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On the extinction route of a stochastic population model under heteroclinic bifurcation

Acta Mechanica Sinica 2022年第1期38卷 59-69页

作者： Qing Yu Yang Li Xianbin Liu State Key Laboratory of Mechanics and Control for Mechanical Structures College of Aerospace EngineeringNanjing University of Aeronautics and AstronauticsNanjing 210016China School of Automation Nanjing University of Science and TechnologyNanjing 210094China

The noise-induced transition of the augmented Lotka-Volterra system is investigated under vanishingly small *** will ultimately go extinct because of intrinsic noise,and different extinction routes may occur due to the Freidlin-Wentzell large deviation *** relation between the most probable extinction route(MPER)and heteroclinic bifurcation is studied in this *** MPERs and the quasi-potentials in different regimes of parameters are analyzed in *** the bifurcation,the predator goes extinct,and the prey will survive for a long ***,the heteroclinic bifurcation changes the MPER wherein both species go *** heteroclinic cycle plays a role in transferring the most probable extinction ***,the analyses of the weak noise limit can contribute to predicting the stochastic behavior under finite small *** the heteroclinic bifurcation and the rotational deterministic vector field can reduce the action necessary for the MPER.

关键词： Lotka-Volterra system Most probable extinction route heteroclinic bifurcation Quasi-potential

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Codimension 2 reversible heteroclinic bifurcations with inclination flips

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Science China Mathematics 2009年第7期52卷 1801-1814页

作者： XU YanCong ZHU DeMing DENG GuiFeng Department of Mathematics Hangzhou Normal UniversityHangzhou(Xiasha)310036China Department of Mathematics East China Normal UniversityShanghai 200241China School of Mathematics and Information Shanghai Lixiu University of CommerceShanghai 201620China

In this paper, the heteroclinic bifurcation problem with real eigenvalues and two incli- nation-flips is investigated in a four-dimensional reversible system. We perform a detailed study of this case by using the method originally established in the papers "Problems in Homoclinic bifurcation with Higher Dimensions" and "bifurcation of heteroclinic Loops," and obtain fruitful results, such as the existence and coexistence of R-symmetric homoclinic orbit and R-symmetric heteroclinic loops, R-symmetric homoclinic orbit and R-symmetric periodic orbit. The double R-symmetric homoclinic bifurcation (i.e., two-fold R-symmetric homoclinic bifurcation) for reversible heteroclinic loops is found, and the existence of infinitely many R-symmetric periodic orbits accumulating onto a homoclinic orbit is demonstrated. The relevant bifurcation surfaces and the existence regions are also located.

关键词： heteroclinic bifurcation inclination flips reversible system 37G40 37C29 34C37

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CHAOTIC OSCILLATION OF ANONLINEAR POWER SYSTEM

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Applied Mathematics and Mechanics(English Edition) 1999年第10期20卷 1175-1182页

作者：张伟年张卫东 Department of Mathematics Sichuan University Chengdu P R China Chongqing Power College Chongqing P R China

For a nonlinear power transmission system, the residue calculus method is introduced and applied to study its heteroclinic bifurcation. There a cone region and a strip region of parameters are obtained, in which the power transmission system displays chaotic oscillation. This gives a theoretic analysis and a computational method for the purpose to control the nonlinear system with deviation stably running.

关键词： power transmission system nonlinear heteroclinic bifurcation chaotic oscillation

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External bifurcations of Double Heterodimensional Cycles with One Orbit Flip

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Applied Mathematics 2021年第4期12卷 348-369页

作者： Huimiao Dong Tiansi Zhang College of Science University of Shanghai for Science and Technology Shanghai China

In this paper, external bifurcations of heterodimensional cycles connecting three saddle points with one orbit flip, in the shape of “∞”, are studied in three-dimensional vector field. We construct a poincaré return map between returning points in a transverse section by establishing a locally active coordinate system in the tubular neighborhood of unperturbed double heterodimensional cycles, through which the bifurcation equations are obtained under different conditions. Near the double heterodimensional cycles, the authors prove the preservation of “∞”-shape double heterodimensional cycles and the existence of the second and third shape heterodimensional cycle and a large 1-heteroclinic cycle connecting with P1 and P3. The coexistence of a 1-fold large 1-heteroclinic cycle and the “∞”-shape double heterodimensional cycles and the coexistence conditions are also given in the parameter space.

关键词： Double heteroclinic Loops Orbit Flip heteroclinic bifurcation bifurcation Theory

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