70,675 research outputs found
Stabilization by Unbounded-Variation Noises
In this paper, we claim the availability of deterministic noises for
stabilization of the origins of dynamical systems, provided that the noises
have unbounded variations. To achieve the result, we first consider the system
representations based on rough path analysis; then, we provide the notion of
asymptotic stability in roughness to analyze the stability for the systems. In
the procedure, we also confirm that the system representations include
stochastic differential equations; we also found that asymptotic stability in
roughness is the same property as uniform almost sure asymptotic stability
provided by Bardi and Cesaroni. After the discussion, we confirm that there is
a case that deterministic noises are capable of making the origin become
asymptotically stable in roughness while stochastic noises do not achieve the
same stabilization results.Comment: 22 pages, 5 figure
Classification and stability of simple homoclinic cycles in R^5
The paper presents a complete study of simple homoclinic cycles in R^5. We
find all symmetry groups Gamma such that a Gamma-equivariant dynamical system
in R^5 can possess a simple homoclinic cycle. We introduce a classification of
simple homoclinic cycles in R^n based on the action of the system symmetry
group. For systems in R^5, we list all classes of simple homoclinic cycles. For
each class, we derive necessary and sufficient conditions for asymptotic
stability and fragmentary asymptotic stability in terms of eigenvalues of
linearisation near the steady state involved in the cycle. For any action of
the groups Gamma which can give rise to a simple homoclinic cycle, we list
classes to which the respective homoclinic cycles belong, thus determining
conditions for asymptotic stability of these cycles.Comment: 34 pp., 4 tables, 30 references. Submitted to Nonlinearit
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