14 research outputs found
On dynamical connection between continuous and tropical discretized dynamical systems in one-dimensional (Mathematical structures of integrable systems, their developments and applications)
No full textWe discuss the relationship of stability and the local bifurcations between one-dimensional differential equations and their tropically discretized ones. The discretized time interval is introduced as a bifurcation parameter in the tropical discretized equation, and emergence condition of an additional bifurcation, flip bifurcation, is revealed. By reviewing the continuous dynamical systems with saddle node and pitchfork bifurcations treated in our previous study (S. Ohmori and Y. Yamazaki, J. Math. Phys. 61 122702 (2020)), correspondence of their dynamics to their tropical discrete equations is discussed
A product space of of an abstract polycrystal (Research Trends on Set-theoretic and Geometric Topology and their cooperation with various branches)
No full textOn dynamical connection between continuous and tropical discretized dynamical systems in one-dimensional (Mathematical structures of integrable systems, their developments and applications)
No full textWe discuss the relationship of stability and the local bifurcations between one-dimensional differential equations and their tropically discretized ones. The discretized time interval is introduced as a bifurcation parameter in the tropical discretized equation, and emergence condition of an additional bifurcation, flip bifurcation, is revealed. By reviewing the continuous dynamical systems with saddle node and pitchfork bifurcations treated in our previous study (S. Ohmori and Y. Yamazaki, J. Math. Phys. 61 122702 (2020)), correspondence of their dynamics to their tropical discrete equations is discussed
Construction of topological representation of geometric patterns using Cantor self-similar set
No full textUniversal representation of geometric patterns of disordered matters is
investigated with the aid of general topology. By utilizing the result obtained
in the previous study (S. Ohmori, et.al., Phys. Scr. 94, 105213 (2019)) that
any patterns can be represented by a specific topological space, a construction
of topological representation of patterns using Cantor set is shown. The
obtained topological representations are then demonstrated by the contractions
that characterize the self-similarity of Cantor set. For some practical
geometric patterns, e.g., network, dendritic, and clusterized patterns, their
topological representations are focused on
Topologically Representation of Cantor Cube Model for Geometric Patterns (Research Trends on General Topology and its Related Fields)
No full textThe characteristic geometric structures found in disordered matters are discussed from the viewpoint of general topology. Cantor cube which is a topological space consisting of the infinite product space of O and 1 provides specific decomposition spaces representing topologically geometric patterns of matters such as graphs, clusterized structures, dendrites
Ultradiscrete equations for bifurcations in low-dimensional dynamical systems (Recent Developments in Dynamical Systems and their Applications)
No full textWe discuss ultradiscrete equations in low-dimensional, especially, one-dimensional dynamical systems. The ultradiscrete equations are derived from the normal forms of the local bifurcations i.e., saddle-node, transcretical and pitchfork bifurcations, of one-dimensional continuous dynamical systems. With the aid of the graphical analysis, the dynamical properties of the obtained ultradiscrete equations are revealed. In particular, we show that these ultradiscrete equations exhibit the bifurcations characterized by piecewise linearity, say ultradiscrete bifurcations
Rigged Hilbert Space Approach for Non-Hermite Systems with Positive Definite Metric
No full textWe investigate Dirac's bra-ket formalism based on a rigged Hilbert space for
a non-Hermite quantum system with a positive-definite metric. First, the rigged
Hilbert space, characterized by positive-definite metric, is established. With
the aid of the nuclear spectral theorem for the obtained rigged Hilbert space,
spectral expansions are shown for the bra-kets by the generalized eigenvectors
of a quasi-Hermite operator. The spectral expansions are utilized to endow the
complete bi-orthogonal system and the transformation theory between the Hermite
and non-Hermite systems. As an example of application, we show a specific
description of our rigged Hilbert space treatment for some parity-time
symmetrical quantum systems.Comment: 10 page
Emergence of ultradiscrete states due to phase lock caused by saddle-node bifurcation in discrete limit cycles
No full textDynamical properties of limit cycles in a tropically discretized negative
feedback model are numerically investigated. This model has a controlling
parameter , which corresponds to time interval for the time evolution of
phase in the limit cycles. By considering as a bifurcation parameter, we
find that ultradiscrete state emerges due to phase lock caused by saddle-node
bifurcation. Furthermore, focusing on limit cycles for the max-plus negative
feedback model, it is found that the unstable limit cycle in the max-plus model
corresponds to the unstable fixed points emerging by the saddle-node
bifurcation in the tropically discretized model
Dynamical properties of discrete negative feedback models
No full textDynamical properties of tropically discretized and max-plus negative feedback
models are investigated. Reviewing the previous study [S. Gibo and H. Ito, J.
Theor. Biol. 378, 89 (2015)], the conditions under which the Neimark-Sacker
bifurcation occurs are rederived with a different approach from their previous
one. Furthermore, for limit cycles of the tropically discretized model, it is
found that ultradiscrete state emerges when the time interval in the model
becomes large. For the max-plus model, we find the two limit cycles; one is
stable and the other is unstable. The dynamical properties of these limit
cycles can be characterized by using the Poincar\'e map method. Relationship
between ultradiscrete limit cycle states for the tropically discretized and the
max-plus models is also discussed
Relation of stability and bifurcation properties between continuous and ultradiscrete dynamical systems via discretization with positivity: one dimensional cases
No full textStability and bifurcation properties of one-dimensional discrete dynamical
systems with positivity, which are derived from continuous ones by tropical
discretization, are studied. The discretized time interval is introduced as a
bifurcation parameter in the discrete dynamical systems, and emergence
condition of an additional bifurcation, flip bifurcation, is identified.
Correspondence between the discrete dynamical systems with positivity and the
ultradiscrete ones derived from them is discussed. It is found that the derived
ultradiscrete max-plus dynamical systems can retain the bifurcations of the
original continuous ones via tropical discretization and ultradiscretization
