{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,5]],"date-time":"2022-04-05T23:52:58Z","timestamp":1649202778530},"reference-count":14,"publisher":"World Scientific Pub Co Pte Lt","issue":"12","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Bifurcation Chaos"],"published-print":{"date-parts":[[2015,11]]},"abstract":"<jats:p> In previous papers [Isagi et al., 1997; Satake &amp; Iwasa, 2000], a forest model was proposed. The authors demonstrated numerically that the mature forest could possibly exhibit annual reproduction (fixed point synchronization), periodic and chaotic synchronization as the energy depletion constant [Formula: see text] is gradually increased. To understand such rich synchronization phenomena, we are led to study global dynamics of a piecewise smooth map [Formula: see text] containing two parameters [Formula: see text] and [Formula: see text]. Here [Formula: see text] is the energy depletion quantity and [Formula: see text] is the coupling strength. In particular, we obtain the following results. First, we prove that [Formula: see text] has a chaotic dynamic in the sense of Devaney on an invariant set whenever [Formula: see text], which improves a result of [Chang &amp; Chen, 2011]. Second, we prove, via the Schwarzian derivative and a generalized result of [Singer, 1978], that [Formula: see text] exhibits the period adding bifurcation. Specifically, we show that for any [Formula: see text], [Formula: see text] has a unique global attracting fixed point whenever [Formula: see text] ([Formula: see text]) and that for any [Formula: see text], [Formula: see text] has a unique attracting period [Formula: see text] point whenever [Formula: see text] is less than and near any positive integer [Formula: see text]. Furthermore, the corresponding period [Formula: see text] point instantly becomes unstable as [Formula: see text] moves pass the integer [Formula: see text]. Finally, we demonstrate numerically that there are chaotic dynamics whenever [Formula: see text] is in between and away from two consecutive positive integers. We also observe the route to chaos as [Formula: see text] increases from one positive integer to the next through finite period doubling. <\/jats:p>","DOI":"10.1142\/s0218127415501576","type":"journal-article","created":{"date-parts":[[2015,12,3]],"date-time":"2015-12-03T01:30:18Z","timestamp":1449106218000},"page":"1550157","source":"Crossref","is-referenced-by-count":0,"title":["Bifurcation and Chaos in Synchronous Manifold of a Forest Model"],"prefix":"10.1142","volume":"25","author":[{"given":"Chun-Ming","family":"Huang","sequence":"first","affiliation":[{"name":"Department of Applied Mathematics, National Chiao Tung University, Hsinchu, Taiwan, R.O.C."}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jonq","family":"Juang","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics and Center of Mathematical Modeling and Scientific Computing, National Chiao Tung University, Hsinchu, Taiwan, R.O.C."},{"name":"National Center for Theoretical Sciences, Hsinchu, Taiwan, R.O.C."}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2015,12,2]]},"reference":[{"key":"S0218127415501576BIB1","doi-asserted-by":"publisher","DOI":"10.1063\/1.3660662"},{"key":"S0218127415501576BIB2","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-78043-1"},{"key":"S0218127415501576BIB3","volume-title":"A First Course in Chaotic Dynamical Systems: Theory and Experiment","author":"Devaney R. 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