A new class of integrable Lotka–Volterra systems

Helen Christodoulidi; Andrew N. W. Hone; Theodoros E. Kouloukas

doi:10.3934/jcd.2019011

Article Contents

2019, Volume 6, Issue 2: 223-237. Doi: 10.3934/jcd.2019011

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A new class of integrable Lotka–Volterra systems

1.
Research Center for Astronomy and Applied Mathematics, Academy of Athens, Athens 11527, Greece
2.
School of Mathematics, Statistics and Actuarial Science, University of Kent, Canterbury CT2 7FS, UK
3.
School of Mathematics and Statistics, University of New South Wales, Sydney NSW 2052, Australia

^* Corresponding author: Theodoros E. Kouloukas
^* Corresponding author: Theodoros E. Kouloukas

Received: March 2019

Revised: July 2019

Published: November 2019

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Abstract

A parameter-dependent class of Hamiltonian (generalized) Lotka–Volterra systems is considered. We prove that this class contains Liouville integrable as well as superintegrable cases according to particular choices of the parameters. We determine sufficient conditions which result in integrable behavior, while we numerically explore the complementary cases, where these analytically derived conditions are not satisfied.

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Mathematics Subject Classification: Primary: 37K10, 70Hxx; Secondary: 70Kxx.

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Figure 1. The Poincaré surface of section $ x_2 = 1,x_1>1 $ for the Lotka–Volterra system with $ a_i = 1 $ and $ E = 6 $ for various $ k_i $, $ i = 1,2,3,4 $ values: (a) $ (k_1,k_2, k_3, k_4) = (-1, -2,- 1,-1) $ (b) $ (k_1,k_2, k_3, k_4) = (-1, -2,- 1,-2) $, (c) $ (k_1,k_2, k_3, k_4) = (-1,-4,-2,-3) $, (d) $ (k_1,k_2, k_3, k_4) = (-1, -4,- 2,-1) $

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Figure 2. 3D projections on the $ x_2,x_3,x_4 $ plane for the system with $ (k_1,k_2, k_3, k_4) = (-1, -4,- 2,-1) $ and for initial conditions: (a) close to a fixed point of Fig. 1(d) ($ E = 6 $), (b) on an ellipse around the fixed point ($ E = 6 $), (c) randomly chosen from Fig. 1(d) ($ E = 6 $) and (d) randomly chosen at a higher total energy ($ E = 20 $) exhibiting chaotic behavior

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Figure 3. The Poincaré surface of section $ x_2 = 1, x_1>1 $ for the Lotka–Volterra system with $ a_i = 1 $ and $ k_i = -1 $, $ i = 1,2,3,4 $ for the energies: (a) $ E = 4.2 $, (b) $ E = 6 $, (c) $ E = 8 $, (d) $ E = 29 $

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Figure 4. The largest Lyapunov exponent $ \lambda $ for the Lotka–Volterra system with $ a_i = 1 $ and $ k_i = -1 $, $ i = 1,2,3,4 $ for the energies: (a) $ E = 4.2 $ and (b) $ E = 29 $

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Figure 5. The evolution in time of the phase space variables for the integrable cases: (a) $ (k_1,k_2, k_3, k_4) = (0, 0,- 1,-1) $ and (b) $ (k_1,k_2, k_3, k_4) = (-2, -2,- 2,2) $

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Figure 6. The trajectories projected on the 3D plane $ x_1,x_3,x_4 $ plane for the integrable systems: (a) $ (k_1,k_2, k_3, k_4) = (0, 0,- 1,-1) $ and (b) $ (k_1,k_2, k_3, k_4) = (-2, -2,- 2,2) $

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Figure 7. The largest Lyapunov exponent $ \lambda $ for the system with $ (k_1,k_2, k_3, k_4) = (-2, -2,- 2,2) $ at (b) $ E = 10 $ and (c) $ E = 72 $

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