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Bifurcation in the almost periodic $ 2 $D Ricker map

  • * Corresponding author: Brian Ryals

    * Corresponding author: Brian Ryals 

RJS is supported by a University of Southern California, Dornsife School of Letters Arts and Sciences Faculty Development Grant, 12-1855-0032

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  • This paper studies bifurcations in the coupled $ 2 $ dimensional almost periodic Ricker map. We establish criteria for stability of an almost periodic solution in terms of the Lyapunov exponents of a corresponding dynamical system and use them to find a bifurcation function. We find that if the almost periodic coefficients of all the maps are identical, then the bifurcation function is the same as the one obtained in the one dimensional case treated earlier, and that this result holds in $ N $ dimension under modest coupling constraints. In the general two-dimensional case, we compute the Lyapunov exponents numerically and use them to examine the stability and bifurcations of the almost periodic solutions.

    Mathematics Subject Classification: Primary: 39A30, 39A28; Secondary: 37C75.

    Citation:

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  • Figure 1.  The stability region of the coexistence fixed point of equation (4) is shown in the $ p $-$ q $ plane for the coupling values $ a = 0.5 $ and $ b = 0.7 $. The geometry is similar for all $ ab < 1 $, with the stability region bounded by two straight lines and a piece of a hyperbola

    Figure 2.  Plot of the Bifurcation Equation $ B(\gamma,\gamma,b) = 0 $. A bifurcation takes place as the parameter pair $ (b,\log{\gamma}) $ crosses from the stability region ($ B<0 $) to the instability region ($ B>0 $)

    Figure 3.  Plots of the values of the two Lyapunov Exponents $ \chi_1 $ (left plot) and $ \chi_2 $ (right plot) as functions of the parameter $ 0\leq b \leq 0.99 $. Values used were $ \gamma_{01} = e^{1.2} $, $ \gamma_{02} = e^{1.6} $, $ a_{12} = 0.6 $, $ a_{21} = 0.8 $, $ G_1(\theta) = \sin(2\pi \theta) $, $ G_2(\theta) = \sin(2\pi\theta)\cos(2\pi\theta) $, and $ \omega = \frac{e^{\pi}}{25}\approx 0.8984 $. Both are decreasing functions with respect to $ b $. The $ b $ values used were $ 0.01m $ for $ m = 0, 1, \cdots 99 $

    Figure 4.  The border of the stability region for an almost periodic solution is shown for the values $ b = 0 $, $ b = 0.2 $, $ b = 0.4 $, and $ b = 0.6 $, in a neighborhood of $ (e^2,e^2) $. Here the $ x $-axis is $ \log(\gamma_{01}) $ and the $ y $-axis is $ \log(\gamma_{02}) $. The curves are ordered from bottom left to top right in order of increasing $ b $, so that the stability region is growing with $ b $. Values used were $ a_{12} = 0.5 $, $ a_{21} = 0.7 $, $ G_1(\theta) = \sin(2\pi \theta) $, $ G_2(\theta) = \sin(2\pi\theta)\cos(2\pi\theta) $, and $ \omega = \frac{e^{\pi}}{25}\approx 0.8984 $

    Figure 5.  The top left image shows the long-term dynamics of $ (x_1(t),x_2(t),\gamma_1(t)) $ for $ b = 0.4 $, while the top right shows the long-term dynamics of $ (x_1(t),x_2(t),\gamma_2(t)) $ for the same value of $ b $. The bottom row uses $ b = 0.2 $ instead where the AP solution is unstable. Values used were $ a_{12} = 0.5 $, $ a_{21} = 0.7 $ and the almost periodic sequences are $ G_1(\theta_k) = \sin(2\pi \theta_k) $, $ G_2(\theta_k) = \sin(2\pi\theta_k)\cos(2\pi\theta_k) $ where $ \theta_k = \theta+k\omega $ with $ \omega = \frac{e^{\pi}}{25}\approx 0.8984 $

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