The evolution equation governing the positions of two interacting nonpropagating solitons is found by using the variational principle approach. Furthermore, the potential which characterizes the interaction between solitons is derived, from which some important results are obtained: (1) Two solitons with the same polarity always attract each other, while those with opposite polarity repel. (2) The repulsion is much weaker than the attraction and both attraction and repulsion are of short distance. (3) The period of mutual oscillation caused by interaction between two solitons with the same polarity increases as the drive amplitude rises; especially, the period can spread from zero to infinity as the initial distance between solitons increases. All the above results are well coincident with experiments.
In this paper we mainly investigate two codimension-two bifurcations of a second-order difference equation from macroeconomics. Applying the center manifold theorem and the normal form analysis, we firstly give the parameter conditions for the generalized flip bifurcation, and prove that the system does not produce a strong resonance. Then, we compute the normal forms to obtain the parameter conditions for the Neimark-Sacker bifurcation, from which we present the conditions for the Chenciner bifurcation. In order to verify the correctness of our results, we also numerically simulate a half stable invariant circle and two invariant circles, one stable and one unstable, arising from the Chenciner bifurcation.
In this paper, we discuss the dynamics of a discrete-time leafeating herbivores model. First of all, to investigate the bifurcations of the model, we study the qualitative properties of a fixed point, including hyperbolic and non-hyperbolic. Secondly, applying the center manifold theorem, we give the conditions that the model produces a supercritical flip bifurcation and a subcritical flip bifurcation respectively, from which we find a generalized flip bifurcation point. And then, we prove rigorously that the model undergoes a generalized flip bifurcation and give three parameter regions that the model possesses two period-two cycles, one period-two cycles and none respectively. Next, computing the normal form, we prove that the model undergoes a subcritical Neimark-Sacker bifurcation and produces a unique unstable invariant circle near the fixed point. Finally, by numerical simulations, we not only verify our results but also show a saddle period-five cycle and a saddle period-six cycle on the invariant circle.
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