| Summary: | <p>Bursting dynamics of mappings is investigated in this paper. We first present stability analysis of the mappings' equilibria with various parameters. Then for three mappings <mml:math alttext="$P$"> <mml:mi>P</mml:mi> </mml:math>, <mml:math alttext="$ar{P}$"> <mml:mover accent='true'> <mml:mi>P</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:math>, and <mml:math alttext="$widehat{P}$"> <mml:mover accent='true'> <mml:mi>P</mml:mi> <mml:mo>^</mml:mo> </mml:mover> </mml:math> with different parameters, we study their powers <mml:math alttext="$P^4$"> <mml:msup> <mml:mi>P</mml:mi> <mml:mn>4</mml:mn> </mml:msup> </mml:math>, <mml:math alttext="$ar{P}^6$"> <mml:msup> <mml:mover accent='true'> <mml:mi>P</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> <mml:mn>6</mml:mn> </mml:msup> </mml:math>, and <mml:math alttext="$widehat{P}^4$"> <mml:msup> <mml:mover accent='true'> <mml:mi>P</mml:mi> <mml:mo>^</mml:mo> </mml:mover> <mml:mn>4</mml:mn> </mml:msup> </mml:math>. We show that the mappings thus obtained are chaotic by giving a rigorous verification of existence of horseshoes in these mappings. Precisely, we prove that the mapping <mml:math alttext="$ar{P}^6$"> <mml:msup> <mml:mover accent='true'> <mml:mi>P</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> <mml:mn>6</mml:mn> </mml:msup> </mml:math> is semiconjugate to the 3-shift mapping; the mappings <mml:math alttext="$P^4$"> <mml:msup> <mml:mi>P</mml:mi> <mml:mn>4</mml:mn> </mml:msup> </mml:math> and <mml:math alttext="$widehat{P}^4$"> <mml:msup> <mml:mover accent='true'> <mml:mi>P</mml:mi> <mml:mo>^</mml:mo> </mml:mover> <mml:mn>4</mml:mn> </mml:msup> </mml:math> are semiconjugate to the 4-shift mapping.</p>
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