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<p>Quadratic operators are known to be in one-to-one correspondence with cubic matrices, and those associated with stochastic cubic matrices preserve the simplex. It is widely believed that quadratic stochastic operators do not exhibit chaotic dynamics on the simplex. We show that this belief fails in the non-stochastic case. Specifically, we provide explicit constructions of one-dimensional and two-dimensional non-stochastic quadratic operators displaying chaotic behavior for certain parameter values. Our results are obtained via rigorous dynamical analysis and are illustrated by numerical simulations.</p>

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Chaotic dynamics generated by quadratic operators on the two-dimensional simplex

  • 期刊论文

  • 发表日期：

    2026年7月22日

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  • Z. S. Boxonov,
  • S. S. Khudayarov,
  • E. B. Dilmurodov

摘要

Quadratic operators are known to be in one-to-one correspondence with cubic matrices, and those associated with stochastic cubic matrices preserve the simplex. It is widely believed that quadratic stochastic operators do not exhibit chaotic dynamics on the simplex. We show that this belief fails in the non-stochastic case. Specifically, we provide explicit constructions of one-dimensional and two-dimensional non-stochastic quadratic operators displaying chaotic behavior for certain parameter values. Our results are obtained via rigorous dynamical analysis and are illustrated by numerical simulations.