| Citation: |
Yuxuan Yang, Li Xu. BIFURCATION ANALYSIS AND CHAOS CONTROL FOR A DISCRETE SPACE–TIME LOKTA–VOLTERRA DIFFUSION EQUATION WITH NEUMANN BOUNDARY CONDITIONS[J]. Journal of Applied Analysis & Computation, 2027, 17(2): 1623-1648. doi: 10.11948/20260099
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BIFURCATION ANALYSIS AND CHAOS CONTROL FOR A DISCRETE SPACE–TIME LOKTA–VOLTERRA DIFFUSION EQUATION WITH NEUMANN BOUNDARY CONDITIONS
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Abstract
This paper investigates a discrete Lotka–Volterra competition diffusion system subject to Neumann boundary conditions. By employing normal form theory and center manifold reduction, we show that the system exhibits a flip bifurcation at the unique positive equilibrium, whereas neither a Neimark–Sacker nor a fold bifurcation is observed, moreover, a Turing bifurcation is identified. To regulate the bifurcation behavior, both direct and indirect state feedback control strategies are implemented. Under appropriate parameter conditions, these control schemes effectively stabilize the system by suppressing chaos and bifurcation-induced instabilities. Extensive numerical simulations are conducted to validate the theoretical results. The bifurcation analysis framework developed herein is generalizable to other discrete diffusion systems with different boundary configurations.
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Keywords:
- Bifurcation /
- chaos control /
- stability /
- center manifold reduction
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References
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Yuxuan Yang, Li Xu. BIFURCATION ANALYSIS AND CHAOS CONTROL FOR A DISCRETE SPACE–TIME LOKTA–VOLTERRA DIFFUSION EQUATION WITH NEUMANN BOUNDARY CONDITIONS[J]. Journal of Applied Analysis & Computation, 2027, 17(2): 1623-1648. doi: 10.11948/20260099
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Figure 1.
Flip bifurcation diagram and maximum Lyapunov exponent for
$ u_{1} $ -
Figure 2.
The 3-dimensional bifurcation diagram for
$ (u_{1},d,a) $ $ (u_{1},d,r) $ -
Figure 3.
The sequence diagram or bifurcation diagram on direct control for
$ u_{1} $ $ r=1.95,a=0.76 $ $ s_{1}=0.2,s_{2}=0.2 $ $ s_{1}=0.5,s_{2}=0.45 $ $ s_{1}=0.5,s_{2}=0.53 $ $ s_{1}=0.5,s_{2}=0.56 $ $ s_{1}=0.5,s_{2}=0.62 $ $ s_{1}=0.5, s_{2}=0.82. $ -
Figure 4.
The bifurcation diagram on indirect control for
$ u_{1} $ $ b_{1}=0.1,e_{1}=0.1,e_{2}=0.1,eta _{1}=0.1,\eta _{2}=0.1 $ $ b_{2}=0.11 $ $ b_{2}=0.31 $ $ b_{2}=0.75 $ -
Figure 5.
The selected patterns. (a)
$ r=0.1,a=0.1,d=0.5 $ $ r=1.3,a=0.2,d=0.3 $ $ r=1.99,a=0.3,d=0.01 $ -
Figure 6.
The stable patterns on direct control when
$ r=0.1,a=0.1,d=0.5 $ $ s_{1}=s_{2}=0.1. $ -
Figure 7.
The selected patterns on direct control when
$ r=0.1,a=0.1,d=0.5 $ $ s_{1}=s_{2}=0.71 $ $ s_{1}=s_{2}=0.81 $ $ s_{1}=s_{2}=0.88 $ -
Figure 8.
The selected patterns on indirect control when
$ r=0.1,a=0.1,d=0.5 $