There are two parameters, the Peclet number (Pe) and mobility ratio (M), that determine the stability characteristics. The result shows that at short times, both the growth rate and the wavelength of fingers are in good agreement with predictions from our previous linear stability theory. However, as the time goes on, the nonlinear behavior of fingers becomes important. There are always a few dominant fingers that spread and shield the growth of other fingers. The spreading and shielding effects are caused by a spanwise secondary instability, and are aided by the transverse dispersion. It is shown that once a finger becomes large enough, the concentration gradient of its front becomes steep as a result of stretching caused by the cross‐flow, in turn causing the tip of the finger to become unstable and split. The splitting phenomenon in miscible displacement is studied by the authors for the first time. A study of the averaged one‐dimensional axial concentration profile is also presented, which indicates that the mixing length grows linearly in time, and that effective one‐dimensional models cannot describe the nonlinear fingering.

Soil-structure-interaction analysis in time domain

Abstract: The procedures to perform nonlinear soil-structure-interaction analysis in the time domain are summarized. The nonlinearity is restricted to the structure and possibly an adjacent irregular soil region. The unbounded soil (far field) must remain linear in this formulation. Besides the direct method where local frequency-independent boundary conditions are enforced on the artificial boundary, various formulations based on the substructure method are addressed, ranging from a discrete model with springs, dashpots and masses to boundary-element methods with convolution integrals involving either the dynamic-stiffness coefficients or the Green's functions in the time domain via the iterative hybrid-frequency-time-domain analysis procedure with the nonlinearities affecting only the right-hand side of the equations of motion.

Viscous flow with large free surface motion

Abstract: An arbitrary Lagrangian-Eulerian (ALE) Petrov-Galerkin finite element technique is developed to study nonlinear viscous fluids under large free surface wave motion. A review of the kinematics and field equations from an arbitrary reference is presented and since the major challenge of the ALE description lies in the mesh rezoning algorithm, various methods, including a new mixed formulation, are developed to update the mesh and map the moving domain in a more rational manner. Moreover, the streamline-upwind/Petrov-Galerkin formulation is implemented to accurately describe highly convective free surface flows. The effectiveness of the algorithm is demonstrated on a tsunami problem, the dam-break problem where the Reynolds number is taken as high as 3000, and a large-amplitude sloshing problem.

Nonlinear physics : from the pendulum to turbulence and chaos

Abstract: Part 1 Particles: the elements of dynamics - phase space, systems with one degree of freedom, an example - the nonlinear pendulum, two more examples of nonlinear oscillations, Poincare's integral invariants, multidimensional integrable systems, mappings, some remarks in conclusion approximate methods - perturbation theory, the averaging method, adiabatic invariants, charged particles in a magnetic field, linear analogues of adiabatic invariance special methods - nonlinear resonance, the Kolmogorov-Arnold-Moser (KAM), structural properties of phase trajectories, simple bifurcations ergodic theory and chaos - ergodicity and mixing, K-systems, examples, recurrences and periodic orbits chaos in detail - a universal mapping for nonlinear oscillations, overlapping of resonances, formation of a stochastic layer, destruction of the integrals of motion, stochastic attractors, examples of stochastic attractors, general notes on the onset of chaos elements of kinetics - the Fokker-Planck-Kolmogorov equation, kinetics in dissipative mappings, stochastic acceleraton and "heating" of particles fractal properties of chaos - fractals, fractals and chaos. Part 2 Waves: nonlinear stationary waves - steepening of waves, stationary waves, examples of stationary waves, collision-free shock waves Hamiltonian description of waves - variational principles, resonance interaction of waves, nonlinear wave resonances, interaction of nonlinear waves chaos in wave fields - weakly nonlinear fields, the fermi-pasta-ulam (FPU) problem, turbulence of a weekly nonlinear field, stochastic instability of a nonlinear wave strong turbulence - Lorenz model, convective cells, features of the onset of turbulence, Langmuir turbulence, soliton turbulence exactly integrable wave equations - integration of the KdV equation, integrable equations. Part 3 Examples: motion of particles in wave fields - regular and stochastic dynamics of particles, motion in a magnetic field and the field of a wave packet, the paradox of the disappearance of Landau damping, stochastic web billiards - mixing billiards, nonlinear-ray dynamics nonlinear optics - nonlinear geometrical optics, nonlinear co-operative phenomena structural properties of one dimensional chains - atom chains, spin chains, excitation in chains of molecules perturbations in Kepler's problem - nonlinear dynamics in a coulomb field, excitation and ionization of a hydrogen atom, diffusion of the eccentricity of orbits in the gravitational field of planets, diffusion of comets from the oort cloud. Part 4 Numerical simulation: nonlinear physics in colour - general notes on the pictures diskettes the ATRS program.

