The restricted three-body problem is a classic model in gravitational dynamics. It considers two primary bodies—in this case, Earth and the Moon—that orbit one another, while a third body has negligible mass and does not alter their motion. Despite its apparently simple setup, the equations are nonlinear and can produce a remarkable range of behavior, including escape trajectories, temporary capture, unstable passages, libration-point orbits and long-lived periodic motions. In the circular restricted three-body problem, or CRTBP, Earth and Moon are assumed to travel in circular orbits. The elliptic version, known as the ERTBP, allows their separation and orbital speed to vary as they follow an ellipse. That seemingly modest change removes an important symmetry and makes the search for repeating trajectories substantially more difficult.

Nagai’s work focuses on resonant periodic orbits, in which different components of a spacecraft’s motion return to their original configuration after a precise number of cycles. The key idea is frequency matching. A trajectory can oscillate horizontally around the Earth–Moon system while also moving above and below the orbital plane. When the vertical oscillation frequency bears a rational relationship to the principal orbital frequency, the motion can close after a finite period instead of drifting indefinitely. The study concentrates on 1:n resonances, a class in which the relevant frequencies are related by an integer ratio. Such resonances are central to periodic-orbit theory because they transform what might otherwise be a quasiperiodic, non-repeating path into a closed orbit with a recognizable geometric pattern.

To identify these paths, the study begins with a simplified approximation of the equations of motion near the Moon. In that region, the gravitational influence of the Moon dominates the local motion, while Earth’s field and the rotating reference frame continue to shape the trajectory. The approximation makes it possible to understand the essential structure before confronting the full nonlinear equations. Nagai shows that the in-plane part of the approximate solutions is consistent with the rose curves associated with the Italian mathematician Guido Grandi, who studied these curves in the early eighteenth century. In polar-style form, the radial distance varies sinusoidally while the angular position advances with time. The resulting trajectory repeatedly expands and contracts, creating lobes or “petals” around a central region. The number and arrangement of these petals depend on the ratio of the frequencies and on the initial phases.

The mathematical connection is more than a visual coincidence. A rose curve can be written parametrically so that its radial amplitude follows one sinusoidal function while the direction of motion follows another. In Nagai’s formulation, the coordinates contain a factor of the form (\sin((n/N)t-\phi_1)), multiplied by the rotating directional terms (\sin(t-\phi_2)) and (\cos(t-\phi_2)). Here, (n) and (N) are positive integers, and the phase parameters determine the initial orientation and timing of the pattern. When the frequencies are commensurate—meaning their ratio is rational—the curve repeats. In the restricted three-body setting, however, the physical orbit is not merely a two-dimensional textbook curve. The rose-like form is the projection of a dynamical solution, and the vertical component must satisfy its own resonance condition for the full three-dimensional motion to become periodic.

The approximate trajectories serve as initial guesses for a numerical single-shooting procedure. This is a standard but delicate technique in periodic-orbit computation. A trial state—typically including position and velocity—is integrated forward for a proposed period. At the end of that integration, the numerical state is compared with the starting state. If the position and velocity do not match, the initial conditions and, when necessary, the period are adjusted. An iterative correction process then seeks a solution for which the final and initial states coincide within a specified tolerance. In effect, the method solves a boundary-value problem by repeatedly asking the equations of motion to “shoot” from one point and return precisely to it. Using the rose-like approximation as a guide greatly improves the chances of converging on the desired family rather than landing on an unrelated orbit.

The first accurate solutions are computed in the Earth–Moon CRTBP, where the primaries move on circular paths and the rotating frame provides a comparatively stable environment for numerical analysis. Once the 1:n resonant periodic orbits have been found there, Nagai continues them into the ERTBP by gradually increasing the eccentricity of the Earth–Moon orbit. This continuation strategy avoids trying to discover every elliptic solution from scratch. Instead, a known periodic orbit at zero eccentricity is used as the starting point, and the equations are modified in small increments. At each step, the preceding solution supplies the initial estimate for the next one. The procedure traces how the orbit’s shape, period and stability evolve as the idealized circular model becomes more realistic.

