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Conical intersection (CI) leads to fast electronic energy transfer. However, Hamm and Stock [Phys. Rev. Lett. 109, 173201 (2012)] showed the existence of a vibrational CI and its role in vibrational energy relaxation. In this paper, we further investigate the vibrational energy relaxation using an isolated model Hamiltonian system of four vibrational modes with two distinctively different timescales (two fast modes and two slow modes). We show that the excitation of the slow modes plays a crucial role in the energy relaxation mechanism. We also analyze the system from a mixed quantum-classical (surface hopping method) and a completely classical point of view. Notably, surface hopping and even classical simulations also capture fast energy relaxation, which is a signature of CI's existence. [ABSTRACT FROM AUTHOR]

By using the recently derived universal discrete imaginary-time propagator of the harmonic oscillator, both thermodynamic and Hamiltonian energies can be given analytically and evaluated numerically at each imaginary time step for any short-time propagator. This work shows that, using only currently known short-time propagators, the Hamiltonian energy can be optimized to the twelfth-order, converging to the ground state energy of the harmonic oscillator in as few as three beads. This study makes it absolutely clear that the widely used second-order primitive approximation propagator, when used in computing thermodynamic energy, converges extremely slowly with an increasing number of beads. [ABSTRACT FROM AUTHOR]

Simulations of laser-induced electron dynamics in a molecular system are performed using time-dependent (TD) equation-of-motion (EOM) coupled-cluster (CC) theory. The target system has been chosen to highlight potential shortcomings of truncated TD-EOM-CC methods [represented in this work by TD-EOM-CC with single and double excitations (TD-EOM-CCSD)], where unphysical spectroscopic features can emerge. Specifically, we explore driven resonant electronic excitations in magnesium fluoride in the proximity of an avoided crossing. Near the avoided crossing, the CCSD similarity-transformed Hamiltonian is defective, meaning that it has complex eigenvalues, and oscillator strengths may take on negative values. When an external field is applied to drive transitions to states exhibiting these traits, unphysical dynamics are observed. For example, the stationary states that make up the time-dependent state acquire populations that can be negative, exceed one, or even complex-valued. [ABSTRACT FROM AUTHOR]

In this paper, we present an extended dissipaton equation of motion for studying the dynamics of electronic impurity systems. Compared with the original theoretical formalism, the quadratic couplings are introduced into the Hamiltonian accounting for the interaction between the impurity and its surrounding environment. By exploiting the quadratic fermionic dissipaton algebra, the proposed extended dissipaton equation of motion offers a powerful tool for studying the dynamical behaviors of electronic impurity systems, particularly in situations where nonequilibrium and strongly correlated effects play significant roles. Numerical demonstrations are carried out to investigate the temperature dependence of the Kondo resonance in the Kondo impurity model. [ABSTRACT FROM AUTHOR]

In the seminal paper, Chirikov (Phys Rep 52:263–379, 1979) showed that the standard map does not exhibit a boundary to chaos, but rather that there are small islands ("islets") of stability for arbitrarily large values of the nonlinear perturbation. In this context, he established that the area of the islets in the phase space and the range of parameter values where they exist should decay following power laws with exponents - 2 and - 1 , respectively. In this paper, we carry out a systematic numerical search for islets of stability and we show that the power laws predicted by Chirikov hold. Furthermore, we use high-resolution 3D islets to reveal that the islets' volume decays following a similar power law with exponent - 3 . [ABSTRACT FROM AUTHOR]

The time dependence of one-dimensional quantum mechanical probability densities is presented when the potential in which a particle moves is suddenly changed, called a quench. Quantum quenches are mainly addressed, but a comparison with results for the dynamics in the framework of classical statistical mechanics is useful. Analytical results are presented when the initial and final potentials are harmonic oscillators. When the final potential vanishes, the problem reduces to the broadening of wave packets. A simple introduction to the concept of the Wigner function is presented, which allows a better understanding of the dynamics of general wave packets. It is pointed out how special the broadening of Gaussian wave packets is, the only example usually presented in quantum mechanics textbooks. Quantum quenches occur when the Hamiltonian of a system is suddenly changed, so that the system is driven out of equilibrium. This paper, appropriate for graduate-level quantum mechanics classes, examines the similarities and differences of quantum quenches compared to their classical counterparts for three one-dimensional examples: the harmonic oscillator, the linear potential, and the zero potential for which the broadening of a non-Gaussian wave packet is addressed. [ABSTRACT FROM AUTHOR]

In this paper, we focus on an extended nonlinear Schrödinger equation describing the pulse propagation in a nonlinear Schrödinger equation with self‐steepening and magneto‐optic effects. The existence of periodic and solitary solutions are proved based on the bifurcation method, and also, the Hamiltonian properties and the classification of its equilibrium points are obtained. The chirped solutions of the extended nonlinear Schrödinger equations are obtained by using the complete discrimination system for polynomial method, and under specific parameter conditions, three types of optical wave patterns are obtained to visualize the model. In particular, we consider the external perturbation terms to analyze the chaotic behavior of this equation. [ABSTRACT FROM AUTHOR]

