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Boundary-equilibrium bifurcations in piecewise-smooth slow-fast systems
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10.1063/1.3596708
/content/aip/journal/chaos/21/2/10.1063/1.3596708
http://aip.metastore.ingenta.com/content/aip/journal/chaos/21/2/10.1063/1.3596708
View: Figures

Figures

Image of FIG. 1.
FIG. 1.

Phase portraits of the reduced system (28) and (32) about the origin for ɛ = 0; (a) Case I with the origin being a repeller of the reduced system, (b) Case II with the origin a saddle node of the reduced system, (c) Case III with the origin an attractor of the reduced system, and (d) Case IV with the origin a saddle node of the reduced system.

Image of FIG. 2.
FIG. 2.

Phase portraits of the slow-fast systems (28) and (32) about the origin for ɛ > 0 in the case when there exist (a) an admissible fixed point of f and g (black dot), and a virtual fixed point of f +, g + (small circle), and (b) an admissible fixed point of f + and g + (black dot), and a virtual fixed point of f , g (small circle).

Image of FIG. 3.
FIG. 3.

Phase portraits of the slow-fast systems (28) and (32) about the origin for ɛ > 0 in the case when there exist (a) no admissible fixed points of f , g and f +, g +, and (b) there exist an admissible fixed point of f +, g + and of f , g (an attracting fixed point on the left and a saddle point on the right).

Image of FIG. 4.
FIG. 4.

Phase portraits of the slow-fast systems (28) and (32) about the origin for ɛ > 0 in the case when there exist (a) an admissible fixed points of f , g and a virtual fixed point of f +, g +, and (b) an admissible fixed point of f +, g + and a virtual fixed point of f , g .

Image of FIG. 5.
FIG. 5.

Schematic diagram of the box model. Variables with subscripts e are in the low latitude (equatorial) region and variables with subscripts p are in the higher latitude (polar) region (after Dijkstra15).

Image of FIG. 6.
FIG. 6.

Phase portraits of the slow-fast systems (28) and (32) about the origin for ɛ > 0 in the case when there exist (a) an admissible fixed points of f , g and f +, g + (an attracting fixed point on the right and a saddle point on the left), and (b) only virtual fixed points of f +, g + and of f , g .

Image of FIG. 7.
FIG. 7.

Sketch of the graphs of the right hand side of (44) for (a) A < 1; and (b) A > 1.

Image of FIG. 8.
FIG. 8.

Fixed points of system (42) on the slow manifold x = 1 approximated to for A > 1. Note the existence of three equilibrium states for 1 > μ > (1 + A)/2A which is the region of bi-stability. The dashed line denotes the unstable and the solid line the stable equilibrium points, and the dash-dotted lines denote the points where the saddle-node bifurcations occur.

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/content/aip/journal/chaos/21/2/10.1063/1.3596708
2011-06-24
2014-04-18
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Boundary-equilibrium bifurcations in piecewise-smooth slow-fast systems
http://aip.metastore.ingenta.com/content/aip/journal/chaos/21/2/10.1063/1.3596708
10.1063/1.3596708
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