Stability is one of the most important questions surrounding any periodic orbit. A trajectory may close perfectly in a mathematical model but be so sensitive to small disturbances that a spacecraft could not remain near it without frequent correction. Nagai analyzes the linear stability of the rose-like periodic orbits during the continuation in eccentricity. In practical terms, linear stability examines how tiny deviations from the reference orbit grow or shrink over time. This information is commonly extracted from the state-transition or monodromy matrix, which maps a small perturbation through one complete period. Its eigenvalues, often called characteristic multipliers, indicate whether perturbations remain bounded, oscillate or expand. The study therefore does not stop at drawing unusual trajectories; it follows their dynamical response as the Earth–Moon model changes.

The work also places these solutions within a long history of three-dimensional periodic orbits in the restricted three-body problem. Earlier studies identified halo orbits, vertical self-resonant satellite orbits and other families that pass near the Earth–Moon libration points. Such trajectories have influenced both theoretical celestial mechanics and mission design, including concepts for spacecraft operating near gravitational balance regions. Rose-like orbits belong to a different visual and dynamical category, but they emerge from the same fundamental principle: nonlinear gravitational systems can support organized families of repeating motion. Their existence illustrates how planar oscillations and vertical resonances can combine to create geometry that is simultaneously simple to recognize and difficult to derive.

The potential significance of the results lies in the bridge they create between classical geometry, modern numerical dynamics and spaceflight applications. The rose curve was developed centuries ago as a mathematical object; here, a related pattern appears naturally in a gravitational model involving the Earth and Moon. That connection could make complicated resonant behavior easier to classify and communicate, while also supplying useful starting points for searches through the enormous catalogue of possible periodic trajectories. The study does not claim that every rose-like orbit is immediately suitable for a mission, nor does it provide operational designs for a spacecraft. Instead, it establishes a computational pathway: approximate the local dynamics, identify resonant structure, refine the orbit in the circular problem, continue it into the elliptic problem and test its stability. As future missions increasingly explore cislunar space, families of structured periodic orbits may become valuable maps of what gravity can make possible—and of where a spacecraft can repeatedly go without simply following an ordinary Keplerian ellipse.

Subject of Research: Resonant rose-like periodic orbits in the Earth–Moon circular and elliptic restricted three-body problems

Article Title: Rose-like periodic orbits in the restricted three-body problem

Article References: Nagai, Y. “Rose-like periodic orbits in the restricted three-body problem.” Celestial Mechanics and Dynamical Astronomy 138, article 52 (2026).

Image Credits: AI Generated

DOI: 10.1007/s10569-026-10327-w

Keywords: Rose curve; periodic orbit; stability; circular restricted three-body problem (CRTBP); elliptic restricted three-body problem (ERTBP)

At the core of this investigation lies a two-dimensional time-modulated lattice subject to nonlinear effects where solitons serve as the primary agents of transport. The researchers demonstrate that the soliton’s net displacement over a complete driving cycle is not arbitrary but is topologically quantized. This quantization stems from distinct Chern numbers, which are fundamental topological invariants traditionally associated with band theory in condensed matter physics. Crucially, the work extends beyond the established first Chern number—typical of one-dimensional linear pumps—introducing higher-order Chern invariants that govern transport in multi-dimensional, nonlinear systems.

Topological pumping refers to the phenomenon where a wave packet or particle systematically shifts across a lattice as a system parameter evolves cyclically in time. In linear regimes, this transport is discretized and quantified by an integer number corresponding to a first Chern number, reflecting the global topological properties of the band structure. However, introducing nonlinearity into such driven lattices significantly enriches the transport dynamics. Here, the soliton pumping is influenced by multiple Chern numbers in higher dimensions, including second Chern numbers, which offer a refined classification of the soliton’s quantum transport behavior in the two-dimensional lattice.

