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Dynamics of a piecewise linear map with a gap
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| Zeitschriftentitel: | Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences |
|---|---|
| Personen und Körperschaften: | , , |
| In: | Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 463, 2007, 2077, S. 49-65 |
| Format: | E-Article |
| Sprache: | Englisch |
| veröffentlicht: |
The Royal Society
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| Schlagwörter: |
| author_facet |
Hogan, S.J Higham, L Griffin, T.C.L Hogan, S.J Higham, L Griffin, T.C.L |
|---|---|
| author |
Hogan, S.J Higham, L Griffin, T.C.L |
| spellingShingle |
Hogan, S.J Higham, L Griffin, T.C.L Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences Dynamics of a piecewise linear map with a gap General Physics and Astronomy General Engineering General Mathematics |
| author_sort |
hogan, s.j |
| spelling |
Hogan, S.J Higham, L Griffin, T.C.L 1364-5021 1471-2946 The Royal Society General Physics and Astronomy General Engineering General Mathematics http://dx.doi.org/10.1098/rspa.2006.1735 <jats:p> In this paper, we consider periodic solutions of discontinuous non-smooth maps. We show how the fixed points of a general piecewise linear map with a discontinuity (‘a map with a gap’) behave under parameter variation. We show in detail all the possible behaviours of period 1 and period 2 solutions. For positive gaps, we find that period 2 solutions can exist independently of period 1 solutions. Conversely, for negative gaps, period 1 and period 2 solutions can coexist. Higher periodic orbits can also exist and be stable and we give several examples of how these solutions behave under parameter variation. Finally, we compare our results with those of Jain & Banerjee (Jain & Banerjee 2003 <jats:italic>Int. J. Bifurcat. Chaos</jats:italic> <jats:bold>13</jats:bold> , 3341–3351) and Banerjee <jats:italic>et al</jats:italic> . (Banerjee <jats:italic>et al</jats:italic> . 2004 <jats:italic>IEEE Trans. Circ. Syst. II</jats:italic> <jats:bold>51</jats:bold> , 649–654) and explain their numerical simulations. </jats:p> Dynamics of a piecewise linear map with a gap Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences |
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| title |
Dynamics of a piecewise linear map with a gap |
| title_unstemmed |
Dynamics of a piecewise linear map with a gap |
| title_full |
Dynamics of a piecewise linear map with a gap |
| title_fullStr |
Dynamics of a piecewise linear map with a gap |
| title_full_unstemmed |
Dynamics of a piecewise linear map with a gap |
| title_short |
Dynamics of a piecewise linear map with a gap |
| title_sort |
dynamics of a piecewise linear map with a gap |
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General Physics and Astronomy General Engineering General Mathematics |
| url |
http://dx.doi.org/10.1098/rspa.2006.1735 |
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2007 |
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49-65 |
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<jats:p>
In this paper, we consider periodic solutions of discontinuous non-smooth maps. We show how the fixed points of a general piecewise linear map with a discontinuity (‘a map with a gap’) behave under parameter variation. We show in detail all the possible behaviours of period 1 and period 2 solutions. For positive gaps, we find that period 2 solutions can exist independently of period 1 solutions. Conversely, for negative gaps, period 1 and period 2 solutions can coexist. Higher periodic orbits can also exist and be stable and we give several examples of how these solutions behave under parameter variation. Finally, we compare our results with those of Jain & Banerjee (Jain & Banerjee 2003
<jats:italic>Int. J. Bifurcat. Chaos</jats:italic>
<jats:bold>13</jats:bold>
, 3341–3351) and Banerjee
<jats:italic>et al</jats:italic>
. (Banerjee
<jats:italic>et al</jats:italic>
. 2004
<jats:italic>IEEE Trans. Circ. Syst. II</jats:italic>
<jats:bold>51</jats:bold>
, 649–654) and explain their numerical simulations.
</jats:p> |
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| description | <jats:p> In this paper, we consider periodic solutions of discontinuous non-smooth maps. We show how the fixed points of a general piecewise linear map with a discontinuity (‘a map with a gap’) behave under parameter variation. We show in detail all the possible behaviours of period 1 and period 2 solutions. For positive gaps, we find that period 2 solutions can exist independently of period 1 solutions. Conversely, for negative gaps, period 1 and period 2 solutions can coexist. Higher periodic orbits can also exist and be stable and we give several examples of how these solutions behave under parameter variation. Finally, we compare our results with those of Jain & Banerjee (Jain & Banerjee 2003 <jats:italic>Int. J. Bifurcat. Chaos</jats:italic> <jats:bold>13</jats:bold> , 3341–3351) and Banerjee <jats:italic>et al</jats:italic> . (Banerjee <jats:italic>et al</jats:italic> . 2004 <jats:italic>IEEE Trans. Circ. Syst. II</jats:italic> <jats:bold>51</jats:bold> , 649–654) and explain their numerical simulations. </jats:p> |
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| spelling | Hogan, S.J Higham, L Griffin, T.C.L 1364-5021 1471-2946 The Royal Society General Physics and Astronomy General Engineering General Mathematics http://dx.doi.org/10.1098/rspa.2006.1735 <jats:p> In this paper, we consider periodic solutions of discontinuous non-smooth maps. We show how the fixed points of a general piecewise linear map with a discontinuity (‘a map with a gap’) behave under parameter variation. We show in detail all the possible behaviours of period 1 and period 2 solutions. For positive gaps, we find that period 2 solutions can exist independently of period 1 solutions. Conversely, for negative gaps, period 1 and period 2 solutions can coexist. Higher periodic orbits can also exist and be stable and we give several examples of how these solutions behave under parameter variation. Finally, we compare our results with those of Jain & Banerjee (Jain & Banerjee 2003 <jats:italic>Int. J. Bifurcat. Chaos</jats:italic> <jats:bold>13</jats:bold> , 3341–3351) and Banerjee <jats:italic>et al</jats:italic> . (Banerjee <jats:italic>et al</jats:italic> . 2004 <jats:italic>IEEE Trans. Circ. Syst. II</jats:italic> <jats:bold>51</jats:bold> , 649–654) and explain their numerical simulations. </jats:p> Dynamics of a piecewise linear map with a gap Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences |
| spellingShingle | Hogan, S.J, Higham, L, Griffin, T.C.L, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Dynamics of a piecewise linear map with a gap, General Physics and Astronomy, General Engineering, General Mathematics |
| title | Dynamics of a piecewise linear map with a gap |
| title_full | Dynamics of a piecewise linear map with a gap |
| title_fullStr | Dynamics of a piecewise linear map with a gap |
| title_full_unstemmed | Dynamics of a piecewise linear map with a gap |
| title_short | Dynamics of a piecewise linear map with a gap |
| title_sort | dynamics of a piecewise linear map with a gap |
| title_unstemmed | Dynamics of a piecewise linear map with a gap |
| topic | General Physics and Astronomy, General Engineering, General Mathematics |
| url | http://dx.doi.org/10.1098/rspa.2006.1735 |