Soliton switching in fiber nonlinear directional couplers.

Abstract: We predict all-optical switching of solitons between the two linear modes of a nonlinear coherent coupler made from a dual-core fiber operating in the anomalous dispersion regime.

Modelling of inductance machines for electric drives

Abstract: The author presents several linear and nonlinear models suitable for transient and steady-state analyses of induction machine drives The models are presented in equivalent circuit form to preserve the identity of nonlinear parameters The circuits, designated as Gamma or inverse Gamma form, are simpler than the conventional T-form circuit Their parameters are readily determined from terminal measurements The major effects of magnetic nonlinearity are included in the models in a way that is more accurate than that usually obtained with the conventional T-form circuits Modeling of time harmonics is also discussed >

Subspace methods for large inverse problems with multiple parameter classes

Abstract: SUMMARY Most nonlinear inverse problems can be cast into the form of determining the minimum of a misfit functional of model parameters. This functional determines the misfit between observations and the corresponding theoretical predictions, subject to some regularization conditions on the form of the model. When there is only one type of parameter in the model, methods based on gradient techniques work well, especially when information on rate of change of gradients is included. In the case of problems depending on multiparameter classes, simple gradient methods mix parameters of different character and physical dimensionality. This may lead to rather poor convergence and strong dependence on the scaling of the different parameter types. These difficulties can be overcome by replacing a gradient step by a local minimization in a subspace spanned by a limited number of vectors in model space. The basis vectors for the subspace should be chosen in the directions determined by the variation of the misfit functional with respect to each of the parameter types, with supplementation if required by additional vectors representing the rate of change of the gradient partitions. The construction of the perturbation requires the inversion of a matrix with the dimensions of the subspace which is easily accomplished. Such a subspace scheme takes into account the different functional dependences on the various parameter types in a balanced way. The update to the current model does not depend on the scaling of the individual parameter classes. The subspace method is flexible and can be adapted to a wide range of choices of misfit criterion and modes of representation of the parameter classes. This style of iterative subspace procedure is well adapted to nonlinear problems with dependence on many parameters and can be successfully applied in a variety of problems, e.g. seismic reflection tomography, the simultaneous nonlinear determination of earthquake locations and velocity fields and in the inversion of full seismic waveforms.

Nonlinear gyrokinetic theory for finite‐beta plasmas

Abstract: A self‐consistent and energy‐conserving set of nonlinear gyrokinetic equations, consisting of the averaged Vlasov and Maxwell’s equations for finite‐beta plasmas, is derived. The method utilized in the present investigation is based on the Hamiltonian formalism and Lie transformation. The resulting formulation is valid for arbitrary values of k⊥ρi and, therefore, is most suitable for studying linear and nonlinear evolution of microinstabilities in tokamak plasmas as well as other areas of plasma physics where the finite Larmor radius effects are important. Because the underlying Hamiltonian structure is preserved in the present formalism, these equations are directly applicable to numerical studies based on the existing gyrokinetic particle simulation techniques.