[Paper Review] Finding the Rikitake's attractors by parameter switching

Marius‐F. Danca, S. Codreanu|arXiv (Cornell University)|Feb 10, 2011

Stability and Controllability of Differential Equations25 references3 citations

TL;DR

This paper introduces a parameter switching algorithm to synthesize any attractor in the Rikitake dynamical system, extending the method beyond hyperbolic equilibria. By periodically or randomly switching the control parameter $ p $, the algorithm generates target attractors that match those obtained at the mean parameter value, even for chaotic or stable limit cycle dynamics, and reveals a previously unnoticed long-lasting transient attractor due to $ x_3 $-axis instability.

ABSTRACT

In this paper the attractors synthesis algorithm for a class of dissipative dynamical systems with hyperbolic equilibria, presented in [1], is applied to generate any attractor of the Rikitake system. By switching periodically, or even randomly, the control parameter inside a given set of values, during any finite time interval while the attractor is numerically approximated, any attractor can be generated. Beside the extension of the synthesis algorithm to systems with non-hyperbolic equilibria, we have found for the Rikitake system, a new intriguing transient which, occurring for a long time interval, is difficult to be numerically found due to the known system instability along the x3-axis.

Motivation & Objective

• To extend the attractor synthesis algorithm to systems with non-hyperbolic equilibria, such as the Rikitake system.
• To numerically demonstrate that any Rikitake attractor can be generated via periodic or random switching of the control parameter $ p $.
• To identify and characterize a long-lived transient attractor (TA) arising from instability along the $ x_3 $-axis, which is difficult to capture with standard numerical methods.
• To validate the equivalence between attractors generated by switching and those at the mean parameter value using superimposed phase portraits, histograms, and Poincaré sections.

Proposed method

• The attractor synthesis algorithm applies piecewise-constant switching of the control parameter $ p(t) $ over finite time intervals during numerical integration of the Rikitake system.
• The switching rule uses periodic or random sequences of parameter values $ p_1, p_2, \dots $, with the mean $ p^* $ determining the target attractor.
• The method relies on the convexity property of the attractor set, ensuring that the synthesized attractor $ A^* $ matches the attractor $ A_{p^*} $ at the mean parameter value.
• Numerical integration uses the fourth-order Runge-Kutta method with careful handling of stiffness to resolve the transient attractor.
• Validation is performed via superimposed phase portraits, histograms of state distribution, and Poincaré sections to confirm attractor identity.
• The transient attractor is identified through long-time simulations and geometric symmetry analysis, despite numerical challenges from $ x_3 $-axis instability.

Experimental results

Research questions

• RQ1 Can the attractor synthesis algorithm be successfully extended to systems with non-hyperbolic equilibria, such as the Rikitake system?
• RQ2 Does parameter switching—periodic or random—allow the generation of any desired attractor in the Rikitake system?
• RQ3 What causes the emergence of a long-lasting transient attractor in the Rikitake system, and why is it difficult to detect numerically?
• RQ4 To what extent is the synthesized attractor $ A^* $ equivalent to the attractor $ A_{p^*} $ at the mean parameter value $ p^* $?
• RQ5 Can the transient attractor be rigorously interpreted as a dynamical component rather than a numerical artifact?

Key findings

• The attractor synthesis algorithm successfully generates any Rikitake attractor using periodic or random switching of the control parameter $ p $, even when individual parameter values yield chaotic or stable dynamics.
• For the deterministic scheme $[1p_1, 1p_2]$ with $ p_1 = 9.66 $, $ p_2 = 12 $, the synthesized attractor $ A^* $ at $ p^* = 10.83 $ matches the attractor $ A_{p^*} $, confirmed by superimposed phase portraits and histograms.
• With the scheme $[1p_1, 2p_2, 1p_3]$ using $ p_1 = 5 $, $ p_2 = 7 $, $ p_3 = 24.32 $, the synthesized attractor $ A^* $ at $ p^* = 10.83 $ is identical to $ A_{p^*} $, as shown by overlapping Poincaré sections and histograms.
• When $ p_1 = 17 $, $ p_2 = 23 $, the algorithm produces a chaotic attractor $ A^* $ identical to $ A_{p^*} $ at $ p^* = 20 $, verified via Poincaré sections and histogram comparison.
• For $ p_1 = 26 $, $ p_2 = 28 $, both individual attractors are stable limit cycles, and all switching schemes produce only a stable limit cycle, matching $ A_{p^*} $ at $ p^* = 27 $.
• Using random switching with uniform distribution over $ p_1 = 14.5 $, $ p_2 = 20.7 $, the synthesized attractor $ A^* $ matches $ A_{p^*} $ at $ p^* = 17.6 $, confirmed by overlapping phase portraits and histograms.