Dynamical Systems Techniques Research Articles

A numerical technique for variable-order fractional functional nonlinear dynamic systems

This paper provides an efficient technique to discretize the variable-order fractional functional nonlinear differential equations. The proposed technique is based on a piecewise integro quadratic spline interpolation and finite difference approximation. To reveal the performance and accuracy of the proposed method, the behavioral responses of the suitcase with two wheels and pantograph models with variable-order fractional order are analyzed.

Global dynamics of non-smooth Filippov pest-natural enemy system with constant releasing rate.

Modelling integrated pest management (IPM) with a threshold control strategy can be achieved with a non-smooth Filippov dynamical system coupled by an untreated subsystem and a treated subsystem which includes chemical and biological tactics. The releasing constant of natural enemies related to biological control generates the complex dynamics. Comprehensive qualitative analyses reveal that the treated subsystem exists with transcritical, saddle-node, Hopf and Bogdanov-Takens bifurcations, for which the threshold conditions and bifurcation curves are provided. Further, by applying techniques of non-smooth dynamical systems including the Filippov convex method and sliding bifurcation techniques, we first obtain the sliding dynamic equation, and then we analyze the existence and stability of regular/virtual equilibria, pseudo-equilibria, boundary equilibria, sliding segments and sliding bifurcations. In particular, if we choose the economic threshold (ET) as the bifurcation parameter, then interesting dynamical behaviors, including boundary equilibrium → pseudo-homoclinic → touching → buckling → crossing bifurcations, occur in succession. It is interesting to note that although the number of pests in the untreated subsystem could increase and exceed the economic injury level (EIL), the final size could be less than ET and stabilizes at a relative low level due to side effects of the pesticide on natural enemies. However, the side effects can be effectively avoided by increasing the releasing constant, which can maintain the number of pests below the EIL always and thus achieve the control purpose.

Cosmological dynamics of brane gravity: A global dynamical system perspective

The braneworld model of gravity is well-known for several notable cosmological features such as self-acceleration originating from a geometric and not matter source, effective dark energy behavior with phantom characteristics but not leading to a Big-Rip singularity, rough resemblance to the $\Lambda$CDM evolution, etc. The dynamical system tools usually allow us to obtain generic conclusions on the global dynamics of a system over a wide range of initial conditions. With this motivation, in order to recover the important features of the braneworld model from a more global perspective, here, we investigate the global cosmological dynamics of the braneworld model using dynamical system techniques. We first analyze the case where there is just a normal matter on the brane and then extend the analysis to the case with an extra scalar field also trapped on the brane. In the presence of a scalar field, potentials belonging to different classes are considered. The stability behavior of critical points is examined using linear stability analysis and when necessary center manifold theory as well as numerical perturbation techniques are also used. To understand the global dynamics of a dynamical system, we utilized the Poincar\'e compactification method to capture the properties of all possible critical points. Applying dynamical system analysis, we found that brane gravity is consistent with observed actions of the Universe. In particular, our analysis shows that important cosmological behaviors like the long-lasting matter-dominated era, late time acceleration as well as the avoidance of Big-Rip singularity can be realized in brane gravity for a wide range of initial conditions.

An Error Bound for Low Order Approximation of Dynamical Systems Subjected to Initial Conditions

In recent years, a great effort has been taken focused on the development of reduced order modeling techniques of dynamical systems. This necessity is pushed by the requirement for efficient numerical techniques for simulations of dynamical systems arising from structural dynamics, controller design, circuit simulation, fluid dynamics and micro electromechanical systems.We introduce a method to calculate the minimum upper $\mathcal{L}_2$ error bound of a linear time invaritant reduced order model considering any possible unitary initial conditions (IC). As a consequence, the proposed method calculates the unitary IC vector which leads to the maximum $\mathcal{L}_2$ norm of the error. Based on this error bound, it is discussed the capacity of a reduced order system to approximate the free transient response in the worst case scenario.

An Error Bound for Low Order Approximation of Dynamical Systems Subjected to Initial Conditions

In recent years, a great effort has been taken focused on the development of reduced order modeling techniques of dynamical systems. This necessity is pushed by the requirement for efficient numerical techniques for simulations of dynamical systems arising from structural dynamics, controller design, circuit simulation, fluid dynamics and micro electromechanical systems. We introduce a method to calculate the minimum upper $\mathcal{L}_2$ error bound of a linear time invaritant reduced order model considering any possible unitary initial conditions (IC). As a consequence, the proposed method calculates the unitary IC vector which leads to the maximum $\mathcal{L}_2$ norm of the error. Based on this error bound, it is discussed the capacity of a reduced order system to approximate the free transient response in the worst case scenario.

Standard map-like models for single and multiple walkers in an annular cavity

Recent experiments on walking droplets in an annular cavity showed the existence of complex dynamics including chaotically changing velocity. This article presents models, influenced by the kicked rotator/standard map, for both single and multiple droplets. The models are shown to achieve both qualitative and quantitative agreement with the experiments, and make predictions about heretofore unobserved behavior. Using dynamical systems techniques and bifurcation theory, the single droplet model is analyzed to prove dynamics suggested by the numerical simulations.