This paper develops a theoretical basis and a systematic process for resolving all inertia forces along generalized coordinates from the overall energy equation of a dynamical system. The theory is developed for natural systems with scleronomic constraints, where the potential energy is independent of generalized velocities. The process involves expansion of the energy equation, and specifically a special expansion of the kinetic energy term, from which the inertia forces emerge. The expansion uses fundamental kinematic identities of the phase space. It is also guided by insights drawn from the structure of the Hamiltonian function. The resulting equation has the structure of the D'Alembert's equation but involving generalized coordinates, from which the Lagrange's equations of motion are obtained. The expansion process elucidates how certain inertia forces manifest in the energy equation as composite terms that must be accurately resolved along different generalized coordinates. The process uses only the system energy equation, and neither the Hamiltonian nor the Lagrangian function are required. Extension of this theory to non-autonomous and non-holonomic systems is an area of future research. [ABSTRACT FROM AUTHOR]

An exact similarity solution is worked out for fluid motion in the vicinity of a cylindrical surface in an electrically conducting viscoelastic fluid obeying the Jeffrey model. Cylindrical surface expands axially with linear velocity and has a prescribed surface temperature. The present analysis focuses on the partial slip assumption which gives rise to a nonlinear case Robin-type (mixed) boundary condition in axial velocity. In contrast to a previously published study (J. Mech., vol 31, pp. 69–78, 2015) which only reported local similarity solutions in no-slip case, the present work provides an exact similarity analysis by defining a quadratic surface temperature. The effects of the magnetic field and Joule heating on the momentum and thermal transport phenomena are also scrutinized. A numerical routine is used to solve the equation accurately for a wide range of fluid parameters, including ranges that could not be handled by previous approximate analytical methods that were limited to the no-slip limit. The influences of viscoelasticity and partial slip on the flow behavior are investigated, and the coefficient of skin friction for a Jeffrey fluid is calculated. Obtained results are consistent with the previously published works under special situations. The case of flat surface, which can be obtained as a limiting case of the model, is not even explored in the past. [ABSTRACT FROM AUTHOR]

This paper studies the summability of the transseries solution of a nonintegrable Hamiltonian system. Since our system has a resonance and is not integrable a general transseries theory does not work well as far as the author knows. In order to construct a formal transseries solution and prove its summability our main idea is to use the superintegrability of a Hamiltonian system in a class of transseries. More precisely we first show the superintegrability of a Hamiltonian system in the category of transseries via the key Lemmas 1 and 4 which follow. By virtue of the superintegrability we show the existence of a formal transseries solution. Then its summability is proved via the superintegrability. We note that the argument based on the superintegrability is elementary. [ABSTRACT FROM AUTHOR]

Dynamical systems subject to perturbations that decay over time are relevant in describing many physical models, e.g. when considering the effect of a laser pulse on a molecule, in epidemiological studies, and celestial mechanics. For this purpose, we consider time-dependent Hamiltonian vector fields that are the sum of two components. The first has an invariant torus supporting quasiperiodic solutions, and the second decays as time tends to infinity. The time decay is modelled by functions satisfying suitable conditions verified by a proper polynomial decay in time. We prove the existence of orbits converging as time tends to infinity to the quasiperiodic solutions associated with the unperturbed system. The proof of this result relies on a new strategy based on a refined analysis of the Banach spaces and the functionals involved in the resolution of suitable nonlinear invariant equations. This result is proved for finite differentiable and real-analytic Hamiltonians. Analogous statements for time-dependent vector fields on the torus are also obtained as a corollary. These results extend a previous work of Canadell and de la Llave, where only exponential decay in time is considered. The relaxation of the decay in time makes the results in the present paper suited for applications in many physical problems, such as celestial dynamics. [ABSTRACT FROM AUTHOR]

We investigate the existence of families of symmetric periodic solutions of second kind as continuation of the elliptical orbits of the two-dimensional Kepler problem for certain symmetric differentiable perturbations using Delaunay coordinates. More precisely, we characterize the sufficient conditions for its existence and its type of stability is studied. The estimate on the characteristic multipliers of the symmetric periodic solutions is the new contribution to the field of symmetric periodic solutions. In addition, we present some results about the relationship between our symmetric periodic solutions and those obtained by the averaging method for Hamiltonian systems. As applications of our main results, we get new families of periodic solutions for: the perturbed hydrogen atom with stark and quadratic Zeeman effect, for the anisotropic Seeligers two-body problem and to the planar generalized Størmer problem. [ABSTRACT FROM AUTHOR]

