A Study of the Circular Pursuit Dynamics using Bifurcation Theoretic Computational Approach

Abstract: A circular pursuit guidance problem involving pursuer-target engagement is studied in this paper using a bifurcation theory based numerical approach. While target is modeled as a point mass moving around in a circle with certain velocity, pursuer dynamics is driven by the relative position and orientation with respect to the target. A planar case is currently considered. A mathematical model representing the engagement scenario is derived and two cases are presented, one without and the other with a basic model for pursuer speed dynamics accounting for limitations imposed by available force. Analytical and simulation results are presented to elucidate the novel approach. Advantages of using this approach for arriving at laws for pursuer-target engagement are highlighted.

• The paper demonstrates that engagement feasibility in circular pursuit is governed by the pursuer-to-target speed ratio and available thrust.
• It employs bifurcation theory and numerical continuation to establish stability transitions and determine capture thresholds.
• Time-domain simulations confirm that approximately 65% of maximum thrust is required for successful target engagement.

Bifurcation-Theoretic Computational Analysis of Circular Pursuit Dynamics

Introduction and Problem Formulation

This paper addresses the circular pursuit problem—a fundamental guidance scenario ubiquitous in aerospace and related engineering disciplines—through the lens of bifurcation theory and numerical continuation. The scenario involves a pursuer (e.g., an unmanned aerial vehicle or missile) tracking a moving target constrained to circular motion. While traditional analyses model these as two-body kinematic systems, this work advances the state-of-the-art by explicitly incorporating nonlinear pursuer dynamics and leveraging computational bifurcation analysis to obtain equilibrium states and their stability properties as function of key parameters, e.g., pursuer-to-target speed ratio and available thrust.

The pursuit dynamics are initially developed for a planar case, leading to nonlinear ODEs representing the evolutions of relative distance and orientation (line-of-sight) between pursuer and target. The target executes uniform circular motion; the pursuer is modeled first as a constant-speed agent pointing toward the target, and later with explicit nonlinear dynamics incorporating thrust and drag, typical of aerospace vehicles.

Analytical and Bifurcation-Theoretic Approach

The main technical contribution is the formulation and analysis of the engagement dynamics using bifurcation theory—a methodology suited to study parametric variation and qualitative changes in nonlinear dynamical systems. The modeling distinguishes two scenarios:

1. Kinematic Pursuer: The pursuer's speed is fixed, only geometric variables evolve.
2. Dynamic Pursuer: The pursuer's acceleration is limited by available thrust and vehicle drag; speed evolves as a system state.

For the kinematic scenario, the system is reduced to two nonlinear state equations for normalized distance ($r$) and orientation ($\varphi$), parameterized by speed ratio $k = v_1/v_2$. Equilibrium analysis yields explicit conditions:

• $\sin\varphi^* = k$
• $r^* = \cos\varphi^* = \sqrt{1 - k^2}$

Real (physically meaningful) equilibrium occurs for $0 \leq k \leq 1$. Local stability is established via linearization (Jacobian analysis), with qualitative change (bifurcation) in equilibrium type (focus vs. node) at a critical speed ratio ($k_c$). At $k = 1$, the pursuer can sustain engagement with the target; below this, capture is impossible.

Phase-plane simulations confirm bifurcation analysis: oscillatory transient behavior (stable focus) transitions to monotonic exponential convergence (stable node) as $k$ approaches unity. The computed bifurcation diagrams rigorously delineate regions of parameter space where engagement is dynamically feasible.

Augmented Model Incorporating Pursuer Dynamics

The more realistic scenario models the pursuer's speed as a state variable, evolving under a force balance involving thrust and quadratic aerodynamic drag. The dimensionless system expands to:

• Distance and orientation evolution, as before
• Nonlinear ordinary differential equation for speed (throttle input $n$ as control parameter)

The critical insight is that pursuer engagement is possible only if available thrust supports a terminal speed at least matching the target's. The bifurcation diagrams, this time parametrized by $\varphi$0, establish the threshold throttle necessary to achieve $\varphi$1 ("terminal maneuvering capability"). The analysis reveals that approximately 65% of maximum thrust is required to engage a target on the specified trajectory for the sample aircraft parameters.

Eigenvalue tracking of the linearized system across the parameter sweep confirms a bifurcation in equilibrium type, matching theoretical predictions for nonlinear second- and third-order dynamical systems. The bifurcation-theoretic computational methodology provides strong numerical evidence that the stability of the engagement is fundamentally dictated by accessible physical resources (thrust, drag).

Numerical Simulations and Practical Implications

Time-domain simulations illustrate the transient and asymptotic behaviors under step increases in throttle. If the throttle reaches or exceeds the derived threshold, the pursuer closes the gap and achieves engagement, with the relative distance decaying nearly exponentially. For sub-threshold throttle, engagement is asymptotically unreachable.

These results have immediate practical implications:

• Design of Pursuit Guidance Laws: The approach enables computational synthesis of pursuit strategies that explicitly respect vehicle propulsion and maneuvering constraints, leading to more robust and physically realizable guidance algorithms.
• Performance Assessment: By mapping the effective reachability space in terms of physical limits, the method allows for systematic assessment of which maneuvering targets are actually interceptable by a given airframe/powerplant configuration.
• Guideline for Minimum Required Resources: The bifurcation diagrams provide actionable charts for practitioners to determine, prior to engagement, whether an interception maneuver is feasible for a known target profile.

Theoretical Implications and Future Directions

The bifurcation-theoretic approach extends the legacy of dynamical systems theory in control and guidance by providing a systematic computational method for direct analysis of nonlinear, high-dimensional engagement models. The findings—particularly the explicit relationship between physical power limits, target maneuvering parameters (radius, angular speed), and engagement feasibility—are broadly generalizable. As vehicle models become more complex (higher-order, including 3D maneuvers, actuator dynamics, or more elaborate guidance logic), these techniques scale naturally using available numerical continuation packages (e.g., AUTO, MATCONT).

Potential avenues for future work include:

• Extension to full 6-DOF pursuit models incorporating three-dimensional geometry and actuator limitations.
• Integration with optimal control theory to determine minimal time-to-intercept strategies under resource constraints.
• Application to swarming guidance, heterogeneous teams (multiple pursuers, cooperative/competitive strategies), and adversarial motion planning.

Conclusion

This work provides a rigorous computational methodology, based on bifurcation and continuation theory, for analyzing and designing pursuit engagement dynamics with explicit incorporation of nonlinear vehicle dynamics and physical constraints (2604.09065). The approach offers both analytical and numerical tools to delineate engagement feasibility, transient behavior, and equilibrium properties as functions of physically meaningful parameters. The results are directly relevant for the development of robust, resource-aware guidance laws in aerospace and related fields, and the method sets a foundation for the analysis of more complex pursuit-evasion scenarios in future research.