2,184 research outputs found
Ultimate boundary estimations and topological horseshoe analysis of a new 4D hyper-chaotic system
In this paper, we first estimate the boundedness of a new proposed 4-dimensional (4D) hyper-chaotic system with complex dynamical behaviors. For this system, the ultimate bound set Ω1 and globally exponentially attractive set Ω2 are derived based on the optimization method, Lyapunov stability theory and comparison principle. Numerical simulations are presented to show the effectiveness of the method and the boundary regions. Then, to prove the existence of hyper-chaos, the hyper-chaotic dynamics of the 4D nonlinear system is investigated by means of topological horseshoe theory and numerical computation. Based on the algorithm for finding horseshoes in three-dimensional hyper-chaotic maps, we finally find a horseshoe with two-directional expansions in the 4D hyper-chaotic system, which can rigorously prove the existence of the hyper-chaos in theory
Backwards theory supports modelling via invariant manifolds for non-autonomous dynamical systems
This article establishes the foundation for a new theory of
invariant/integral manifolds for non-autonomous dynamical systems. Current
rigorous support for dimensional reduction modelling of slow-fast systems is
limited by the rare events in stochastic systems that may cause escape, and
limited in many applications by the unbounded nature of PDE operators. To
circumvent such limitations, we initiate developing a backward theory of
invariant/integral manifolds that complements extant forward theory. Here, for
deterministic non-autonomous ODE systems, we construct a conjugacy with a
normal form system to establish the existence, emergence and exact construction
of center manifolds in a finite domain for systems `arbitrarily close' to that
specified. A benefit is that the constructed invariant manifolds are known to
be exact for systems `close' to the one specified, and hence the only error is
in determining how close over the domain of interest for any specific
application. Built on the base developed here, planned future research should
develop a theory for stochastic and/or PDE systems that is useful in a wide
range of modelling applications
Strange Attractors in Dissipative Nambu Mechanics : Classical and Quantum Aspects
We extend the framework of Nambu-Hamiltonian Mechanics to include dissipation
in phase space. We demonstrate that it accommodates the phase space
dynamics of low dimensional dissipative systems such as the much studied Lorenz
and R\"{o}ssler Strange attractors, as well as the more recent constructions of
Chen and Leipnik-Newton. The rotational, volume preserving part of the flow
preserves in time a family of two intersecting surfaces, the so called {\em
Nambu Hamiltonians}. They foliate the entire phase space and are, in turn,
deformed in time by Dissipation which represents their irrotational part of the
flow. It is given by the gradient of a scalar function and is responsible for
the emergence of the Strange Attractors.
Based on our recent work on Quantum Nambu Mechanics, we provide an explicit
quantization of the Lorenz attractor through the introduction of
Non-commutative phase space coordinates as Hermitian matrices in
. They satisfy the commutation relations induced by one of the two
Nambu Hamiltonians, the second one generating a unique time evolution.
Dissipation is incorporated quantum mechanically in a self-consistent way
having the correct classical limit without the introduction of external degrees
of freedom. Due to its volume phase space contraction it violates the quantum
commutation relations. We demonstrate that the Heisenberg-Nambu evolution
equations for the Quantum Lorenz system give rise to an attracting ellipsoid in
the dimensional phase space.Comment: 35 pages, 4 figures, LaTe
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