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For several decades now it has been known that systems with shearless invariant tori, nontwist Hamiltonian systems, possess barriers to chaotic transport. These barriers are resilient to breakage under perturbation and therefore regions where they occur are natural places to look for barriers to transport. We describe a novel kind of effective barrier that persists after the shearless torus is broken. Because phenomena are generic, for convenience we study the Standard Nontwist Map (SNM), an area-preserving map that violates the twist condition locally in the phase space. The novel barrier occurs in nontwist systems when twin even period islands are present, which happens for a broad range of parameter values in the SNM. With a phase space composed of regular and irregular orbits, the movement of chaotic trajectories is hampered by the existence of both shearless curves, total barriers, and a network of partial barriers formed by the stable and unstable manifolds of the hyperbolic points. Being a degenerate system, the SNM has twin islands and, consequently, twin hyperbolic points. We show that the structures formed by the manifolds intrinsically depend on period parity of the twin islands. For this even scenario the novel structure, named a torus free barrier, occurs because the manifolds of different hyperbolic points form an intricate chain atop a dipole configuration and the transport of chaotic trajectories through the chain becomes a rare event. This structure impacts the emergence of transport, the escape basin for chaotic trajectories, the transport mechanism and the chaotic saddle. The case of odd periodic orbits is different: we find for this case the emergence of transport immediately after the breakup of the last invariant curve, and this leads to a scenario of higher transport, with intricate escape basin boundary and a chaotic saddle with non-uniformly distributed points.

We consider a dissipative version of the standard nontwist map. Nontwist systems present a robust transport barrier, called the shearless curve, that becomes the shearless attractor when dissipation is introduced. This attractor can be regular or chaotic depending on the control parameters. Chaotic attractors can undergo sudden and qualitative changes as a parameter is varied. These changes are called crises, and at an interior crisis the attractor suddenly expands. Chaotic saddles are nonattracting chaotic sets that play a fundamental role in the dynamics of nonlinear systems, they are responsible for chaotic transients, fractal basin boundaries, chaotic scattering and they mediate interior crises. In this work we discuss the creation of chaotic saddles in a dissipative nontwist system and the interior crises they generate. We show how the presence of two saddles increase the transient times and analyze the phenomenon of crisis induced intermittency.

Chaotic transport is a subject of paramount importance in a variety of problems in plasma physics, specially those related to anomalous transport and turbulence. On the other hand, a great deal of information on chaotic transport can be obtained from simple dynamical systems like two-dimensional area-preserving (symplectic) maps, where powerful mathematical results like KAM theory are available. In this work we review recent works on transport barriers in area-preserving maps, focusing on systems which do not obey the so-called twist property. For such systems KAM theory no longer holds everywhere and novel dynamical features show up as non-resistive reconnection, shearless curves and shearless bifurcations. After presenting some general features using a standard nontwist mapping, we consider magnetic field line maps for magnetically confined plasmas in tokamaks.

A method, called beatification, is presented for rapidly extracting weakly nonlinear Hamiltonian systems that describe the dynamics near equilibria for systems possessing Hamiltonian form in terms of noncanonical Poisson brackets. The procedure applies to systems like fluids and plasmas in terms of Eulerian variables that have such noncanonical Poisson brackets, i.e., brackets with nonstandard and possibly degenerate form. A collection of examples of both finite and infinite dimensions is presented.

A method for extracting finite-dimensional Hamiltonian systems from a class of 2+1 Hamiltonian mean field theories is presented. These theories possess noncanonical Poisson brackets, which normally resist Hamiltonian truncation, but a process of beatification by coordinate transformation near a reference state is described in order to perturbatively overcome this difficulty. Two examples of four-wave truncation of Euler's equation for scalar vortex dynamics are given and compared: one a direct non-Hamiltonian truncation of the equations of motion, the other obtained by beatifying the Poisson bracket and then truncating., Comment: 27 pages, 1 figure; references added

Moment closures of the Vlasov-Amp{\`e}re system, whereby higher moments are represented as functions of lower moments with the constraint that the resulting fluid system remains Hamiltonian, are investigated by using water-bag theory. The link between the water-bag formalism and fluid models that involve density, fluid velocity, pressure and higher moments is established by introducing suitable thermodynamic variables. The cases of one, two and three water-bags are treated and their Hamiltonian structures are provided. In each case, we give the associated fluid closures and we discuss their Casimir invariants. We show how the method can be extended to an arbitrary number of fields, i.e., an arbitrary number of water-bags and associated moments. The thermodynamic interpretation of the resulting models is discussed. Finally, a general procedure to derive Hamiltonian N-field fluid models is proposed.

