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افتخار مضر طالب الشرع
5/12/2011 7:12:05 AM
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The bifurcation theory is the mathematical study of how and when the solution to a problem changes from there only being one possible solution, to there being two, which is called a bifurcation. Most commonly used in the mathematical study of dynamical systems, the bifurcation occurs when a small smooth change made to the parameter values (the bifurcation parameters) of a system causes a sudden "qualitative" or topological change in its long term dynamical behavior.
In this work, we will recall one parameter of one dimensional vector field to undergo a saddle node bifurcation, transcritical bifurcation and pitchfork bifurcation. Also, one parameter of two dimensional vector fields to undergo a Hopf bifurcation.
Lyapunov exponents measure the rate at which nearby orbits converge or diverge. There are as many Lyapunov exponents as there are dimensions in the state space of the system, but the largest is usually the most important. The goal of our work is to calculate Lyapunov exponent to types of local bifurcation by Mathlab program .We get the saddle node bifurcation has positive Lyapunov exponent if ,for all the domain.
Also, the transcritical bifurcation has positive Lyapunov exponent if ,for all the domain.But,the pitchfork bifurcation has negative Lyapunov exponent, for all ,for all the domain. The last bifurcation is the Hopf bifurcation has positive Lyapunov exponent at (0,0) if ,but, otherwise the Hopf bifurcation t has negative Lyapunov exponents.
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- The bifurcation divided into two principal classes: local bifurcations and global bifurcations. Local bifurcations,
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