Kosambi-Cartan-Chern perspective on chaos: Unveiling hidden attractors in nonlinear autonomous systems

This article confronts the formidable task of exploring chaos within hidden attractors in nonlinear three-dimensional autonomous systems, highlighting the lack of established analytical and numerical methodologies for such investigations. As the basin of attraction does not touch the unstable manifold, there are no straightforward numerical processes to detect those attractors and one has to implement special numerical and analytical strategies. In this article we present an alternative approach that allows us to predict the basin of attraction associated with hidden attractors, overcoming the existing limitations. The method discussed here is based on the Kosambi-Cartan-Chern theory which enables us to conduct a comprehensive theoretical analysis by means of evaluating geometric invariants and instability exponents, thereby delineating the regions encompassing chaotic and periodic zones. Our analytical predictions are thoroughly validated by numerical results.

Figure 1

Attractor of the system of Eqs. (3.1)–(3.3) with the initial condition (0,0,0).

Figure 2

Components of the deviation curvature tensor $P_j^i$, Eq. (3.10), as it varies with time for the system of Eqs. (3.1)–(3.3) with the initial condition (0,0,0). (a) $P_1^1$ ($t$) vs time ($t$). (b) $P_2^1$ ($t$) vs time ($t$). (c) $P_1^2$ ($t$) vs time ($t$). (d) $P_2^2$ ($t$) vs time ($t$).

Figure 3

Basin of attraction of the system Eqs. (3.1)–(3.3). (a) Basin ($x_0,y_0$) is numerically determined by the estimation of maximum Lyapunov exponent. (b) Basin is determined by calculating the instability exponent $\delta$ with the help of the deviation vector computed from Eqs. (3.15) and (3.16).

Figure 4

Deviation vectors are numerically calculated from Eqs. (3.15) and (3.16). (a) Variation of the deviation vector ${\xi }^1$ with time. Initial conditions are $\dot{{\xi }^1}\left(0\right)={10}^{-10}$ and $\dot{{\xi }^2}\left(0\right)={10}^{-10}$. (b) Variation of the deviation vector ${\xi }^2$ with time. (c) Phase portrait in the ${\xi }^1\text{−}{\xi }^2$ plane.

Figure 5

(a) Critical time ${\tau }_0$ at which the curvature $\kappa$ is zero, i.e., ${\xi }^1\left({\tau }_0\right)={\xi }^2\left({\tau }_0\right)$, is approximately 7.5. (b) Variation of curvature $\kappa$ with time ($t$). It clearly shows that the curvature changes its sign before the critical time, which is a quantitative indication of chaos.

Figure 6

(a) Instability exponent near the equilibrium point $S(x_0,y_0,z_0)$ for the system of Eqs. (3.1)–(3.3) calculated by Eq. (2.12) and ${\xi }_{10}={10}^{-10}$. (b) Variation of $\xi \left(t\right)$ and $t^2$ against $t$.

Figure 7

Phase plot ($y\text{−}z$) of the system of Eqs. (4.1)–(4.3) with two different initial conditions (0,1,1) and (0,5,1).

Figure 8

Components of the deviation curvature tensor $P_j^i$, Eq. (4.12), of the system of Eqs. (4.1)–(4.3) with the initial condition (0,1,1) forming a nested tori. (a) $P_1^1$ ($t$) vs time ($t$). (b) $P_2^1$ ($t$) vs time ($t$). (c) $P_1^2$ ($t$) vs time ($t$). (d) $P_2^2$ ($t$) vs time ($t$).

Figure 9

Components of deviation curvature tensor $P_j^i$, Eq. (4.12), of the system of Eqs. (4.1)–(4.3) with the initial condition (0,5,1) forming a chaotic attractor. (a) $P_1^1$ ($t$) vs time ($t$). (b) $P_2^1$ ($t$) vs time ($t$). (c) $P_1^2$ ($t$) vs time ($t$). (d) $P_2^2$ ($t$) vs time ($t$).

Figure 10

Basin of attraction of the system of Eqs. (4.1)–(4.3). (a) Basin ($x_0,y_0$) is numerically determined by the estimation of the maximum Lyapunov exponent. (b) Basin determined by calculating the instability exponent $\delta$ with the help of the deviation vector computed from Eqs. (4.18) and (4.19).