TY - JOUR
T1 - A numerical verification method to specify homoclinic orbits as application of local Lyapunov functions
AU - Nitta, Koki
AU - Yamamoto, Nobito
AU - Matsue, Kaname
N1 - Funding Information:
We would like to thank Profs. Kosuke Kuto and Tomoyuki Miyaji for fruitful discussions. Prof. Kuto helped us to prove Theorem and Prof. Miyaji showed us the second example problem in Sect. . KN and NY are partially supported by JST CREST Grant Number JPMJCR14D4, Japan. NY is partially supported by JSPS KAKENHI Grant Number JP18K03410. KM is partially supported by JSPS KAKENHI Grant Number JP17K14235, JP20H01801, JP21H01001 and JP21K03360.
Publisher Copyright:
© 2022, The Author(s).
PY - 2022
Y1 - 2022
N2 - We propose a verification method for specification of homoclinic orbits as application of our previous work for constructing local Lyapunov functions by verified numerics. Our goal is to specify parameters appeared in the given systems of ordinary differential equations (ODEs) which admit homoclinic orbits to equilibria. Here we restrict ourselves to cases that each equilibrium is independent of parameters. The feature of our methods consists of Lyapunov functions, integration of ODEs by verified numerics, and Brouwer’s coincidence theorem on continuous mappings. Several techniques for constructing continuous mappings from a domain of parameter vectors to a region of the phase space are shown. We present numerical examples for problems in 3 and 4-dimensional cases.
AB - We propose a verification method for specification of homoclinic orbits as application of our previous work for constructing local Lyapunov functions by verified numerics. Our goal is to specify parameters appeared in the given systems of ordinary differential equations (ODEs) which admit homoclinic orbits to equilibria. Here we restrict ourselves to cases that each equilibrium is independent of parameters. The feature of our methods consists of Lyapunov functions, integration of ODEs by verified numerics, and Brouwer’s coincidence theorem on continuous mappings. Several techniques for constructing continuous mappings from a domain of parameter vectors to a region of the phase space are shown. We present numerical examples for problems in 3 and 4-dimensional cases.
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U2 - 10.1007/s13160-022-00502-5
DO - 10.1007/s13160-022-00502-5
M3 - Article
AN - SCOPUS:85127262451
JO - Japan Journal of Industrial and Applied Mathematics
JF - Japan Journal of Industrial and Applied Mathematics
SN - 0916-7005
ER -