Chattering extremals in Hamiltonian systems with control in a square

The dynamical system is a notion for any fixed map, which describes the time dependence of a position in its space of states. At any given time a dynamical system has a state given by a vector x, which can be represented by a point in an appropriate state space. Infinitesimal changes in the state of the system correspond to small changes in the vectors. The evolution map of the dynamical system is a fixed rule that describes what future states follow from the current state. The notions of gradient and Hamiltonian systems arise in dynamical systems theory (Hirsh and Smale, 1974; Dubrovin et al., 1992; Vilasi, 2001). In this chapter, generalizations of gradient and Hamiltonian systems are suggested. We use differential forms and exterior derivatives of fractional orders. Its allow us to define Hamiltonian and gradient dynamical systems (Gilmor, 1981; Dubrovin et al., 1992; Vilasi, 2001; Godbillon, 1969) of non-integer (fractional) orders. In the general case, the fractional Hamiltonian (or gradient) systems cannot be considered as Hamiltonian (gradient) systems. The suggested class of fractional gradient and Hamiltonian systems is wider (Tarasov, 2005a,b) than the usual class of gradient and Hamiltonian dynamical systems. The systems of gradient and Hamiltonian type can be considered as a special case of fractional gradient and Hamiltonian systems.

We establish the Weyl‐Titchmarsh theory for singular linear Hamiltonian dynamic systems on a time scale 𝕋, which allows one to treat both continuous and discrete linear Hamiltonian systems as special cases for 𝕋 = ℝ and 𝕋 = ℤ within one theory and to explain the discrepancies between these two theories. This paper extends the Weyl‐Titchmarsh theory and provides a foundation for studying spectral theory of Hamiltonian dynamic systems. These investigations are part of a larger program which includes the following: (i) M(λ) theory for singular Hamiltonian systems, (ii) on the spectrum of Hamiltonian systems, (iii) on boundary value problems for Hamiltonian dynamic systems.

Roughly speaking, the Poincaré disc D2 is the closed disc centered at the origin of the coordinates of R2, where the whole of R2 is identified with the interior of D2 and the circle of the boundary of D2 is identified with the infinity of R2, because in the plane R2, we can go to infinity in as many directions as points have the circle. The phase portraits of the quadratic Hamiltonian systems in the Poincaré disc were classified in 1994. Since then, no new interesting classes of Hamiltonian systems have been classified on the Poincaré disc. In this paper, we determine the phase portraits in the Poincaré disc of five classes of homogeneous Hamiltonian polynomial differential systems of degrees 1, 2, 3, 4, and 5 with finitely many equilibria. Moreover, all these phase portraits are symmetric with respect to the origin of coordinates. We showed that these polynomial differential systems exhibit precisely 2, 2, 3, 3, and 4 topologically distinct phase portraits in the Poincaré disc. Of course, the new results are for the homogeneous Hamiltonian polynomial differential systems of degrees 3, 4, and 5. The tools used here for obtaining these phase portraits also work for obtaining any phase portrait of a homogeneous Hamiltonian polynomial differential system of arbitrary degree.

Two sets of vectors, covariant Lyapunov vectors (CLVs) and orthogonal Lyapunov vectors (OLVs), are currently used to characterize the linear stability of chaotic systems. A comparison is made to show their similarity and difference, especially with respect to the influence on hydrodynamic Lyapunov modes (HLMs). Our numerical simulations show that in both Hamiltonian and dissipative systems HLMs formerly detected via OLVs survive if CLVs are used instead. Moreover, the previous classification of two universality classes works for CLVs as well, i.e., the dispersion relation is linear for Hamiltonian systems and quadratic for dissipative systems, respectively. The significance of HLMs changes in different ways for Hamiltonian and dissipative systems with the replacement of OLVs with CLVs. For general dissipative systems with nonhyperbolic dynamics the long-wavelength structure in Lyapunov vectors corresponding to near-zero Lyapunov exponents is strongly reduced if CLVs are used instead, whereas for highly hyperbolic dissipative systems the significance of HLMs is nearly identical for CLVs and OLVs. In contrast the HLM significance of Hamiltonian systems is always comparable for CLVs and OLVs irrespective of hyperbolicity. We also find that in Hamiltonian systems different symmetry relations between conjugate pairs are observed for CLVs and OLVs. Especially, CLVs in a conjugate pair are statistically indistinguishable in consequence of the microreversibility of Hamiltonian systems. Transformation properties of Lyapunov exponents, CLVs, and hyperbolicity under changes of coordinate are discussed in appendices.