The nonlinear dynamics carve out distinct transport regimes. In one regime, solitons exhibit integer-quantized motion, moving an exact integer multiple of unit cells per driving cycle. This integer displacement is intricately tied to the quantization dictated by the cumulative Chern invariants of the system’s underlying topological bands. In an alternative regime, the researchers uncover fractional-quantized soliton pumping, where the displacement per cycle appears as a rational fraction of the lattice constant. This fractional quantization signals the emergence of subtle topological phases and nonlinear effects coalescing to produce transport phenomena not explained by conventional linear theories.

Beyond quantization, the soliton’s mobility is sensitive to the lattice band structure and the strength of nonlinear interactions. At strong nonlinearities, solitons become localized, entering a trapped regime wherein their position remains nearly stationary throughout the driving period. This nontrivial localization hints at a competition between nonlinear self-focusing effects and topological driving forces. Furthermore, anisotropic transport behavior was observed, where soliton displacement differs along perpendicular spatial directions. Such anisotropy results in complex mixed regimes featuring different topological quantization on different lattice axes, adding layers of control in engineering wave-packet motion through nonlinear lattices.

To experimentally confirm these theoretical predictions, the team designed nonlinear topolectrical circuits mimicking the time-modulated lattice dynamics with inherent nonlinearity. These topolectrical circuits, composed of nonlinear circuit elements arranged in time-varying networks, serve as versatile platforms to emulate the nonlinear wave dynamics and measure soliton transport properties with high fidelity. The experiments successfully captured integer and fractional quantized soliton pumping, the onset of soliton trapping, and anisotropic transport phenomena, affirming the theoretical framework and the robustness of topological invariants in nonlinear settings.

The implications of this work stretch far beyond the immediate physical system studied. By revealing how higher-order topological invariants dictate nonlinear wave transport, it opens new avenues to control and harness localized excitations in various engineered media. Topological concepts traditionally confined to linear, electronic systems now find application in nonlinear optics, acoustics, and circuit platforms, where dynamic control over wave localization and pumping can lead to breakthroughs in signal processing, energy delivery, and quantum information transfer.

Delving into the mathematical structure, the involvement of higher-dimensional Chern numbers corresponds to sophisticated geometric phases accumulated by the soliton’s wavefunction during one driving cycle in parameter space. These phases encode global topological information inaccessible through local band parameters alone. The addition of nonlinearity effectively couples the soliton’s internal degrees of freedom to the geometry of the lattice’s topological bands, resulting in a rich tapestry of dynamical responses modulated by these topological invariants.

Furthermore, the fractional quantization regime represents a subtle form of topological pumping where the soliton’s trajectory embodies a rational winding number. This regime challenges conventional understandings based predominantly on linear theory and integer-valued invariants, suggesting that nonlinearities and multi-dimensional topology may host unexplored fractionalized transport phenomena. Understanding these effects could illuminate parallels with fractional quantum Hall states and other exotic topological phases in condensed matter physics.

The study also emphasizes the precision with which topological invariants control not only the magnitude but the directionality of the soliton’s movement. The observed anisotropic pumping behavior hints at the possibility of designing waveguiding devices where solitons can be steered along preferred lattice directions by tuning lattice parameters or nonlinear interactions. Such controllability adds functional versatility to topological insulator analogs in nonlinear regimes, enabling purpose-built pathways for information or energy transmission.

In summary, this research pioneers a new paradigm where nonlinear wave physics, high-dimensional topology, and artificial lattice engineering converge to produce controlled, quantized transport of robust localized wave-packets. The integration of experimental topolectrical circuits confirms the practical feasibility of harnessing these effects, setting the stage for future explorations in larger, more complex lattices and alternative wave platforms such as photonic and acoustic metamaterials. These developments promise transformative applications in modern wave-based technologies, offering robust and tunable transport mechanisms operating beyond the linear regime.

