Based on the Silnikov criterion, this paper studies a chaotic system of cubic polynomial ordinary differential equations in three dimensions. Using the Cardano formula, it obtains the exact range of the value of the p...Based on the Silnikov criterion, this paper studies a chaotic system of cubic polynomial ordinary differential equations in three dimensions. Using the Cardano formula, it obtains the exact range of the value of the parameter corresponding to chaos by means of the centre manifold theory and the method of multiple scales combined with Floque theory. By calculating the manifold near the equilibrium point, the series expression of the homoclinic orbit is also obtained. The space trajectory and Lyapunov exponent are investigated via numerical simulation, which shows that there is a route to chaos through period-doubling bifurcation and that chaotic attractors exist in the system. The results obtained here mean that chaos occurred in the exact range given in this paper. Numerical simulations also verify the analytical results.展开更多
The existence of Silnikovs orbits in a four-dimensional dynamical system is discussed. The existence of Silnikovs orbit resulting in chaotic dynamics is established by the fiber structure of invariant manifold and hig...The existence of Silnikovs orbits in a four-dimensional dynamical system is discussed. The existence of Silnikovs orbit resulting in chaotic dynamics is established by the fiber structure of invariant manifold and high-dimensional Melnikov method. Numerical simulations are given to demonstrate the theoretical analysis.展开更多
The existence of Silnikov's orbits in one coupled Duffing equation is discussed by using the fiberstructure of invariant manifold and high-dimensional Melnikov's method. Example and numerical simulationresults...The existence of Silnikov's orbits in one coupled Duffing equation is discussed by using the fiberstructure of invariant manifold and high-dimensional Melnikov's method. Example and numerical simulationresults are also given to demonstrate the theoretical analysis.展开更多
A 3D continuous autonomous chaotic system is reported, which contains a cubic term and six system parameters. Basic dynamic properties of the new Van der Pol Jerk system are studied by means of theoretical analysis an...A 3D continuous autonomous chaotic system is reported, which contains a cubic term and six system parameters. Basic dynamic properties of the new Van der Pol Jerk system are studied by means of theoretical analysis and numerical simulation. Based on the Silnikov theorem, the chaotic characterisitics of the dynamic system are discussed. Using Cardano formula and series solution of differential equation, eigenvalue problem and the existence of homoclinic orbit are studied. Furthermore, a rigorous proof for the existence of Silnikov-sense Smale horseshoes chaos is presented and some conditions which lead to the chaos are obtained. The formation mechanism indicates that this chaotic system has impulsive homoclinic chaos, and numerical simulation demonstrates that there is a route to chaos.展开更多
In this paper we study the singularity at the origin with three–fold zeroeigenvalue for symmetric vector fields with nilpotent linear part and 3–jet C~∞–equivalent toy(partial deriv)/(partial deriv) + z(partial de...In this paper we study the singularity at the origin with three–fold zeroeigenvalue for symmetric vector fields with nilpotent linear part and 3–jet C~∞–equivalent toy(partial deriv)/(partial deriv) + z(partial deriv)/(partial deriv)y + ax^2y (partialderiv)/(partial deriv)/z with a ≠ 0. We first obtain several subfamilies of the symmetric versalunfoldings of this singularity by using the normal form and blow–up methods under some conditions,and derive the local and global bifurcation behavior, then prove analytically the existence of theSilnikov homoclinic bifurcation for some subfamilies of the symmetric versal unfoldings of thissingularity, by using the generalized Melnikov methods of a homoclinic orbit to a hyperbolic ornon–hyperbolic equilibrium in a highdimensional space.展开更多
基金Project supported by the National Natural Science Foundation of China (Grant No.10872141)the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No.20060056005)
文摘Based on the Silnikov criterion, this paper studies a chaotic system of cubic polynomial ordinary differential equations in three dimensions. Using the Cardano formula, it obtains the exact range of the value of the parameter corresponding to chaos by means of the centre manifold theory and the method of multiple scales combined with Floque theory. By calculating the manifold near the equilibrium point, the series expression of the homoclinic orbit is also obtained. The space trajectory and Lyapunov exponent are investigated via numerical simulation, which shows that there is a route to chaos through period-doubling bifurcation and that chaotic attractors exist in the system. The results obtained here mean that chaos occurred in the exact range given in this paper. Numerical simulations also verify the analytical results.
基金Supported by National Key Basic Research Special Foundation (No.G1998020307)the Youth Foundation of BUCT (No.QN0138).
文摘The existence of Silnikovs orbits in a four-dimensional dynamical system is discussed. The existence of Silnikovs orbit resulting in chaotic dynamics is established by the fiber structure of invariant manifold and high-dimensional Melnikov method. Numerical simulations are given to demonstrate the theoretical analysis.
基金This work was supported by the National Key Basic Research Special Fund (No. G.1998020304).
文摘The existence of Silnikov's orbits in one coupled Duffing equation is discussed by using the fiberstructure of invariant manifold and high-dimensional Melnikov's method. Example and numerical simulationresults are also given to demonstrate the theoretical analysis.
基金Supported by National Natural Science Foundation of China(No. 10872141)Doctoral Foundation of Ministry of Education of China(No. 20060056005)Natural Science Foundation of Tianjin University of Science and Technology (No. 20070210)
文摘A 3D continuous autonomous chaotic system is reported, which contains a cubic term and six system parameters. Basic dynamic properties of the new Van der Pol Jerk system are studied by means of theoretical analysis and numerical simulation. Based on the Silnikov theorem, the chaotic characterisitics of the dynamic system are discussed. Using Cardano formula and series solution of differential equation, eigenvalue problem and the existence of homoclinic orbit are studied. Furthermore, a rigorous proof for the existence of Silnikov-sense Smale horseshoes chaos is presented and some conditions which lead to the chaos are obtained. The formation mechanism indicates that this chaotic system has impulsive homoclinic chaos, and numerical simulation demonstrates that there is a route to chaos.
基金Project supported by the National Natural Science Foundation of China(No.10171044)the Foundation for University Key Teachers of the Ministry of Education 34C05,34C15,58F14,58F30
文摘In this paper we study the singularity at the origin with three–fold zeroeigenvalue for symmetric vector fields with nilpotent linear part and 3–jet C~∞–equivalent toy(partial deriv)/(partial deriv) + z(partial deriv)/(partial deriv)y + ax^2y (partialderiv)/(partial deriv)/z with a ≠ 0. We first obtain several subfamilies of the symmetric versalunfoldings of this singularity by using the normal form and blow–up methods under some conditions,and derive the local and global bifurcation behavior, then prove analytically the existence of theSilnikov homoclinic bifurcation for some subfamilies of the symmetric versal unfoldings of thissingularity, by using the generalized Melnikov methods of a homoclinic orbit to a hyperbolic ornon–hyperbolic equilibrium in a highdimensional space.