This thesis is concerned with extending the well-established theory of pseudoholomorphic curve methods for finite-dimensional symplectic manifolds, to the setting of infinite-dimensional symplectic spaces. Our new results are then used to prove the existence of periodic orbits of infinite-dimensional Hamiltonian systems. The systems we consider are non-linear Hamiltonian systems, and a big portion of the material is devoted to the precise description of the type of nonlinearities we allow for, and we show that our results do not hold for other nonlinearities. Examples of systems we consider are nonlinear Hamiltonian differential equations, as well as Hamiltonian particle-field systems. A main problem that arises in generalizing the finite-dimensional results to infinite dimensions, is that of small divisors. This core problem is prevalent in all of our results and it is concisely described how we overcome it. In the first paper we establish a type of compactness result, where we show the existence of Floer curves in an infinite-dimensional symplectic Hilbert space for admissible Hamiltonians. We then show that these curves give rise to periodic orbits of the Hamiltonian system. We extend this result by coupling the linear theory to a finite-dimensional closed symplectic manifold and prove a cuplength estimate. In the second paper we extend the latter result for Hamiltonian particle-field systems where the particle is restricted to a closed submanifold of the $n$-torus and the phase space is the cotangent bundle of this submanifold. The third paper establishes the Fredholm theory, as a step in our program of defining an infinite-dimensional Floer theory. We describe how the small divisor problem permeates the theory and how we overcome it by introducing a modified norm and Sobolev completion related to the small divisors.

This paper considers Hamiltonian structures related to quantum mechanics. We do not discuss structures used in the quantization of classical Hamil� tonian or Lagrangian systems (which was first done by Heisenberg and Feynman); on the contrary, we are mainly interested in structures which make it possible to consider quantum systems as classical (although infinitedimensional in the most natural cases) Hamiltonian (or Lagrangian) systems. The introduc� tion of such structures can be called the dequantiza� tion of a quantum system. We emphasize that, whereas the quantization of a classical (Hamiltonian or Lagrangian) system sup� poses the introduction of fundamentally new mathe� matical structures (usually related to the passage from ordinary differential equations to partial differential equations), dequantization means only the passage to an equivalent description of the same quantum sys� tem. To obtain the result of two successive operations (usual quantization and subsequent dequantization), it suffices to know only properties of the initial classi� cal system; therefore, it can be said that the passage from the initial classical Hamiltonian (or Lagrangian) system to the Hamiltonian system being the result of the corresponding dequantization is yet another quan� tization method of the initial system, different from quantizations in the sense of Schrodinger, Heisenberg, and Feynman. We refer to this quantization as Hamil� tonian quantization (precise definitions are given below).

The world, its many subsystems and all their theories, starting with logic, can be reduced to two related functions: a combinatorial system generator and a hamiltonian system organizer. These can be derived, in turn, from an Axiom of Lawfulness, the expansion being guided by pseudo‐category and pseudo‐functor analysis to produce an axiomatic theory of the world or general theory of evolution. Specifically, world evolution is generated by a constrained combinatorial world generator, F:G(X), deduced from two related axioms: I. The Axiom of World Lawfulness and II. The Axiom of World Constraint Constants, c = c1, c2, of primordial physical combinatee (substance), c1, and physical combinator (motion), c2. Axiom I postulates a lawful analysis by an analyzer adhering to appropriate coordinate systems, CS, of a lawful analysand obeying a conservation law, X = X. The analysand consists of a base combinatee (the set and elements), X = {x1, x2,… xn}, and a base combinator, namely, the universal Boolean operator, NO...

