A new method of dynamical stability, i.e. fractional generalized Hamiltonian method, and its applications

Fractional generalized Hamiltonian method of dynamical stability is presented.Six criterions for the new method of dynamical stability are given.The new method's applications are studied.Five kinds of actual fractional dynamical models are constructed.Dynamical stability of the five actual fractional dynamical models is explored. In the paper, we present fractional generalized Hamiltonian method of dynamical stability, in terms of Riesz-Riemann-Liouville derivative, and study its applications. For an actual dynamical system, the fractional generalized Hamiltonian method of constructing a fractional dynamical model is given, and then the six criterions for fractional generalized Hamiltonian method of dynamical stability are presented. As applications, by using the fractional generalized Hamiltonian method, we construct five kinds of actual fractional dynamical models, which include a fractional Euler-Poinsot model of rigid body that rotates with respect to a fixed-point, a fractional Hojman-Urrutia model, a fractional Lorentz-Dirac model, a fractional Whittaker model and a fractional Robbins-Lorenz model, and we explore the dynamical stability of these models, respectively. This work provides a general method for studying the dynamical stability of an actual fractional dynamical system that is related to science and engineering.

[1]  Mathieu Moze,et al.  LMI Characterization of Fractional Systems Stability , 2007 .

[2]  Lin Li,et al.  Fractional generalized Hamiltonian equations and its integral invariants , 2013 .

[3]  Dingheng Pi,et al.  On the stability of a second order retarded differential equation , 2015, Appl. Math. Comput..

[4]  Lin Li,et al.  A Lie symmetrical basic integral variable relation and a new conservation law for generalized Hamiltonian systems , 2013 .

[5]  Qi-Ming Zhang,et al.  Lyapunov inequalities and stability for discrete linear Hamiltonian systems , 2011, Appl. Math. Comput..

[6]  Wenan Jiang,et al.  Stability for manifolds of equilibrium states of generalized Hamiltonian systems , 2012 .

[7]  Jun Zhang,et al.  Lie symmetry analysis of the time fractional KdV-type equation , 2014, Appl. Math. Comput..

[8]  Víctor Lanchares,et al.  Lyapunov stability for a generalized Hénon-Heiles system in a rotating reference frame , 2015, Appl. Math. Comput..

[9]  Sergio A. Hojman,et al.  On the inverse problem of the calculus of variations , 1981 .

[10]  Xiaohuan Wang.  Stability of planar waves in a Lotka-Volterra system , 2015, Appl. Math. Comput..

[11]  Frederick E. Riewe,et al.  Mechanics with fractional derivatives , 1997 .

[12]  T. Atanacković,et al.  An extremum variational principle for classical Hamiltonian systems , 1984 .

[13]  Agnieszka B. Malinowska,et al.  Fractional Variational Calculus with Classical and Combined Caputo Derivatives , 2011, 1101.2932.

[14]  Ruggero Maria Santilli,et al.  Foundations of Theoretical Mechanics I: The Inverse Problem in Newtonian Mechanics , 1978 .

[15]  Baiying He,et al.  Hamiltonian forms of the two new integrable systems and two kinds of Darboux transformations , 2014, Appl. Math. Comput..

[16]  F. Mei,et al.  A fractional gradient representation of the Poincaré equations , 2013 .

[17]  Lin Li,et al.  Lie algebraic structure and generalized Poisson conservation law for fractional generalized Hamiltonian systems , 2014 .

[18]  On stability of motion of nonholonomic systems , 1967 .

[19]  Luo Shao-kai.  New Types of the Lie Symmetries and Conserved Quantities for a Relativistic Hamiltonian System , 2003 .

[20]  Lin Li,et al.  Fractional generalized Hamiltonian mechanics , 2013 .

[21]  Mohammad Saleh Tavazoei,et al.  Stability criteria for a class of fractional order systems , 2010 .

[22]  Lin Li,et al.  Stability for manifolds of equilibrium state of generalized Hamiltonian system with additional terms , 2013 .

[23]  Lin Li,et al.  Fractional generalized Hamiltonian mechanics and Poisson conservation law in terms of combined Riesz derivatives , 2013 .

[24]  Joseba Mendiguren,et al.  A generalised fractional derivative model to represent elastoplastic behaviour of metals , 2012 .

[25]  Vasily E. Tarasov,et al.  Fractional Dynamics of Relativistic Particle , 2010 .

[26]  Dumitru Baleanu,et al.  Hamiltonian formulation of systems with linear velocities within Riemann–Liouville fractional derivatives , 2005 .

[27]  V. Arnold.  Mathematical Methods of Classical Mechanics , 1974 .

[28]  Lin Li,et al.  A new Lie symmetrical method of finding a conserved quantity for a dynamical system in phase space , 2012 .

[29]  Hamiltonian Formulation and Action Principle for the Lorentz-Dirac System , 2006 .

[30]  S. Luo,et al.  Fractional Lorentz-Dirac Model and Its Dynamical Behaviors , 2015 .

[31]  D. Gitman,et al.  The action principle for a system of differential equations , 2007, 0710.4532.

[32]  W. Pauli.  On the hamiltonian structure of non-local field theories , 1953 .

[33]  Shao-Kai Luo,et al.  A new type of non-Noether exact invariants and adiabatic invariants of generalized Hamiltonian systems , 2012 .

[34]  S. Luo,et al.  Stability for manifolds of the equilibrium state of fractional Birkhoffian systems , 2015 .

[35]  Om Prakash Agrawal,et al.  Generalized Variational Problems and Euler-Lagrange equations , 2010, Comput. Math. Appl..

[36]  Malgorzata Klimek,et al.  Fractional sequential mechanics — models with symmetric fractional derivative , 2001 .

[37]  Wang Peng,et al.  Hojman Exact Invariants and Adiabatic Invariants of Hamilton System , 2007 .

[38]  Shao-Kai Luo,et al.  Fractional Birkhoffian mechanics , 2015 .

[39]  G. Zaslavsky,et al.  Nonholonomic constraints with fractional derivatives , 2006, math-ph/0603067.

[40]  C. Sparrow.  The Fractal Geometry of Nature , 1984 .

[41]  R. Santilli.  Foundations of Theoretical Mechanics II: Birkhoffian Generalization of Hamiltonian Mechanics , 1983 .

[42]  Edmund Whittaker,et al.  A Treatise on the Analytical Dynamics of Particles and Rigid Bodies: THE REDUCTION OF THE PROBLEM OF THREE BODIES , 1988 .

[43]  J. L. Martin,et al.  Generalized classical dynamics, and the ‘classical analogue’ of a Fermioscillator , 1959, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[44]  Delfim F. M. Torres,et al.  Fractional Optimal Control in the Sense of Caputo and the Fractional Noether's Theorem , 2007, 0712.1844.