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Updated: Mar 8, 2026
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Published on: September 23, 2025
655
Partially controlling transient chaos in the Lorenz equations
Rubén Capeáns1, Juan Sabuco1, Miguel A F Sanjuán2
1Departamento de Física, Universidad Rey Juan Carlos, Tulipán s/n, 28933 Móstoles, Madrid, Spain.
Summary
This study introduces partial control to prevent undesirable trajectory escapes in transient chaos, a phenomenon in nonlinear dynamics. The method is successfully applied to the 3D Lorenz system, offering permanent control over chaotic behavior.
Area of Science:
- Nonlinear Dynamics
- Chaos Theory
- Cybernetical Physics
Background:
- Transient chaos describes temporary chaotic behavior in dynamical systems before trajectories escape to attractors.
- Uncontrolled escapes from chaotic regions can be detrimental in various scientific and engineering applications.
- Controlling such escapes is crucial for maintaining desired system behavior.
Purpose of the Study:
- To introduce and demonstrate the application of partial control to prevent trajectory escapes in transient chaos.
- To extend the application of partial control to three-dimensional systems for the first time.
- To analyze different methods for implementing partial control in the Lorenz system.
Main Methods:
- Application of partial control to prevent trajectory escapes in nonlinear systems.
- Utilizing the Lorenz system as a paradigmatic example exhibiting transient chaos.
- Developing three distinct implementation strategies: 1D map, 2D Poincaré section, and 3D fixed-time map.
Main Results:
- Partial control effectively prevents trajectories from escaping the chaotic region in the Lorenz system.
- Successful extension of partial control to a three-dimensional system.
- Demonstration of three feasible methods for implementing the control strategy.
Conclusions:
- Partial control is a viable method for permanently confining trajectories within chaotic regions.
- The three-dimensional implementation offers practical advantages, such as fixed time intervals for control application.
- This work represents a significant advancement in controlling transient chaos, particularly in higher dimensions.
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