We construct integrable Hamiltonian systems with magnetic fields of the ellipsoidal, paraboloidal and conical type, i.e. systems that generalize natural Hamiltonians separating in the respective coordinate systems to include nonvanishing magnetic field. In the ellipsoidal and paraboloidal case each this classification results in three one–parameter families of systems, each involving one arbitrary function of a single variable and a parameter specifying the strength of the magnetic field of the given fully determined form. In the conical case the results are more involved, there are two one–parameter families like in the other cases and one class which is less restrictive and so far resists full classification. [ABSTRACT FROM AUTHOR]

We study the weak KAM theory for finite co-dimensional Hamiltonian with relative equilibrium. On the one hand, for the Tonelli (strictly convex, superlinear) finite co-dimensional Hamiltonian, we utilize Evans' approximate variational principle and show that the Hamilton-Jacobi equation with relative equilibrium has a weak KAM solution. It as a generating function can transform the original Hamiltonian system into an integrable one on the relative equilibrium. On the other hand, when the finite co-dimensional Hamiltonian is non-convex, we give a decomposition formula of the corresponding effective Hamiltonian. This formula consists of effective Hamiltonians of quasi-convex Hamiltonians. [ABSTRACT FROM AUTHOR]

The dynamic behavior of a physical system often originates from its spectral properties. In open systems, where the effective non-Hermitian description enables a wealth of spectral structures in the complex plane, the concomitant dynamics are significantly enriched, whereas the identification and comprehension of the underlying connections are challenging. Here we experimentally demonstrate the correspondence between the transient self-acceleration of local excitations and the non-Hermitian spectral topology using lossy photonic quantum walks. Focusing first on one-dimensional quantum walks, we show that the measured short-time acceleration of the wave function is proportional to the area enclosed by the eigenspectrum. We then reveal a similar correspondence in two-dimension quantum walks, where the self-acceleration is proportional to the volume enclosed by the eigenspectrum in the complex parameter space. In both dimensions, the transient self-acceleration crosses over to a long-time behavior dominated by a constant flow at the drift velocity. Our results unveil the universal correspondence between spectral topology and transient dynamics, and offer a sensitive probe for phenomena in non-Hermitian systems that originate from spectral geometry. The strong connection between the dynamics of a physical system and its Hamiltonian's spectrum has scarcely been applied in the non-Hermitian case. Here, the authors use a photonic quantum walk to confirm and expand previous theoretical analyses connecting self-acceleration dynamics with non-trivial point-gap topology. [ABSTRACT FROM AUTHOR]

To analyze and study the behaviour of the shallow water waves, the perturbed Boussinesq equation has acquired fundamental importance. The principal objective of this paper is to manifest the exact traveling wave solution of the perturbed Boussinesq equation by two well known techniques named as, two variables (G′G,1G) expansion method and generalized projective Riccati equations method. A diverse array of soliton solutions, encompassing periodic, bright solitons, singular solitons and bright singular solitons are obtained by the applications of proposed techniques. The constraint conditions for newly constructed solutions are also specified. To enhance comprehension, the numerical illustrations of constructed solutions have been represented using surface plots, 2D plots and density plots. The results delineated in this paper transcend existing analysis, offering a novel, well-structured, and modern perspective. The solutions obtained not only enrich understanding of shallow water wave models but also exhibit efficacy in providing detailed descriptions of their dynamics. [ABSTRACT FROM AUTHOR]

In this work, a singular left-definite Hamiltonian system is considered and the characteristic-matrix theory for this Hamiltonian system is constructed. Using the results of this theory we introduce a lower bound for the number of Dirichlet-integrable solutions. Moreover we share a relation between the kernel of the solution of the nonhomogeneous boundary value problem and the characteristic-matrix. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following elliptic system of Hamiltonian type on a bounded domain: (0.1) { − Δ u = K 1 (| y |) | v | p − 1 v , in B 1 (0) , − Δ v = K 2 (| y |) | u | q − 1 u , in B 1 (0) , u = v = 0 on ∂ B 1 (0) , where K 1 (r) and K 2 (r) are positive bounded functions, B 1 (0) is the unit ball in R N , (p , q) is a pair of positive numbers lying on the critical hyperbola: 1 p + 1 + 1 q + 1 = N − 2 N. We investigate the local uniqueness and the nondegeneracy of the bubbling solution for problem (0.1). Our proof is based on the local Pohozaev identities, blow-up analysis, and the properties of Greens function. Our results are quite different from single equation, which is mainly caused by the non-cooperative nonlinear terms and (p , q) lying on the critical hyperbola. We believe that the various new ideas and technique computations that we used in this paper would be very useful to deal with other related problems of Hamiltonian type with interact critical exponents. [ABSTRACT FROM AUTHOR]

