Introduction to Dynamical Systems

Kuznetsov, Yuri A.

doi:10.1007/978-1-4757-2421-9_1

Cited by 37 publications

(93 citation statements)

References 13 publications

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“…The construction of the return map involves solving for the zeros of a function which can be treated in almost precisely the same way as the standard bifurcation theory for fixed points (e.g. [30, chs 4 and 8]). Let P⊂R be the parameter space.…”

Section: Tangencies Leading To Gapsmentioning

confidence: 99%

“…Lemma 3.1 is a re-interpretation of the cusp bifurcation (see [30, §8.2]). The up map T u is xfalse→x+um+τ∗−W112x+W113μW111, where u = u m is the smallest u value satisfying (3.7); hence, the visible fold corresponds to σ = −1 in lemma 3.1.…”

Section: Tangencies Leading To Gapsmentioning

confidence: 99%

“…The size of the discontinuity is the change in τ values from the value on the visible fold to the other value of the cubic (3.7). On the visible fold u 2 = −(1/3) b > 0 to lowest order, and if we choose the negative solution, u=−(−b)/3 then substituting back into (3.7) gives a=−false(2/33false)|bfalse|3/2 [30]. At the fold, there are only two distinct solutions.…”

Section: Tangencies Leading To Gapsmentioning

confidence: 99%

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Creation of discontinuities in circle maps

2021

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Circle maps frequently arise in mathematical models of physical or biological systems. Motivated by Cherry flows and ‘threshold’ systems such as integrate and fire neuronal models, models of cardiac arrhythmias, and models of sleep/wake regulation, we consider how structural transitions in circle maps occur. In particular, we describe how maps evolve near the creation of a discontinuity. We show that the natural way to create discontinuities in the maps associated with both threshold systems and Cherry flows results in a singularity in the derivative of the map as the discontinuity is approached from either one or both sides. For the threshold systems, the associated maps have square root singularities and we analyse the generic properties of such maps with gaps, showing how border collisions and saddle-node bifurcations are interspersed. This highlights how the Arnold tongue picture for tongues bordered by saddle-node bifurcations is amended once gaps are present. We also show that a loss of injectivity naturally results in the creation of multiple gaps giving rise to a novel codimension two bifurcation.

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Section: Tangencies Leading To Gapsmentioning

confidence: 99%

Section: Tangencies Leading To Gapsmentioning

confidence: 99%

Section: Tangencies Leading To Gapsmentioning

confidence: 99%

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Creation of discontinuities in circle maps

2021

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“…First remarkable works for calculation of different cable systems were published in the 1950s [ [1][2][3] ]. Some monographs were published in the 1960s [ [4][5][6][7][8][9][10]. A number of conference reports about suspension structures were published in different proceedings and collections [ [11][12][13][14][15][16][17][18][19][20][21][22].…”

Section: A Review Of International Experience Of Cable Structure Analysismentioning

confidence: 99%

“…In the field of spatial suspension roof structures, several publications deal with the determination of the initial form of the network [ 8,10,23 ]. However, we have found some inaccuracies in the postulates used.…”

Section: A Review Of International Experience Of Cable Structure Analysismentioning

confidence: 99%