As a culmination, the experimental realization of quantized soliton pumping via multiple Chern numbers reflects a profound understanding of nonlinear topological wave transport. This breakthrough bridges condensed matter theory, nonlinear dynamics, and applied physics, charting a course for innovations that leverage the robust topological nature of nonlinear excitations. The capacity to manipulate soliton trajectories with topological precision holds promise for scalable implementations in cutting-edge wave technologies and inspires theoretical pursuits in nonlinear topological phenomena.

Subject of Research: Quantized soliton pumping in nonlinear, time-modulated two-dimensional lattices governed by high-dimensional topological invariants including first and second Chern numbers.

Article Title: Quantized Soliton Pumping Governed by High-Dimensional Chern Numbers

Web References: DOI: 10.1093/nsr/nwag007

Image Credits: ©Science China Press

Keywords: soliton pumping, nonlinear lattices, topological transport, Chern numbers, topolectrical circuits, high-dimensional topology, fractional quantization, nonlinear dynamics, anisotropic transport, wave-packet localization, time-modulated lattices, topological invariants

The researchers delve into the fascinating intricacies of the (\beta)-fractional ((n+1))-dimensional generalized KP model, which is designed to capture nonlinear waves traveling through dispersive media. With its roots in the classic KP equation, the model extends established mathematical methodologies to encompass the effects of fractional derivatives. This adaptation is crucial, as it allows traditional models to address non-local phenomena, a feature that is particularly relevant in many physical systems ranging from fluid dynamics to plasma physics.

One of the key conclusions drawn from the study is the identification of unique nonlinear dynamical behaviors attributable to the (\beta)-fractional generalized KP model. By employing advanced analytical techniques, the authors meticulously illustrate how this model can reveal new solutions and dynamic patterns that are not possible with integer-order models. The introduction of fractional derivatives adds a layer of complexity and is instrumental in capturing the subtleties of wave propagation and interaction in higher-dimensional spaces.

The analytical wave structures generated by the model are both captivating and pivotal for future applications. Not only do these structures facilitate a deeper understanding of wave phenomena, but they also provide a rich canvas for exploring stability and bifurcation scenarios in nonlinear systems. The authors highlight the significance of identifying bifurcation points, which signal qualitative changes in the dynamics of the system. Such insights can have profound implications for understanding phenomena in various fields, including meteorology, oceanography, and even biological systems.

In order to provide a comprehensive perspective on the model’s capabilities, the researchers perform a series of numerical simulations alongside their analytical findings. This dual approach allows them to validate theoretical predictions and explore the parameter space more extensively. By doing so, they investigate the model’s sensitivity to different initial conditions and external disturbances, making their contributions both robust and relevant to real-world applications.

Sensitivity analysis represents a critical aspect of the research, shedding light on how slight variations in parameters can lead to markedly different outcomes. This sensitivity provides a powerful tool for predicting system behavior and for designing control strategies that can mitigate adverse effects in practical scenarios. The research emphasizes the necessity of understanding these nuances in order to develop advanced models that can accurately represent complex behaviors in nonlinear systems.

Moreover, the implications of the (\beta)-fractional generalized KP model extend beyond immediate academic interest. The potential applications span multiple disciplines, including materials science, chemical engineering, and environmental modeling. As the authors aptly note, the intersection of fractional calculus with nonlinear wave dynamics opens new avenues for research and innovation, with the potential to address pressing global challenges.

From a broader perspective, this work contributes to the growing body of literature highlighting the importance of fractional calculus in modern scientific inquiry. Traditionally, differential calculus has been the cornerstone of mathematical modeling. However, the emergence of fractional calculus as a complementary tool signifies a paradigm shift, enabling researchers to tackle problems previously considered intractable due to their complexity.

In examining the research methods employed, it becomes apparent that the authors are adept at leveraging both analytical and numerical techniques cohesively. They utilize perturbative methods to derive solutions under certain conditions, while also employing advanced computational techniques to explore cases that resist simple analytical treatment. This comprehensive strategy underlines the richness and depth of their investigation into the (\beta)-fractional generalized KP model.