We consider two-dimensional, non-relativistic quantum system with asymptotically straight soft waveguide. We show that the local deformation of the symmetric waveguide can lead to the emerging of the embedded eigenvalues living in the continuous spectrum. The main problem of this paper is devoted to the analysis of weak perturbation of the symmetric system. We show that the original embedded eigenvalues turn to the second sheet of the resolvent analytic continuation and constitute resonances. We describe the asymptotics of the real and imaginary components of the complex resonant pole depending on deformation. Finally, we generalize the problem to three dimensional system equipped with a soft layer. [ABSTRACT FROM AUTHOR]

The linearization of nonlinear differential equations represents a robust approach to solution derivation, typically achieved through Lie symmetry analysis. This study adopts a geometric methodology grounded in the Eisenhart lift, revealing transformative techniques that linearize a set of second-order ordinary differential equations. The research underscores the effectiveness of this geometric approach in the linearization of a class of Newtonian systems that cannot be linearized through symmetry analysis. [ABSTRACT FROM AUTHOR]

This paper extends a kinetic-potential energy shaping method to stochastic mechanical port-Hamiltonian systems. The kinetic-potential energy shaping brings a new class of Lyapunov function candidates involving a cross term of the position and the momentum without solving partial differential equations for deterministic port-Hamiltonian systems. However, the conventional kinetic-potential energy shaping does not necessarily work for stochastic port-Hamiltonian systems due to energy increase by stochastic noise. Therefore, we first provide a modification properly compensated by using stochastic generalized canonical transformations. Then, two stochastic stability results are presented. The first result shows a necessary condition for stochastic asymptotic stability for an equilibrium state. The other one shows a necessary condition for stochastic bounded stability for a target state, which is not necessarily an equilibrium point. [ABSTRACT FROM AUTHOR]

Quantum tunneling in a two-dimensional integrable map is studied. The orbits of the map are all confined to the curves specified by the one-dimensional Hamiltonian. It is found that the behavior of tunneling splitting for the integrable map and the associated Hamiltonian system is qualitatively the same, with only a slight difference in magnitude. However, the tunneling tails of the wave functions, obtained by superposing the eigenfunctions that form the doublet, exhibit significant differences. To explore the origin of the difference, we observe the classical dynamics in the complex plane and find that the existence of branch points appearing in the potential function of the integrable map could play the role of yielding non-trivial behavior in the tunneling tail. The result highlights the subtlety of quantum tunneling, which cannot be captured in nature only by the dynamics in the real plane. [ABSTRACT FROM AUTHOR]

We investigate how the phase space structure of a Three-Dimensional (3D) autonomous Hamiltonian system evolves across a series of successive Two-Dimensional (2D) and 3D pitchfork and period-doubling bifurcations, as the transition of the parent families of Periodic Orbits (POs) from stability to simple instability leads to the creation of new stable POs. Our research illustrates the consecutive alterations in the phase space structure near POs as the stability of the main family of POs changes. This process gives rise to new families of POs within the system, either maintaining the same or exhibiting higher multiplicity compared to their parent families. Tracking such a phase space transformation is challenging in a 3D system. By utilizing the color and rotation technique to visualize the Four-Dimensional (4D) Poincaré surfaces of section of the system, i.e. projecting them onto a 3D subspace and employing color to represent the fourth dimension, we can identify distinct structural patterns. Perturbations of parent and bifurcating stable POs result in the creation of tori characterized by a smooth color variation on their surface. Furthermore, perturbations of simple unstable parent POs beyond the bifurcation point, which lead to the birth of new stable families of POs, result in the formation of figure-8 structures of smooth color variations. These figure-8 formations surround well-shaped tori around the bifurcated stable POs, losing their well-defined forms for energies further away from the bifurcation point. We also observe that even slight perturbations of highly unstable POs create a cloud of mixed color points, which rapidly move away from the location of the PO. Our study introduces, for the first time, a systematic visualization of 4D surfaces of section within the vicinity of higher multiplicity POs. It elucidates how, in these cases, the coexistence of regular and chaotic orbits contributes to shaping the phase space landscape. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following perturbed fractional Hamiltonian systems { t D ∞ α (− ∞ D t α u (t)) + L (t) u (t) = ∇ u W (t , u (t)) + ∇ u G (t , u (t)) , t ∈ ℝ , u ∈ H α (ℝ , ℝ N) , where α ∈ (1 / 2 , 1 ] , L ∈ C (ℝ , ℝ N × N) is symmetric and not necessarily required to be positive definite, W ∈ C 1 (ℝ × ℝ N , ℝ) is locally subquadratic and locally even near the origin, and perturbed term G ∈ C 1 (ℝ × ℝ N , ℝ) maybe has no parity in u. Utilizing the perturbed method improved by the authors, a sequence of nontrivial homoclinic solutions is obtained, which generalizes previous results. [ABSTRACT FROM AUTHOR]

