5. Discussion

G

�exf sð Þ <sup>α</sup>ð Þþ <sup>s</sup> <sup>β</sup>ð Þ<sup>s</sup> ex <sup>þ</sup> m sð Þey <sup>Δ</sup><sup>t</sup> <sup>þ</sup>

positive <sup>δ</sup>�-periodic solution. □

<sup>x</sup><sup>Δ</sup>ðÞ¼ � <sup>t</sup> ð Þ<sup>1</sup> ln <sup>∣</sup>t<sup>∣</sup>

<sup>y</sup><sup>Δ</sup>ðÞ¼� <sup>t</sup> <sup>0</sup>:<sup>3</sup> <sup>þ</sup>

4. Conclusion

ln ð Þ<sup>q</sup> þ 4 � �

> ð Þ �<sup>1</sup> ln <sup>∣</sup>t<sup>∣</sup> ln ð Þ<sup>q</sup> þ 7 � �

one <sup>δ</sup>� <sup>q</sup><sup>2</sup>; <sup>t</sup> � � periodic solution. Here, mes <sup>δ</sup><sup>þ</sup> <sup>q</sup><sup>2</sup>; <sup>t</sup> � � � � <sup>¼</sup> <sup>2</sup>:

The important results of this study are:

Ð <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup> κ

6 4

x y � � � �

198 Advanced Technologies of Quantum Key Distribution

¼

2 6 4

Take DJG as the determinant of the Jacobian of G: Since

�e<sup>x</sup> <sup>Ð</sup> <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup>

Ð <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup>

Ð <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup>

ð<sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup> κ

degðJVC; <sup>Ω</sup> <sup>∩</sup> KerL; <sup>0</sup>Þ ¼ degðG; <sup>Ω</sup> <sup>∩</sup>KerL; <sup>0</sup>Þ ¼ <sup>X</sup>

Example 1 Let T ¼ f g0 ∪q<sup>Z</sup>: δ�ð Þ q; t is the shift operator and t<sup>0</sup> ¼ 1:

� �ð Þ<sup>1</sup> ln <sup>∣</sup>t<sup>∣</sup>

exp x t ð Þþ ð Þ <sup>2</sup> exp y t ð Þ ð Þ ,

ln ð Þ<sup>q</sup> þ 0:5 � �

exp x t ð Þ ð Þ

1. The definition of δ�-periodicity notion is adapted to the quantum calculus.

prey dynamic system whose coefficient functions are δ� periodic.

<sup>κ</sup> a sð Þ� b sð Þexp xð ÞΔt

ex ð Þ<sup>2</sup>

All the functions in Jacobian of G is positive; then, signDJG is always positive. Hence,

<sup>κ</sup> d sð Þ� f sð Þexp xð Þ

<sup>α</sup>ð Þþ <sup>s</sup> <sup>β</sup>ð Þ<sup>s</sup> exp xð Þþ m sð Þexp yð ÞΔ<sup>t</sup>

κ

x y � �

<sup>κ</sup> b sð ÞΔt 0

x y � � <sup>∈</sup> <sup>G</sup>�<sup>1</sup> <sup>0</sup> 0 � � � �

exp x t ð Þ� ð Þ exp y t ð Þ ð Þ

f sð Þβð Þs <sup>α</sup>ð Þþ <sup>s</sup> <sup>β</sup>ð Þ<sup>s</sup> ex <sup>þ</sup> m sð Þey � �<sup>2</sup> <sup>Δ</sup><sup>t</sup> � <sup>Ð</sup> <sup>δ</sup>þð Þ <sup>T</sup>;<sup>κ</sup>
