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How do you compute Common-BOA (C-BOA) for Nonlinear Dynamical Systems with Multiple Equilibria?
How do you compute Common-BOA (C-BOA) for Nonlinear Dynamical Systems with Multiple Equilibria?
Paper Title: 2-Contraction and Poincaré-Index Theory-Based Framework for Stability Analysis of Nonlinear Dynamical Systems with Multiple Equilibria
Abstract: This work leverages the 2-contraction theory, an extension of classical contraction theory, to develop a systematic stability framework for systems with multiple equilibria. Coupled with powerful geometric tools such as Poincaré index theory, the 2-contraction theory enables the stability analysis of planar nonlinear systems instead of undertaking it locally around equilibrium points. Using index theory and 2-contraction, we characterize the nature of equilibrium points and delineate regions in R2 where periodic solutions, closed orbits, or stable solutions may exist. A key focus of this work is the identification of regions in the state space where periodic solutions may exist and the regions that guarantee the non-existence of such solutions. Additionally, we address a crucial problem: determining the basin of attraction (BOA) for stable equilibrium points. For systems with multiple equilibria, identifying candidate BOAs is highly nontrivial. We propose a novel methodology adopting the 2-contraction theory to approximate a common BOA for a class of nonlinear systems with multiple equilibria. Theoretical findings are substantiated through benchmark examples and numerical simulations, demonstrating the efficacy of the proposed approach. Furthermore, we extend our framework to analyze networked systems, showcasing its efficacy in an opinion dynamics problem.
Authors: Riddhi Mohan Bora, Bhabani Shankar Dey, Indra Narayan Kar
Journal: International Journal of Control Automation and Systems
Publication Details: Volume 24, Issue: NA, DOI: NA, accepted date: 16-05-2026, publication date: xx:xx:xxxx
C-BOA for a planar system with two stable and one saddle equilibrium points
C-BOA for a planar system with two stable and one saddle equilibrium points
C-BOA for a planar Opinion Dynamics model of interaction between 2-agents
C-BOA for a planar Opinion Dynamics model of interaction between 2-agents
How do you initialize in Iterative Algorithms associated with a non-convex optimization problem?
How do you initialize in Iterative Algorithms associated with a non-convex optimization problem?
Paper Title: A 2-Contraction Framework for Initialization Analysis in Non-Convex Optimization
Abstract: Non-convex optimization problems often exhibit multiple local minima, maxima, and saddle points, making gradient-based methods sensitive to initialization. This paper applies the 2-contraction theory to the gradient-flow dynamics induced by a continuously differentiable non-convex objective function. Using Hessian spectral properties, we characterize the 2-contraction region and remove the saddle region to obtain a candidate region for stable equilibria. A forward-invariant sublevel-set condition is then imposed to construct a certified initialization set. The resulting method provides a-priori initialization guidance for standard gradient descent without modifying its update rule. While focusing on low-dimensional, continuous-time problems, this work also addresses scalability issues and discrete-time implementation for completeness.
Authors: Riddhi Mohan Bora, Bhabani Shankar Dey, Indra Narayan Kar
Conference: IFAC World Congress - 23rd WC 2026
Location: Busan, Republic of Korea
Publication Details: Accepted for Publication.
Landscape of a two-variable non-convex objective function and its corresponding phase space in respective gradient flow dynamics
Landscape of a two-variable non-convex objective function and its corresponding phase space in respective gradient flow dynamics
The certified initialization region for a benchmark non-convex optimization problem
The certified initialization region for a benchmark non-convex optimization problem
Output Regulation problem using I/O Feedback Linearization with MULTI-STABLE ZERO DYNAMICS?
Output Regulation problem using I/O Feedback Linearization with MULTI-STABLE ZERO DYNAMICS?
Paper Title: Input-Output Feedback Linearization: Case Study on 2-Contractive Zero Dynamics
Abstract: Zero dynamics are essential to the design and analysis of input-output feedback linearization control, particularly in nonlinear systems. This study revisits an input-output feedback linearization problem through an illustrative example where the zero dynamics admit multiple equilibrium points. The effect of such zero dynamics on system performance and control input has also been investigated. Conventional tools such as 1-contraction and Lyapunov theory are not directly applicable to the analysis of zero dynamics with multiple equilibria. Therefore, the theory of 2-contraction is employed to perform the analysis effectively. The impact of such zero dynamics on the external control input, tasked with regulating the system's output, is also analyzed. Extensive numerical simulations are conducted to validate and exemplify the theoretical results.
Authors: Riddhi Mohan Bora, Indra Narayan Kar
Conference: 19th IEEE International Conference on Control & Automation (ICCA)
Location: Tallinn, Estonia
Publication Details: https://doi.org/10.1109/ICCA65672.2025.11129836
Control profile when the zero dynamics has multi-stability is not 2-contractive
Control profile when the zero dynamics has multi-stability is not 2-contractive
Control profile when the zero dynamics has multi-stability is 2-contractive
Control profile when the zero dynamics has multi-stability is 2-contractive
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