Analysis and design of anti-controlled higher-dimensional hyperchaotic systems via Lyapunov-exponent generating algorithms

Based on Lyapunov-exponent generation and the Gram-Schimdt orthogonalization, analysis and design of some anti-controlled higher-dimensional hyperchaotic systems are investigated in this paper. First, some theoretical results for Lyapunov-exponent generating algorithms are proposed. Then, the relationship between the number of Lyapunov exponents and the number of positive real parts of the eigenvalues of the Jacobi matrix is qualitatively described and analyzed. By configuring as many as possible positive real parts of the Jacobian eigenvalues, a simple anti-controller of the form $bsin (sigma x)$ for higher-dimensional linear systems is designed, so that the controlled systems can be hyperchaotic with multiple positive Lyapunov exponents. Utilizing the above property, one can resolve the positive Lyapunov exponents allocation problem by purposefully designing the number of positive real parts of the corresponding eigenvalues. Two examples of such anti-controlled higher-dimensional hyperchaotic systems are given for demonstration.