LetHbe a Hamiltonian,e∈H(M) ⊂ ℝ andƐH, ea connected component ofH−1({e}) without singularities. A Hamiltonian system, say a triple (H,e,ƐH, e), is Anosov ifƐH, eis uniformly hyperbolic. The Hamiltonian system (H,e,ƐH, e) is aHamiltonian star systemif all the closed orbits ofƐH, eare hyperbolic and the same holds for a connected component of−1({ẽ}), close toƐH, e, for any Hamiltonian, in someC2-neighbourhood ofH, and ẽ in some neighbourhood ofe.In this paper we show that a Hamiltonian star system, defined on a four-dimensional symplectic manifold, is Anosov. We also prove the stability conjecture for Hamiltonian systems on a four-dimensional symplectic manifold. Moreover, we prove the openness and the structural stability of Anosov Hamiltonian systems defined on a 2d-dimensional manifold,d≥ 2.

Given a control system and a desired property, an abstracted system is a reduced system that preserves the property of interest while ignoring modeling detail. We consider the abstraction problem for Hamiltonian control systems, that is, we preserve the Hamiltonian structure during the abstraction process. We show how the mechanical structure of Hamiltonian control systems can be exploited to simplify the abstraction computations and we provide conditions under which the local accessibility properties of the abstracted Hamiltonian system are equivalent to the local accessibility properties of the original Hamiltonian control system.

A Sturmian theorem for recessive solutions of linear Hamiltonian difference systems

The Kuramoto model constitutes a paradigmatic model for the dissipative collective dynamics of coupled oscillators, characterizing in particular the emergence of synchrony (phase locking). Here we present a classical Hamiltonian (and thus conservative) system with 2N state variables that in its action-angle representation exactly yields Kuramoto dynamics on N-dimensional invariant manifolds. We show that locking of the phase of one oscillator on a Kuramoto manifold to the average phase emerges where the transverse Hamiltonian action dynamics of that specific oscillator becomes unstable. Moreover, the inverse participation ratio of the Hamiltonian dynamics perturbed off the manifold indicates the global synchronization transition point for finite N more precisely than the standard Kuramoto order parameter. The uncovered Kuramoto dynamics in Hamiltonian systems thus distinctly links dissipative to conservative dynamics.

We consider a fractional generalization of Hamiltonian and gradient systems. We use differential forms and exterior derivatives of fractional orders. We derive fractional generalization of Helmholtz conditions for phase space. Examples of fractional gradient and Hamiltonian systems are considered. The stationary states for these systems are derived.

The theory of Hamiltonian (or conservative) systems in the plane is introduced. The differential equations are used to model dynamical systems in which there is no energy loss. Hamiltonian systems are also used extensively when bifurcating limit cycles in the plane (see Chapters 10 and 11).

Abstractions of Hamiltonian control systems

A new class of Hamiltonian dynamical systems with two degrees of freedom is studied, for which the Hamiltonian function is a linear form with respect to moduli of both momenta. For different potentials such systems can be either completely integrable or behave just as normal nonintegrable Hamiltonian systems with two degrees of freedom: one observes many of the phenomena characteristic of the latter ones, such as a breakdown of invariant tori as soon as the integrability is violated; a formation of stochastic layers around destroyed separatrices; bifurcations of periodic orbits, etc. At the same time, the equations of motion are simply integrated on subsequent adjacent time intervals, as in billiard systems; i.e., all the trajectories can be calculated explicitly: Given an initial data, the state of the system is uniquely determined for any moment. This feature of systems in interest makes them very attractive models for a study of nonlinear phenomena in finite-dimensional Hamiltonian systems. A simple representative model of this class (a model with quadratic potential), whose dynamics is typical, is studied in detail. (c) 1997 American Institute of Physics.