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limit cycle
Bifurcation of limit cycle for quadratic system with invariant parabola
      
In a previous paper, we have proved that a planar quadratic system with invariant parabola Γ has at most one limit cycle.
      
It is proved that the quadratic system with a weak focus and a strong focus has a unique limit cycle around one of the two foci, if there exists simultaneously limit cycles around each of the two foci for the system.
      
Limit cycle problem of quadratic system of type (III)m=0, (III)
      
To continue the discussion in (I) and (II), and finish the study of the limit cycle problem for quadratic system (III)m=0 in this paper.
      
Since there is at most one limit cycle that may be created from critical point O by Hopf bifurcation, the number of limit cycles depends on the different situations of separatrix cycle to be formed around O.
      
If it is a homoclinic cycle passing through saddle S1 on 1+ax-y=0, which has the same stability with the limit cycle created by Hopf bifurcation, then the uniqueness of limit cycles in such cases can be proved.
      
If it is a homoclinic cycle passing through saddle N on x=0, which has the different stability from the limit cycle created by Hopf bifurcation, then it will be a case of two limit cycles.
      
The direct criterions for local stability of positive equilibrium and existence of limit cycle are also established when inference parameter of predator is small.
      
Hopf bifurcation and uniqueness of limit cycle for a class of quartic system
      
The theoretical description of the function of stochastic sensitivity is given both for the stationary point and limit cycle.
      
Within the framework of this approach, a positive operator of stochastic stability is assigned to the limit cycle.
      
The spectral radius of this operator characterizes stability of the limit cycle.
      
The constructive potentialities of the results obtained were demonstrated by the example of bifurcational analysis of the stochastic Ressler system at transition to chaos by multiple duplication of the limit cycle period.
      
Using both asymptotic and numerical methods, it is shown that for Pr >amp;gt; 2 the attractor is a two-dimensional invariant torus or a limit cycle; the corresponding convective flows are either quasiperiodic, with two basic frequencies, or periodic.
      
Bifurcation of a limit cycle from the equilibrium submanifold in a system with multiple cosymmetries
      
A linear analysis is made of the stability of the limit cycle in a Q-switched laser with continuous delayed feedback which controls the onset of instability.
      
The bifurcations of limit cycle generation with different periods (tripling of the period is most pronounced) and the passage to a chaotic strange attractor through an intermittency are obtained.
      
It is shown that transition between cycles with different symmetry may result in the spontaneous phase symmetry breaking and the appearance of chaos arising due to the period doubling bifurcation cascade of the asymmetric limit cycle.
      
A computer simulation is used to determine the parameters for which reliable switching takes place in the system and the parameter ranges are found within which the system undergoes a transition from strange attractor to limit cycle.
      
 

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