Long-time behavior of solutions and chaos in reaction-diffusion equations

Kamal N. Soltanov; Anatolij K. Prykarpatski; Denis Blackmore

doi:10.1016/j.chaos.2017.03.057

Long-time behavior of solutions and chaos in reaction-diffusion equations

Kamal N. Soltanov, Anatolij K. Prykarpatski, Denis Blackmore

Mathematical Sciences

Research output: Contribution to journal › Article › peer-review

2 Scopus citations

Abstract

It is shown that members of a class (of current interest with many applications) of non-dissipative reaction-diffusion partial differential equations with local nonlinearity can have an infinite number of different unstable solutions traveling along an axis of the space variable with varying speeds, traveling impulses and also an infinite number of different states of spatio-temporal (diffusion) chaos. These solutions are generated by cascades of bifurcations governed by the corresponding steady states. The behavior of these solutions is analyzed in detail and, as an example, it is explained how space-time chaos can arise. Results of the same type are also obtained in the case of a nonlocal nonlinearity.

Original language	English (US)
Pages (from-to)	91-100
Number of pages	10
Journal	Chaos, Solitons and Fractals
Volume	99
DOIs	https://doi.org/10.1016/j.chaos.2017.03.057
State	Published - Jun 1 2017

All Science Journal Classification (ASJC) codes

Statistical and Nonlinear Physics
General Mathematics
General Physics and Astronomy
Applied Mathematics

Keywords

Behavior of solutions
Chaos
Reaction-diffusion equation
Semilinear PDE

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10.1016/j.chaos.2017.03.057

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title = "Long-time behavior of solutions and chaos in reaction-diffusion equations",

abstract = "It is shown that members of a class (of current interest with many applications) of non-dissipative reaction-diffusion partial differential equations with local nonlinearity can have an infinite number of different unstable solutions traveling along an axis of the space variable with varying speeds, traveling impulses and also an infinite number of different states of spatio-temporal (diffusion) chaos. These solutions are generated by cascades of bifurcations governed by the corresponding steady states. The behavior of these solutions is analyzed in detail and, as an example, it is explained how space-time chaos can arise. Results of the same type are also obtained in the case of a nonlocal nonlinearity.",

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T1 - Long-time behavior of solutions and chaos in reaction-diffusion equations

AU - Soltanov, Kamal N.

AU - Prykarpatski, Anatolij K.

AU - Blackmore, Denis

PY - 2017/6/1

Y1 - 2017/6/1

N2 - It is shown that members of a class (of current interest with many applications) of non-dissipative reaction-diffusion partial differential equations with local nonlinearity can have an infinite number of different unstable solutions traveling along an axis of the space variable with varying speeds, traveling impulses and also an infinite number of different states of spatio-temporal (diffusion) chaos. These solutions are generated by cascades of bifurcations governed by the corresponding steady states. The behavior of these solutions is analyzed in detail and, as an example, it is explained how space-time chaos can arise. Results of the same type are also obtained in the case of a nonlocal nonlinearity.

AB - It is shown that members of a class (of current interest with many applications) of non-dissipative reaction-diffusion partial differential equations with local nonlinearity can have an infinite number of different unstable solutions traveling along an axis of the space variable with varying speeds, traveling impulses and also an infinite number of different states of spatio-temporal (diffusion) chaos. These solutions are generated by cascades of bifurcations governed by the corresponding steady states. The behavior of these solutions is analyzed in detail and, as an example, it is explained how space-time chaos can arise. Results of the same type are also obtained in the case of a nonlocal nonlinearity.

KW - Behavior of solutions

KW - Chaos

KW - Reaction-diffusion equation

KW - Semilinear PDE

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DO - 10.1016/j.chaos.2017.03.057

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AN - SCOPUS:85016795266

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VL - 99

SP - 91

EP - 100

JO - Chaos, Solitons and Fractals

JF - Chaos, Solitons and Fractals

ER -

Long-time behavior of solutions and chaos in reaction-diffusion equations

Abstract

All Science Journal Classification (ASJC) codes

Keywords

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