As the research unfolds, the authors provide a clear narrative that delineates the intricacies of their findings. Their coherent exposition not only serves the academic community but also paves the way for interdisciplinary collaboration. By framing the discussion within the context of real-world phenomena, they invite practitioners from various fields to consider how fractional calculus could inform and enhance their work.

The significance of this research is underscored by its potential to catalyze further studies in fractional calculus and nonlinear dynamics. Scholars and researchers are encouraged to build upon the groundwork laid by Demirbilek and his colleagues, pushing the boundaries of what is known and delving into unexplored territories. The mathematical community stands to gain considerably from such collaborative efforts, as insights from one discipline can significantly influence another.

Looking ahead, the authors express optimism regarding the broader acceptance of fractional calculus in scientific modeling. As challenges grow increasingly complex, the incorporation of fractional derivatives offers a powerful lens through which we can re-examine classical problems. By fostering a culture that embraces novel approaches, the academic community can harness the full potential of these mathematical tools.

In conclusion, the research conducted by Demirbilek, Danladi, and Akbulut represents a significant step forward in the study of nonlinear dynamics and fractional calculus. Their work not only enriches the theoretical landscape but also lays the foundation for practical applications that could revolutionize how we model and understand complex systems. As the scientific community continues to embrace these advanced methodologies, the possibilities for innovative solutions to multifaceted problems are virtually limitless.

Subject of Research: Nonlinear Dynamical Behaviors in Fractional Calculus

Article Title: β-Fractional (n+1)-dimensional generalized KP model: nonlinear dynamical behaviors, analytical wave structures, bifurcation, and sensitivity analysis.

Article References:
Demirbilek, U., Danladi, A., Akbulut, A. et al. β-Fractional (n+1)-dimensional generalized KP model: nonlinear dynamical behaviors, analytical wave structures, bifurcation, and sensitivity analysis. Sci Rep (2025). https://doi.org/10.1038/s41598-025-32261-x

Image Credits: AI Generated

DOI:

Keywords: Nonlinear dynamics, fractional calculus, Korteweg-de Vries model, bifurcation, sensitivity analysis, higher-dimensional models, wave structures.

SIAM, a leading international organization dedicated to applied and computational mathematics, annually selects a highly distinguished group of Fellows who represent the forefront of innovation across diverse industries and academic institutions worldwide. This year, among an extensive membership base of 14,000 professionals across multiple sectors—including academia, government agencies, military, and industry—25 mathematicians were named Fellows. Professor Kang’s inclusion in this select cohort highlights the global recognition of his theoretical and computational expertise.

Kang’s fellowship citation specifically acknowledges his “fundamental theoretical and computational contributions to the analysis, control, and estimation of nonlinear dynamical systems and their applications.” At its core, this research revolves around dynamic systems theory, which deals with mathematically modeling and predicting the behavior of complex systems over time. These systems are often nonlinear, exhibiting behaviors that are intricate and sensitive to initial conditions, which makes their control and estimation particularly challenging but essential in real-world applications such as autonomous vehicles and power grids.

In essence, dynamic systems provide a framework through which the future state of a system can be predicted from its current state, given an underlying set of physical principles. Kang elucidates this by emphasizing the predictive capabilities rooted in fundamental physics, while candidly acknowledging the inherent difficulties in accurate forecasting. The complexity of natural and engineered dynamic systems—especially nonlinear ones—requires sophisticated mathematical tools for effective control, which have been the focus of Kang’s ongoing research efforts.

Control systems, a field where mathematics and engineering converge, are at the heart of Kang’s investigations. These systems are designed to regulate the behavior of dynamic processes, ranging from uncrewed autonomous vehicles to industrial machinery. Kang’s work pushes the boundaries by integrating advanced machine learning, data science, and artificial intelligence techniques to enhance the adaptability and precision of control strategies. This interdisciplinary approach reflects the modern trend of leveraging computational intelligence to solve classical engineering problems.

