A novel hybridization of birds of prey-based optimization with differential evolution mutation and crossover for chaotic dynamics identification – Scientific Reports

Identifying the parameters of chaotic systems like Lorenz, Chen, and Rössler is an infamously brittle task: tiny shifts in coefficients can produce wildly different trajectories. That sensitivity makes conventional time-domain error metrics ill-conditioned and prone to misleading local minima. In response, researchers have explored not only better-behaved objective functions that respect geometric invariants of chaos, but also metaheuristics that maintain diversity and resist premature convergence.

Enter h-BPBODE, a new hybrid optimizer that combines the behavioral intelligence of birds of prey-based optimization (BPBO) with the recombinative power of differential evolution (DE). The result is a more stable, more precise, and faster framework for parameter identification in chaotic dynamics.

Why chaotic identification is so hard

Chaotic flows amplify minute perturbations, so trajectory-matching costs can become jagged and deceptive. Prior research mitigated this by:

• Designing more robust costs (e.g., return-map or Poincaré-based metrics that retain geometric structure).
• Adopting metaheuristics—stochastic, diversity-preserving algorithms—to explore rugged landscapes effectively.

These two threads shifted practice away from strictly deterministic estimators toward robust, hybrid, and sometimes chaos-enhanced optimizers.

What’s new in h-BPBODE

BPBO is built around four complementary behaviors: individual hunting, group hunting, attacking the weakest, and relocation. h-BPBODE retains that adaptive choreography but injects DE mutation and binomial crossover after every phase-wise update. In other words, each candidate solution produced by a BPBO move is immediately refined by DE before acceptance.

This simple change pays off in three ways:

• Recombinative diversity: DE’s mutation and crossover continually rewire the population, curbing collapse around suboptimal basins.
• Balanced search: BPBO’s adaptive phases drive exploration and exploitation, while DE tunes step sizes implicitly through vector differentials.
• Robustness: Boundary constraints are enforced at every refinement step, stabilizing search in high-sensitivity regimes.

Together, these mechanisms translate into faster convergence and tighter accuracy—without sacrificing reliability.

How it works at a glance

• Initialize a population of candidate parameter sets within feasible bounds.
• Iterate through BPBO’s four behaviors to generate updated candidates.
• After each behavioral update, apply DE mutation and binomial crossover to the updated candidates.
• Enforce bounds and accept improvements based on the objective value.
• Stop when convergence criteria are met or a maximum number of iterations is reached.

Benchmarking on Lorenz, Chen, and Rössler

The team tested h-BPBODE by estimating unknown parameters in three classic chaotic systems. The objective was to minimize trajectory mismatches between simulated and ground-truth dynamics under a common experimental protocol.

Head-to-head comparisons with standard BPBO, starfish optimization, hippopotamus optimization, particle swarm optimization (PSO), and DE showed that h-BPBODE:

• Recovered the exact parameter sets with negligible residual error.
• Converged faster than competing methods.
• Exhibited markedly lower variance across independent runs.

Convergence traces, boxplots, and parameter-evolution curves illustrated smooth, stable progress toward ground truth. Statistical summaries reinforced the consistency gains, while cross-study comparisons indicated that h-BPBODE matched or surpassed leading hybrid and chaos-enhanced algorithms reported in the literature.

How it stacks up against the field

Over the last decade, hybrid and diversity-amplifying strategies have steadily improved chaotic identification. Notable examples include quantum-behaved PSO variants, adaptive cuckoo search hybrids, chaotic search–augmented optimizers, cellular automata–infused whale/sine-cosine algorithms, DE-enhanced pollination strategies, and quantum-inspired fruit fly optimizers. These approaches demonstrated that careful balancing of exploration and exploitation—often via hybridization—yields substantial accuracy and reliability gains.

Yet persistent pain points remained: populations could still collapse prematurely, and fixed or poorly adapted step sizes struggled against error surfaces with uneven curvature. By inserting DE’s mutation and crossover into each BPBO phase, h-BPBODE directly targets both issues, preserving diversity while adapting search dynamics continuously.

Key contributions

• A hybrid optimizer that fuses BPBO’s four-phase behavior with DE’s mutation and crossover inside the update loop.
• Improved exploration–exploitation balance through phase-wise recombination and strict boundary handling.
• Demonstrated exact or near-exact parameter recovery on Lorenz, Chen, and Rössler with faster convergence and lower run-to-run variance than BPBO, PSO, DE, and other recent heuristics.
• Evidence of robustness via convergence traces, boxplots, and smooth parameter trajectories across multiple trials.
• Strong cross-method comparability, suggesting readiness for broader nonlinear estimation tasks beyond the three benchmarks.

Why it matters

Accurate parameter identification is a cornerstone for modeling, prediction, synchronization, and control of nonlinear systems. In domains where chaotic dynamics are intrinsic—ranging from secure communications and sensing to mechatronics and bio-inspired control—reliable estimators can unlock better designs and more resilient controllers. h-BPBODE’s stability and efficiency point to a practical path forward, especially when standard estimators falter under sensitivity and noise.

Limitations and next steps

While results are compelling, several questions remain:

• Scalability to higher-dimensional or fractional-order chaotic systems and to large parameter spaces.
• Performance under measurement noise, model mismatch, or unknown initial conditions.
• Real-time feasibility for control-in-the-loop applications.

Future work could incorporate noise-aware cost functions, adaptive population sizing, and problem-specific priors. Extending to controller co-design and synchronization tasks would also test generality.

Bottom line

By weaving DE’s recombination into every step of BPBO’s behavioral search, h-BPBODE delivers a robust and efficient solution to a classic hard problem: identifying parameters in chaotic systems. On Lorenz, Chen, and Rössler, it consistently reaches the right answers faster and more reliably than popular alternatives—an encouraging signal for broader nonlinear estimation challenges.