Fluid reductions of the Vlasov-Amp{\`e}re equations that preserve the Hamiltonian structure of the parent kinetic model are investigated. Hamiltonian closures using the first four moments of the Vlasov distribution are obtained, and all closures provided by a dimensional analysis procedure for satisfying the Jacobi identity are identified. Two Hamiltonian models emerge, for which the explicit closures are given, along with their Poisson brackets and Casimir invariants.

Different variants of hybrid kinetic-fluid models are considered for describing the interaction of a bulk fluid plasma obeying MHD and an energetic component obeying a kinetic theory. Upon using the Vlasov kinetic theory for energetic particles, two planar Vlasov-MHD models are compared in terms of their stability properties. This is made possible by the Hamiltonian structures underlying the considered hybrid systems, whose infinite number of invariants makes the energy-Casimir method effective for determining stability. Equilibrium equations for the models are obtained from a variational principle and in particular a generalized hybrid Grad-Shafranov equation follows for one of the considered models. The stability conditions are then derived and discussed with particular emphasis on kinetic particle effects on classical MHD stability., Comment: 15 pages; no figures

We derive the Hamiltonian structure of the modified Hasegawa-Mima equation from the ion fluid equations applying Dirac's theory of constraints. We discuss the Casimirs obtained from the corresponding Poisson structure.

Hamiltonian bifurcations in the context of noncanonical Hamiltonian matter models are described. First, a large class of 1 + 1 Hamiltonian multi-fluid models is considered. These models have linear dynamics with discrete spectra, when linearized about homogeneous equilibria, and these spectra have counterparts to the steady state and Hamiltonian Hopf bifurcations when equilibrium parameters are varied. Examples of fluid sound waves and plasma and gravitational streaming are treated in detail. Next, using these 1 + 1 examples as a guide, a large class of 2 + 1 Hamiltonian systems is introduced, and Hamiltonian bifurcations with continuous spectra are examined. It is shown how to attach a signature to such continuous spectra, which facilitates the description of the continuous Hamiltonian Hopf bifurcation. This chapter lays the groundwork for Kre\u{i}n-like theorems associated with the CHH bifurcation that are more rigorously discussed in our companion chapter \cite{chaptII}.

Techniques for coordinate changes that depend on both dependent and independent variables are developed and applied to the Maxwell-Vlasov Hamiltonian theory. Particle coordinate changes with a new velocity variable dependent on the magnetic field, with spatial coordinates unchanged, are lifted to transform the noncanonical Poisson bracket and, thus, the field Hamiltonian structure of the Vlasov-Maxwell equation. Several examples are given including magnetic coordinates, where the velocity is decomposed into components parallel and perpendicular to the local magnetic field, and the case of spherical velocity coordinates. An example of the lifting procedure is performed to obtain a simplified version of gyrokinetics, where the magnetic moment is used as a coordinate and the dynamics is reduced by elimination of the electric field energy in the Hamiltonian.

In fluids and plasmas with zonal flow reversed shear, a peculiar kind of transport barrier appears in the shearless region, one that is associated with a proper route of transition to chaos. Using a symplectic nontwist maps, which model such zonal flows, we investigate these barriers. In particular, the standard nontwist map, a paradigm of nontwist systems, is used to analyze the parameter dependence of the transport through a broken shearless barrier. Varying a proper control parameter, we identify the onset of structures with high stickiness that give rise to an effective barrier near the broken shearless curve. Several diagnostics are used: stickiness is measured by escape time and transmissivity plots, while barriers that separate different regions of stickiness are identified by employing two indicators in the phase space, the finite-time rotation number and the finite-time Lyapunov exponent., Comment: 20 pages, 9 figures

The Hamiltonian structures of the incompressible ideal fluid, including entropy advection, and magnetohydrodynamics are investigated by making use of Dirac's theory of constrained Hamiltonian systems. A Dirac bracket for these systems is constructed by assuming a primary constraint of constant density. The resulting bracket is seen to naturally project onto solenoidal velocity fields.

Lagrangian coherent structures are effective barriers, sticky regions, that separate phase space regions of different dynamical behavior. The usual way to detect such structures is via finite-time Lyapunov exponents. We show that similar results can be obtained for single-frequency systems from finite-time rotation numbers, which are much faster to compute. We illustrate our claim by considering examples of continuous and discrete-time dynamical systems of physical interest., Comment: 4 pages, 3 figures

We present a Hamiltonian derivation of a class of reduced plasma two-dimensional fluid models, an example being the Charney-Hasegawa-Mima equation. These models are obtained from the same parent Hamiltonian model, which consists of the ion momentum equation coupled to the continuity equation, by imposing dynamical constraints. It is shown that the Poisson bracket associated with these reduced models is the Dirac bracket obtained from the Poisson bracket of the parent model.