This paper presents a series expansion solution for the vibration question of an orthotropic rectangular thin plate (RTP) with four corners point-supported based on the superposition method and the separation variable method in the Hamiltonian system. First, the original vibration question is decomposed into two sub-vibration questions by analyzing the boundary conditions. After that, the separation variable method in the Hamiltonian system is used to calculate the general solutions for the two sub-vibration questions. Then, these two general solutions are superimposed to obtain the symplectic superposition solution for the original vibration question. Finally, the gained symplectic superposition solution is applied to calculate the vibration questions of isotropic and orthotropic RTPs with four corners point-supported, and the convergence analysis is conducted for the calculation of the orthotropic RTPs using the symplectic superposition solution. [ABSTRACT FROM AUTHOR]

Using high-precision technologies for studying Hamiltonian systems, it is shown that the symmetrical and planar approach of four C164 nanotori with zero spin energy ends in the formation of two kinematic pairs. These pairs are in an intermediate state of dynamic equilibrium until the inclinations of the tori own axes exceed their limiting values. Further, as a result of the connection of the two bases of the tori, a new state of dynamic equilibrium is formed, characterized by the absolute maximum of the van der Waals interaction. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of infinitely many periodic solutions for second-order perturbed Hamiltonian systems. By using variational methods, we obtain new criteria to guarantee the existence of infinitely many periodic solutions using Bolle's perturbation method. Our results improve upon those presented in the existing literature. [ABSTRACT FROM AUTHOR]

Lie–Poisson gauge formalism provides a semiclassical description of noncommutative U(1) gauge theory with Lie algebra type noncommutativity. Using the Dirac approach to constrained Hamiltonian systems, we focus on a class of Lie–Poisson gauge models, which exhibit an admissible Lagrangian description. The underlying noncommutativity is supposed to be purely spatial. Analyzing the constraints, we demonstrate that these models have as many physical degrees of freedom as there are present in the Maxwell theory. [ABSTRACT FROM AUTHOR]

In this work, we implement the bifurcation theory for the planar dynamical system and the complete discrimination system method of polynomials to the integrable Kuralay equations. Solitons generated by these equations are found to be relevant in diverse fields such as ferromagnetic materials, nonlinear optics, and optical fibers. Through specific wave transformations, it undergoes conversion into an ordinary differential equation (ODE), which is subsequently transformed into a planar dynamical system associated with a one-dimensional Hamiltonian function. As per the qualitative theory of the planar dynamical system, phase portraits of the Hamiltonian system are plotted and used to construct some new traveling wave solutions. Numerical examination reveals diverse nonlinear structures in the analytical solutions, encompassing solitary waves, kink waves, and periodic wave profiles. Additionally, the integrable Kuralay equation employs the complete discrimination system method of polynomial for the first time, yielding solutions expressed in trigonometric, exponential, hyperbolic, and Jacobi elliptic functions. Visual representations of the derived solutions include 3-D, 2-D, and contour plots. The reliability and effectiveness are affirmed through the numerical graphs of the solutions. Furthermore, we conducted a numerical investigation into the chaotic and quasiperiodic behavior of the perturbed system by introducing a specific periodic force into the primary system. [ABSTRACT FROM AUTHOR]

The Hamiltonian and hence the equations of motion involving the field operators of two quantum anharmonic oscillators coupled through the angular momentum are constructed. The anharmonic terms are such that the conservative nature of the Hamiltonian is retained. The coupled differential equations involving the field operators are nonlinear and are noncommuting as well. The exact closed form analytical solutions of the field operators are certainly a challenging task to address. The numerical solutions for the field operators are also unlikely because of the presence of nonlinearity and the noncommuting nature of the field operators. We provide the solutions of the field operators by using the short time approximation. The solutions are used to investigate the squeezing and the antibunching of photons for composite coherent light interacting with two oscillators coupled through the angular momentum. The second order variances for the canonically conjugate quadrature of both modes are exhibited as a function of dimensionless coupling and anharmonic terms. By considering the phase angles of composite coherent light as zero, we obtain the squeezing in one of the quadrature components X 1 at the cost of other quadrature P 1 when the frequency of the first oscillator is less than the second one. The squeezing pattern gets reversed when the frequency of the first oscillator is less than the frequency of the second oscillator. The squeezing is increased with the increase of dimensionless coupling time. The monotonic increase of the squeezing with the increase of the dimensionless coupling time is attributed to the shortcomings of short time approximation. However, the dimensionless nonlinear constant causes a decrease in the squeezing effects. The identical behaviour is exhibited for the second mode. The antibunching of photons for both modes is found improbable. Rather, we end up with the bunching of photons for both modes. [ABSTRACT FROM AUTHOR]