Collaborating with his students and various defense research institutions, Kang has contributed to noteworthy projects that demonstrate the transformative power of applied mathematics. His work includes data simulation efforts for numerical weather prediction in partnership with the U.S. Naval Research Laboratory, indicating how mathematical models assist in forecasting atmospheric phenomena critical to naval operations. Additionally, he has been involved in data assimilation studies related to the combustion dynamics of rocket and jet engines through the Air Force Research Laboratory—efforts that are vital for improving propulsion efficiency and reliability.

Furthermore, Kang’s expertise has been instrumental in anomaly detection within power systems, collaborating with the Office of Naval Research’s Next Strategic Technology Evaluation Program (NextSTEP). This research underscores the importance of dynamic system monitoring to maintain the integrity and stability of critical infrastructure, an area of increasing concern as power grids become more complex and integrated with renewable energy sources.

Beyond his applied projects, Kang participates in a multi-institutional initiative funded by the National Science Foundation aimed at exploring the mathematical foundations of machine learning. This collaboration seeks to deepen understanding of the theoretical underpinnings of learning algorithms, which are essential for ensuring robustness, transparency, and efficiency in AI-driven systems, thereby bridging pure mathematics with cutting-edge technological advancements.

In addition to his SIAM Fellowship, Professor Kang is recognized as a Fellow of the Institute of Electrical and Electronics Engineers (IEEE), reflecting his interdisciplinary impact across mathematics, engineering, and computer science. His affiliation with the University of California at Santa Cruz as an adjunct professor further positions him at the nexus of academic innovation and mentorship, where he shapes the next generation of mathematicians and engineers.

Kang values the role of professional societies not only for honoring research excellence but also for fostering community and leadership within the discipline. His contributions extend to organizing and chairing international conferences on systems and controls, as well as serving as vice chair of the systems and controls activity group within SIAM. These service roles amplify his influence in setting agendas and facilitating collaboration among researchers worldwide.

Dr. Ralucca Gera, Chair of the Department of Applied Mathematics at the Naval Postgraduate School, praises Kang’s achievement as a testament to his exceptional research contributions and service to the field. She emphasizes that Professor Kang’s recognition as a SIAM Fellow elevates both his personal standing and the Naval Postgraduate School’s reputation as a hub of innovation and excellence in mathematical research relevant to national defense.

The ascendancy of applied mathematics as a driver of technological innovation is vividly illustrated by Kang’s career, which seamlessly integrates theoretical rigor with practical application. His work addresses some of the most demanding scientific challenges, from predictive modeling and control of autonomous systems to the foundational theory underpinning machine learning, demonstrating how advanced mathematics continues to shape the future of engineering and defense technologies.

As the SIAM community celebrates Professor Wei Kang’s induction into the 2025 class of Fellows, it also acknowledges the increasingly vital role of interdisciplinary collaboration in advancing knowledge. Kang’s visionary integration of applied mathematics with artificial intelligence and engineering exemplifies how the field is evolving to meet complex societal needs, promising exciting developments in dynamic system analysis and control for years to come.

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Subject of Research: Applied Mathematics, Nonlinear Dynamical Systems, Control Theory, Machine Learning Integration, Systems and Controls

Article Title: Naval Postgraduate School’s Wei Kang Named 2025 SIAM Fellow for Pioneering Work in Dynamic Systems and Controls

News Publication Date: Not specified in the content

Web References:
– https://www.siam.org/publications/siam-news/articles/siam-announces-2025-class-of-fellows
– https://nps.edu/web/math
– https://sites.google.com/site/weikangnpsmonterey

Image Credits: U.S. Navy photo by Dan Linehan

Keywords: Applied Mathematics, Control Theory, Dynamic Systems, Nonlinear Systems, Machine Learning, Data Assimilation, Anomaly Detection, Autonomous Vehicles, Numerical Weather Prediction