4,865 research outputs found
The Time Invariance Principle, Ecological (Non)Chaos, and A Fundamental Pitfall of Discrete Modeling
This paper is to show that most discrete models used for population dynamics
in ecology are inherently pathological that their predications cannot be
independently verified by experiments because they violate a fundamental
principle of physics. The result is used to tackle an on-going controversy
regarding ecological chaos. Another implication of the result is that all
continuous dynamical systems must be modeled by differential equations. As a
result it suggests that researches based on discrete modeling must be closely
scrutinized and the teaching of calculus and differential equations must be
emphasized for students of biology
On The Harmonic Oscillator Group
We discuss the maximum kinematical invariance group of the quantum harmonic
oscillator from a view point of the Ermakov-type system. A six parameter family
of the square integrable oscillator wave functions, which seems cannot be
obtained by the standard separation of variables, is presented as an example.
The invariance group of generalized driven harmonic oscillator is shown to be
isomorphic to the corresponding Schroedinger group of the free particle.Comment: 11 pages, no figure
Entropy of nonautonomous dynamical systems
Different notions of entropy play a fundamental role in the classical theory
of dynamical systems. Unlike many other concepts used to analyze autonomous
dynamics, both measure-theoretic and topological entropy can be extended quite
naturally to discrete-time nonautonomous dynamical systems given in the process
formulation. This paper provides an overview of the author's work on this
subject. Also an example is presented that has not appeared before in the
literature
Breathers in inhomogeneous nonlinear lattices: an analysis via centre manifold reduction
We consider an infinite chain of particles linearly coupled to their nearest
neighbours and subject to an anharmonic local potential. The chain is assumed
weakly inhomogeneous. We look for small amplitude discrete breathers. The
problem is reformulated as a nonautonomous recurrence in a space of
time-periodic functions, where the dynamics is considered along the discrete
spatial coordinate. We show that small amplitude oscillations are determined by
finite-dimensional nonautonomous mappings, whose dimension depends on the
solutions frequency. We consider the case of two-dimensional reduced mappings,
which occurs for frequencies close to the edges of the phonon band. For an
homogeneous chain, the reduced map is autonomous and reversible, and
bifurcations of reversible homoclinics or heteroclinic solutions are found for
appropriate parameter values. These orbits correspond respectively to discrete
breathers, or dark breathers superposed on a spatially extended standing wave.
Breather existence is shown in some cases for any value of the coupling
constant, which generalizes an existence result obtained by MacKay and Aubry at
small coupling. For an inhomogeneous chain the study of the nonautonomous
reduced map is in general far more involved. For the principal part of the
reduced recurrence, using the assumption of weak inhomogeneity, we show that
homoclinics to 0 exist when the image of the unstable manifold under a linear
transformation intersects the stable manifold. This provides a geometrical
understanding of tangent bifurcations of discrete breathers. The case of a mass
impurity is studied in detail, and our geometrical analysis is successfully
compared with direct numerical simulations
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