In this paper, we consider a discrete Hamiltonian system on nonnegative integers, and using Sylvester’s inertia indices theory, we construct maximal subspaces on which the Hermitian form has a certain sign. After constructing nested ellipsoids, we introduce a lower bound for the number of linearly independent summable-square solutions of the discrete equation. Finally, we provide a limit-point criterion. [ABSTRACT FROM AUTHOR]

High-resolution simulations of particle-based kinetic plasma models typically require a high number of particles and thus often become computationally intractable. This is exacerbated in multi-query simulations, where the problem depends on a set of parameters. In this work, we derive reduced order models for the semi-discrete Hamiltonian system resulting from a geometric particle-in-cell approximation of the parametric Vlasov–Poisson equations. Since the problem's nondissipative and highly nonlinear nature makes it reducible only locally in time, we adopt a nonlinear reduced basis approach where the reduced phase space evolves in time. This strategy allows a significant reduction in the number of simulated particles, but the evaluation of the nonlinear operators associated with the Vlasov–Poisson coupling remains computationally expensive. We propose a novel reduction of the nonlinear terms that combines adaptive parameter sampling and hyper-reduction techniques to address this. The proposed approach allows decoupling the operations having a cost dependent on the number of particles from those that depend on the instances of the required parameters. In particular, in each time step, the electric potential is approximated via dynamic mode decomposition (DMD) and the particle-to-grid map via a discrete empirical interpolation method (DEIM). These approximations are constructed from data obtained from a past temporal window at a few selected values of the parameters to guarantee a computationally efficient adaptation. The resulting DMD-DEIM reduced dynamical system retains the Hamiltonian structure of the full model, provides good approximations of the solution, and can be solved at a reduced computational cost. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the finite‐time stability (FTS) and H∞$$ {H}_{\infty } $$ control of nonlinear Hamiltonian systems with time‐varying state delays and output delays subject to input saturation. By introducing a new class of Lyapunov–Krasovskii (L‐K) functionals and the assumption about the Hamiltonian function, we present a delay‐dependent FTS criterion. Then, a finite‐time H∞$$ {H}_{\infty } $$ feedback control is designed for the nonlinear Hamiltonian systems. Finally, the validity of the proposed results is demonstrated by a numerical example. [ABSTRACT FROM AUTHOR]

We prove some multiplicity results for Neumann-type boundary value problems associated with a Hamiltonian system. Such a system can be seen as the weak coupling of two systems, the first of which has some periodicity properties in the Hamiltonian function, the second one presenting the existence of a well-ordered pair of lower/upper solutions. [ABSTRACT FROM AUTHOR]

We study the multiplicity of solutions for a two-point boundary value problem of Neumann type associated with a Hamiltonian system which couples a system with periodic Hamiltonian in the space variable with a second one with positively-(p, q)-homogeneous Hamiltonian. The periodic problem is also treated. [ABSTRACT FROM AUTHOR]

In this paper we study coisotropic reduction in different types of dynamics according to the geometry of the corresponding phase space. The relevance of coisotropic reduction is motivated by the fact that these dynamics can always be interpreted as Lagrangian or Legendrian submanifolds. Furthermore, Lagrangian or Legendrian submanifolds can be reduced by a coisotropic one. [ABSTRACT FROM AUTHOR]

This article addresses the finite‐time adaptive stabilization and finite‐time adaptive H∞$$ {H}_{\infty } $$ control problems of a class of nonlinear Hamiltonian singular systems with parametric perturbations. Under the suitable state feedback control scheme, a dissipative differential‐algebraic form of the nonlinear Hamiltonian singular system is obtained. According to this dissipative form, the finite‐time adaptive stabilization of the nonlinear Hamiltonian singular system with unknown parametric perturbations is investigated using the Hamiltonian function method. Then, considering the system with the external disturbances, a finite‐time adaptive H∞$$ {H}_{\infty } $$ controller is designed. Finally, the simulation results of the nonlinear singular circuit system demonstrate the efficacy of the controller designed in this article. [ABSTRACT FROM AUTHOR]

In the present study, the stability limit of the MILD regime of combustion has been studied in a reverse flow combustor using syngas (20% H2, 20% CO, 60% N2) as fuel. Pressure oscillations obtained from the combustor are analyzed using regular approaches from the dynamical systems theory as the oxidizer preheat temperature varies from 772 K to 424 K. Stable MILD combustion is obtained at preheat temperatures ≥724 K; at lower preheat temperatures combustion becomes unstable, leading to high amplitude periodic oscillations. Permutation spectrum test and recurrence plot of the MILD condition unravels the deterministic nature of the pressure oscillations. The correlation dimension confirms that the system has high-dimensional chaotic characteristics. At temperatures <724 K, the correlation dimension becomes close to 1 as the system transitions to combustion instability. In these conditions, the permutation spectrum test shows close to zero standard deviation in all ordinal indices, the phase-space shows doughnut-shaped structures, and recurrence plot consists of long diagonal lines, all confirming the periodic nature of the system. The dominating frequency during combustion instability is attributed to the second axisymmetric mode of the combustor. Finally, three recurrence quantification criteria, namely, Recurrence Rate, Determinism, and Laminarity, and K-value obtained from 0 to 1 test, are tested as indicators for the onset of combustion instability. Determinism and K-value are recommended for capturing the transition in the underlying dynamics. [ABSTRACT FROM AUTHOR]