The Charney-Hasegawa-Mima equation is an infinite-dimensional Hamiltonian system with dynamics generated by a noncanonical Poisson bracket. Here a first principle Hamiltonian derivation of this system, beginning with the ion fluid dynamics and its known Hamiltonian form, is given.

The breakup of the shearless invariant torus with winding number $\omega=\sqrt{2}-1$ is studied numerically using Greene's residue criterion in the standard nontwist map. The residue behavior and parameter scaling at the breakup suggests the existence of a new fixed point of the renormalization group operator (RGO) for area-preserving maps. The unstable eigenvalues of the RGO at this fixed point and the critical scaling exponents of the torus at breakup are computed., Comment: 4 pages, 5 figures

The breakup of shearless invariant tori with winding number $\omega=[0,1,11,1,1,...]$ (in continued fraction representation) of the standard nontwist map is studied numerically using Greene's residue criterion. Tori of this winding number can assume the shape of meanders (folded-over invariant tori which are not graphs over the x-axis in $(x,y)$ phase space), whose breakup is the first point of focus here. Secondly, multiple shearless orbits of this winding number can exist, leading to a new type of breakup scenario. Results are discussed within the framework of the renormalization group for area-preserving maps. Regularity of the critical tori is also investigated., Comment: submitted to Chaos

New global periodic orbit collision/separatrix reconnection scenarios in the standard nontwist map in different regions of parameter space are described in detail, including exact methods for determining reconnection thresholds that are implemented numerically. The results are compared to a break-up diagram of shearless invariant curves. The existence of meanders (invariant tori that are not graphs) is demonstrated numerically for both odd and even period reconnection for certain regions in parameter space, and some of the implications on transport are discussed., Comment: 14 papes, 15 figures

Extending the work of del-Castillo-Negrete, Greene, and Morrison, Physica D {\bf 91}, 1 (1996) and {\bf 100}, 311 (1997) on the standard nontwist map, the breakup of an invariant torus with winding number equal to the inverse golden mean squared is studied. Improved numerical techniques provide the greater accuracy that is needed for this case. The new results are interpreted within the renormalization group framework by constructing a renormalization operator on the space of commuting map pairs, and by studying the fixed points of the so constructed operator., Comment: To be Submitted to Chaos

We describe a new type of top, the twisted top, obtained by appending a cocycle to the Lie-Poisson bracket for the charged heavy top, thus breaking its semidirect product structure. The twisted top has an integrable case that corresponds to the Lagrange (symmetric) top. We give a canonical description of the twisted top in terms of Euler angles. We also show by a numerical calculation of the largest Lyapunov exponent that the Kovalevskaya case of the twisted top is chaotic., Comment: 11 pages, 1 figure. Elsevier style. Published version with a few small corrections

A method is described for predicting statistical properties of turbulence. Collections of Fourier amplitudes are represented by nonuniformly spaced modes with enhanced coupling coefficients. The statistics of the full dynamics can be recovered from the time-averaged predictions of the reduced model. A Liouville theorem leads to inviscid equipartition solutions. Excellent agreement is obtained with two-dimensional forced-dissipative pseudospectral simulations. For the two-dimensional enstrophy cascade, logarithmic corrections to the high-order structure functions are observed., Comment: 12 pages, 4 figures, to appear in Physical Review Letters

Reduction is a process that uses symmetry to lower the order of a Hamiltonian system. The new variables in the reduced picture are often not canonical: there are no clear variables representing positions and momenta, and the Poisson bracket obtained is not of the canonical type. Specifically, we give two examples that give rise to brackets of the noncanonical Lie-Poisson form: the rigid body and the two-dimensional ideal fluid. From these simple cases, we then use the semidirect product extension of algebras to describe more complex physical systems. The Casimir invariants in these systems are examined, and some are shown to be linked to the recovery of information about the configuration of the system. We discuss a case in which the extension is not a semidirect product, namely compressible reduced MHD, and find for this case that the Casimir invariants lend partial information about the configuration of the system., Comment: 11 pages, RevTeX. To appear in Proceedings of the 13th Florida Workshop in Astronomy and Physics

Traditional numerical discretizations of conservative systems generically yield an artificial secular drift of any nonlinear invariants. In this work we present an explicit nontraditional algorithm that exactly conserves these invariants. We illustrate the general method by applying it to the three-wave truncation of the Euler equations, the Lotka--Volterra predator--prey model, and the Kepler problem. This method is discussed in the context of symplectic (phase space conserving) integration methods as well as nonsymplectic conservative methods. We comment on the application of our method to general conservative systems., Comment: 30 pages, postscript (1.3MB). Submitted to SIAM J. Sci. Comput.