| Citation: |
Xianyi Li, Haijun Wang. A THREE-DIMENSIONAL NONLINEAR SYSTEM WITH A SINGLE HETEROCLINIC TRAJECTORY[J]. Journal of Applied Analysis & Computation, 2020, 10(1): 249-266. doi: 10.11948/20190135
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A THREE-DIMENSIONAL NONLINEAR SYSTEM WITH A SINGLE HETEROCLINIC TRAJECTORY
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Abstract
The study for singular trajectories of three-dimensional (3D) nonlinear systems is one of recent main interests. To the best of our knowledge, among the study for most of Lorenz or Lorenz-like systems, a pair of symmetric heteroclinic trajectories is always found due to the symmetry of those systems. Whether or not does there exist a 3D system that possesses a single heteroclinic trajectory? In the present note, based on a known Lorenz-type system, we introduce such a 3D nonlinear system with two cubic terms and one quadratic term to possess a single heteroclinic trajectory. To show its characters, we respectively use the center manifold theory, bifurcation theory, Lyapunov function and so on, to systematically analyse its complex dynamics, mainly for the distribution of its equilibrium points, the local stability, the expression of locally unstable manifold, the Hopf bifurcation, the invariant algebraic surface, and its homoclinic and heteroclinic trajectories, etc. One of the major results of this work is to rigorously prove that the proposed system has a single heteroclinic trajectory under some certain parameters. This kind of interesting phenomenon has not been previously reported in the Lorenz system family (because the huge amount of related research work always presents a pair of heteroclinic trajectories due to the symmetry of studied systems). What's more key, not like most of Lorenz-type or Lorenz-like systems with singularly degenerate heteroclinic cycles and chaotic attractors, the new proposed system has neither singularly degenerate heteroclinic cycles nor chaotic attractors observed. Thus, this work represents an enriching contribution to the understanding of the dynamics of Lorenz attractor. -
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References
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Xianyi Li, Haijun Wang. A THREE-DIMENSIONAL NONLINEAR SYSTEM WITH A SINGLE HETEROCLINIC TRAJECTORY[J]. Journal of Applied Analysis & Computation, 2020, 10(1): 249-266. doi: 10.11948/20190135
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Figure 1. Phase portraits of system (2.2) with
$ (a, b) = (2, 3) $ and different values of parameter$ c $ -
Figure 2. The graph of
$ N_{1}(a, b) $ for$ (a, b) = (0, 3)\times (0, 6) $ -
Figure 3. Phase portraits of system (2.2) with
$ (a, b) = (2, 3) $ , (a)$ c = 5.45 $ ,$ (x_{0}, y_{0}, z_{0}) = (1.3, 1.6, 1.6)\times1e-4 $ , (b)$ c = -5.45 $ and$ (x_{0}, y_{0}, z_{0}) = -(1.3, 1.6, 1.6)\times1e-4 $ . The figures illustrate that system (2.2) has a homoclinic trajectory to$ S_{0} $ when$ (a, b, c)\in W_{212}^{3} $ -
Figure 4. Phase portraits of system (2.2) with (a)
$ (a, c, b) = (2, 19, 6) $ ,$ (x_{0}, y_{0}, z_{0}) = (1.3, 1.6, 1.6)\times1e-4 $ , (b)$ (a, c, b) = (2, -19, 6) $ ,$ (x_{0}, y_{0}, z_{0}) = -(1.3, 1.6, 1.6)\times1e-4 $ . These figures illustrate that system (2.2) has one and only one heteroclinic trajectory to$ S_{0} $ and$ S^{'} $ when$ (a, b, c)\in W_{211} $ -
Figure 5. Phase portraits of system (2.2) with (a)
$ (a, c, b) = (2, 19, 18) $ ,$ (x_{0}, y_{0}, z_{0}) = (1.3, 1.6, 1.6)\times1e-4 $ , (b)$ (a, c, b) = (2, -19, 18) $ ,$ (x_{0}, y_{0}, z_{0}) = -(1.3, 1.6, 1.6)\times1e-4 $ . These figures illustrate that system (2.2) has one and only one heteroclinic trajectory to$ S_{0} $ and$ S^{'} $ when$ (a, b, c)\in W_{211} $ -
Figure 6. Phase portraits of system (2.2) with (a)
$ (a, c, b) = (2, 4, 4) $ ,$ (x_{0}, y_{0}, z_{0}) = (1.3, 1.6, 1.6)\times1e-4 $ , (b)$ (a, c, b) = (2, -4, 4) $ ,$ (x_{0}, y_{0}, z_{0}) = -(1.3, 1.6, 1.6)\times1e-4 $ . These figures illustrate that system (2.2) has one and only one heteroclinic trajectory to$ S_{0} $ and$ S^{'} $ when$ (a, b, c)\in W_{211} $ -
Figure 7. Phase portraits of system (2.2) with (a)
$ (a, c, b) = (2, 4, 5) $ ,$ (x_{0}, y_{0}, z_{0})=(1.3, 1.6, 1.6)\times1e-4 $ , (b)$ (a, c, b) = (2, -4, 5) $ ,$ (x_{0}, y_{0}, z_{0})=-(1.3, 1.6, 1.6)\times1e-4 $ . These figures illustrate that system (2.2) has one and only one heteroclinic trajectory to$ S_{0} $ and$ S^{'} $ when$ (a, b, c)\in W_{211} $ -
Figure 8. Phase portraits of system (2.2) with (a)
$ (a, c, b) = (2, 4, 5) $ ,$ (x_{0}, y_{0}, z_{0}) = (1.3, 1.6, 1.6)\times1e-4 $ , (b)$ (a, c, b) = (2, -4, 5) $ ,$ (x_{0}, y_{0}, z_{0}) = -(1.3, 1.6, 1.6)\times1e-4 $ . These figures illustrate that system (2.2) has one and only one heteroclinic trajectory to$ S_{0} $ and$ S^{'} $ when$ (a, b, c)\in W_{212}^{1} $