The crease flow, replacing the Hamiltonian system used for the evolution of crease sets on black hole horizons, is introduced and its bifurcation properties for null hypersurfaces are discussed. We state the conditions of nondegeneracy and typicality for the crease submanifolds, and find their normal forms and versal unfoldings (codimension 3). The allowed boundary singularities are thus prescribed by the Arnold–Kazaryan–Shcherbak theorem for 3-parameter versal families, and hence identified as swallowtails and Whitney umbrellas of particular kinds. We further present the bifurcation diagrams describing crease evolution at the crossings of the bifurcation sets and elsewhere, and a typical example is studied. Some remarks on the connection of these results to the crease evolution on black hole horizons are also given. [ABSTRACT FROM AUTHOR]

Flows and rotation, particularly E × B rotation, are critical to improving plasma performance, and waves are a primary tool of plasma control. Thus, it is paramount to understand under what conditions waves can drive E × B flows in plasmas. In this didactic review, an invited paper accompanying the 2023 Marshall N. Rosenbluth Doctoral Thesis Award, this question is answered in the context of momentum-conserving quasilinear theory. There are two primary frameworks for momentum-conserving quasilinear theories that can handle both resonant and nonresonant particles: Eulerian averaging theories and oscillation-center Hamiltonian theories. There are also two different paradigmatic wave problems: plane-wave initial value problems, and steady-state boundary value problems. Here, it is shown that each of these frameworks "naturally" works better with a different problem type. By using these theories, one finds a great difference in the behavior of time- vs space-dependent waves. A time-evolving plane wave can only drive flow if the electromagnetic momentum of the wave, given by the Poynting flux, changes. This result precludes flow drive by any planar electrostatic wave. In contrast, a steady-state spatially evolving wave can drive flow whenever there is divergence in the flux of Minkowski momentum, a completely different physical quantity. This review aims to provide a high-level, intuitive understanding of the very different behaviors observed for these two types of problem. [ABSTRACT FROM AUTHOR]

In this work we study the splitting distance of a rapidly perturbed pendulum H(x, y, t) = 1/2y² + (cos(x) 1) + μ (cos(x) - 1)g(t\ω) with g(τ) = ∑|k| >1 g[k]eikτ a 2π-periodic function and μ, ω≪ 1. Systems of this kind undergo exponentially small splitting, and, when μ ≪ 1, it is known that the Melnikov function actually gives an asymptotic expression for the splitting function provided g[±1] ≠ 0. Our study focuses on the case g[±1] = 0, and it is motivated by two main reasons. On the one hand, our study is motivated by the general understanding of the splitting, as current results fail for a perturbation as simple as g(τ) = cos(5τ) + cos(4τ) + cos(3τ). On the other hand, a study of the splitting of invariant manifolds of tori of rational frequency p/q in Arnold's original model for diffusion leads to the consideration of pendulum-like Hamiltonians with g(τ) = sin(p·t\ω)+cos(q·t\ω), where, for most p, q ∈ Z, the perturbation satisfies g[±1] = 0. As expected, the Melnikov function is not a correct approximation for the splitting in this case. To tackle the problem we use a splitting formula based on the solutions of the so-called inner equation and make use of the Hamilton--Jacobi formalism. The leading exponentially small term appears at order μn, where n is an integer determined exclusively by the harmonics of the perturbation. We also provide an algorithm to compute it. [ABSTRACT FROM AUTHOR]

In prior work [Katsanikas & Wiggins, 2021a, 2021b, 2023c, 2023d], we introduced two methodologies for constructing Periodic Orbit Dividing Surfaces (PODS) tailored for Hamiltonian systems possessing three or more degrees of freedom. The initial approach, outlined in [Katsanikas & Wiggins, 2021a, 2023c], was applied to a quadratic Hamiltonian system in normal form having three degrees of freedom. Within this context, we provided a more intricate geometric characterization of this object within the family of 4D toratopes that describe the structure of the dividing surfaces discussed in these papers. Our analysis confirmed the nature of this construction as a dividing surface with the no-recrossing property. All these findings were derived from analytical results specific to the case of the Hamiltonian system discussed in these papers. In this paper, we extend our results for quartic Hamiltonian systems with three degrees of freedom. We prove for this class of Hamiltonian systems the no-recrossing property of the PODS and we investigate the structure of these surfaces. In addition, we compute and study the PODS in a coupled case of quartic Hamiltonian systems with three degrees of freedom. [ABSTRACT FROM AUTHOR]

We study in detail the dynamics of conformal Hamiltonian flows that are defined on a conformal symplectic manifold (this notion was popularized by Vaisman in 1976). We show that they exhibit some conservative and dissipative behaviours. We also build many examples of various dynamics that show simultaneously their difference and resemblance with the contact and symplectic case. [ABSTRACT FROM AUTHOR]

We study singular reduction of contact Hamiltonian systems acted upon properly by a Lie group. The tools we use are the category of differential space. [ABSTRACT FROM AUTHOR]

Recent progress towards understanding the mechanism of dynamical tunneling in Hamiltonian systems with three or more degrees of freedom (DoF) is reviewed. In contrast to systems with two degrees of freedom, the three or more degrees of freedom case presents several challenges. Specifically, in higher-dimensional phase spaces, multiple mechanisms for classical transport have significant implications for the evolution of initial quantum states. In this review, the importance of features on the Arnold web, a signature of systems with three or more DoF, to the mechanism of resonance-assisted tunneling is illustrated using select examples. These examples represent relevant models for phenomena such as intramolecular vibrational energy redistribution in isolated molecules and the dynamics of Bose–Einstein condensates trapped in optical lattices. [ABSTRACT FROM AUTHOR]

This paper presents a quantum formulation for classical abstract dynamical systems (ADS), defined by coupled sets of first-order differential equations. They are referred to as "abstract" because their dynamical variables can be of different interrelated natures, not necessarily corresponding to physics, such as populations, socioeconomic variables, behavioral variables, etc. A classical linear Hamiltonian can be derived for ADS by using Dirac's dynamics for singular Hamiltonian systems. Also, a corresponding first-order Schrödinger equation (which involves the existence of a system Planck constant particular of each system) can be derived from this Hamiltonian. However, Madelung's reinterpretation of quantum mechanics, followed by the Bohm and Hiley work, produces no further information about the mathematical formulation of ADS. However, a classical quadratic Hamiltonian can also be derived for ADS, as well as a corresponding second-order Schrödinger equation. In this case, the Madelung reinterpretation of quantum mechanics provides a quantum Hamiltonian that does provide the quantum formulation for ADS, which provides new quantum variables interrelated dynamically with the classical variables. An application case is presented: the one-dimensional autonomous system given by the logistic dynamics. The differences between the classical and the quantum trajectories are highlighted in the context of this application case. [ABSTRACT FROM AUTHOR]

When confronted with the imminent threat of predation, the prey instinctively employ strategies to avoid being consumed. These anti-predator tactics involve individuals acting collectively to intimidate predators and reduce potential harm during an attack. In the present work, we propose a state-dependent feedback control predator-prey model that incorporates a nonmonotonic functional response, taking into account the anti-predator behavior observed in pest-natural enemy ecosystems within the agricultural context. The qualitative analysis of this model is presented utilizing the principles of impulsive semi-dynamical systems. Firstly, the stability conditions of the equilibria are derived by employing pertinent properties of planar systems. The precise domain of the impulsive set and phase set is determined by considering the phase portrait of the system. Secondly, a Poincaré map is constructed by utilizing the sequence of impulsive points within the phase set. The stability of the order-1 periodic solution at the boundary is subsequently analyzed by an analog of the Poincaré criterion. Additionally, this article presents various threshold conditions that determine both the existence and stability of an order-1 periodic solution. Furthermore, it investigates the existence of order-k ( k ≥ 2 ) periodic solutions. Finally, the article explores the complex dynamics of the model, encompassing multiple bifurcation phenomena and chaos, through computational simulations. [ABSTRACT FROM AUTHOR]

Estimating governing equations from observed time-series data is crucial for understanding dynamical systems. From the perspective of system comprehension, the demand for accurate estimation and interpretable results has been particularly emphasized. Herein, we propose a novel data-driven method for estimating the governing equations of dynamical systems based on machine learning with high accuracy and interpretability. The proposed method enhances the estimation accuracy for dynamical systems using sparse modeling by incorporating physical constraints derived from Hamiltonian mechanics. Unlike conventional approaches used for estimating governing equations for dynamical systems, we employ a sparse representation of Hamiltonian, allowing for the estimation. Using noisy observational data, the proposed method demonstrates a capability to achieve accurate parameter estimation and extraction of essential nonlinear terms. In addition, it is shown that estimations based on energy conservation principles exhibit superior accuracy in long-term predictions. These results collectively indicate that the proposed method accurately estimates dynamical systems while maintaining interpretability. [ABSTRACT FROM AUTHOR]