Interpreting the synchronisation of driven colloidal oscillators via the mean pair interaction

To explore the generic features of synchronisation facilitated by hydrodynamic interactions dominated by viscous forces, we investigate small systems of rowers; rowers are a highly simplified model for motile cilia, with each cilium approximated by a rigid sphere driven by a geometrically updated force. We introduce a new framework to analyse rowers, in which we average the pair-wise interaction of rowers by converting to the natural phase. In doing so the function describing the interaction becomes continuous, dramatically simplifying the system and allowing standard dynamical system techniques to be applied. Through inspection of phase portraits, we capture the broad features of rowers driven by power law forces and quantify their coupling strength. This approach is not limited to power law potentials, and can be applied to any monotonically increasing function. When implemented in systems with more rowers, the phase portraits show more diverse behaviour. By exploring the phase space, small systems of rowers can be designed to demonstrate specific features.

Generalized matter couplings in general relativity

A new class of modified theory of gravity is introduced where the volume form becomes dynamical. This approach is motivated by unimodular gravity and can also be related to Brans-Dicke theory. On the level of the action, the only change made will be through the volume element which is used in the integration. This is achieved by the introduction of a fourth order tensor which connects the spacetime metric to the new volume form. Using dynamical systems techniques, this model is studied in the context of cosmology. The most interesting result is that there exist parameter ranges where this model starts undergoing an epoch of accelerated expansion, followed by a decelerating expansion which evolves to a final epoch of accelerated expansion.

Enduring Lagrangian coherence of a Loop Current ring assessed using independent observations

Ocean flows are routinely inferred from low-resolution satellite altimetry measurements of sea surface height assuming a geostrophic balance. Recent nonlinear dynamical systems techniques have revealed that surface currents derived from altimetry can support mesoscale eddies with material boundaries that do not filament for many months, thereby representing effective transport mechanisms. However, the long-range Lagrangian coherence assessed for mesoscale eddy boundaries detected from altimetry is constrained by the impossibility of current altimeters to resolve ageostrophic submesoscale motions. These may act to prevent Lagrangian coherence from manifesting in the rigorous form described by the nonlinear dynamical systems theories. Here we use a combination of satellite ocean color and surface drifter trajectory data, rarely available simultaneously over an extended period of time, to provide observational evidence for the enduring Lagrangian coherence of a Loop Current ring detected from altimetry. We also seek indications of this behavior in the flow produced by a data-assimilative system which demonstrated ability to reproduce observed relative dispersion statistics down into the marginally submesoscale range. However, the simulated flow, total surface and subsurface or subsampled emulating altimetry, is not found to support the long-lasting Lagrangian coherence that characterizes the observed ring. This highlights the importance of the Lagrangian metrics produced by the nonlinear dynamical systems tools employed here in assessing model performance.

Nonlinear dynamics of displaced non-Keplerian orbits with low-thrust propulsion

This paper discusses the stability, transition and control of displaced non-Keplerian orbits by the spacecraft using low-thrust propulsion. The two-body dynamical model developed in the polar coordinates is parameterized by the thrust pitch angle, and then two of the hyperbolic and elliptic equilibria are solved from it. The bounded motions near two equilibria are investigated by dynamical system techniques to find out all the stable and unstable periodic trajectories, and two scenarios of the resonant periodic trajectory are presented. Regardless of the thrust pitch angle, all the transit orbits are numerically demonstrated to be restricted inside the invariant manifolds of Lyapunov orbit near the hyperbolic equilibrium. Then the transit orbits can be distinguished from non-transit ones by the restriction of three-dimensional invariant manifolds projected onto the Poincaré section or position space. Based on the influence of thrust direction on the system topology, operating the thrust pitch angle is an effective tool to achieve the transfer within different types of KAM tori, or even transfer beyond the KAM tori.

Reduced order surrogate modeling technique for linear dynamic systems

The availability of reduced order models can greatly decrease the computational costs needed for modeling, identification and design of real-world structural systems. However, since these systems are usually employed with some uncertain parameters, the approximant must provide a good accuracy for a range of stochastic parameters variations. The derivation of such reduced order models are addressed in this paper. The proposed method consists of a polynomial chaos expansion (PCE)-based state-space model together with a PCE-based modal dominancy analysis to reduce the model order. To solve the issue of spatial aliasing during mode tracking step, a new correlation metric is utilized. The performance of the presented method is validated through four illustrative benchmarks: a simple mass-spring system with four Degrees Of Freedom (DOF), a 2-DOF system exhibiting a mode veering phenomenon, a 6-DOF system with large parameter space and a cantilever Timoshenko beam resembling large-scale systems.

Bouncing universe of entropy-corrected Friedmann equations

In this paper, we investigate the possibility of obtaining bouncing-oscillating solution in modified Friedmann equations with logarithmic entropy corrected, S=frac{A}{4G}+alpha ln frac{A}{4G}+beta frac{4G}{A}, for positive, negative and zero values of (alpha ,beta ) pre-factors and all kinds of curved universes. The results are argued using the dynamical system techniques and by employing the phase plane analysis for full classification of the nonsingular evolutions. Our analysis indicates that it is possible to have an oscillating universe as well as a bounce universe for k=1 and k=- 1 curvatures. In k=1 case, both positive and negative values of alpha and beta can make bouncing-oscillating solution, while in k=- 1 case, only the positive value of alpha with negative value of beta can make a bounce. Also the flat universe have no bounce solution.

Buffer System Control for Hybrid Plants

The use of multimodal control techniques for batch dynamic systems offers a promising alternative to successfully perform tasks of supervision and control in industrial processes. These techniques seek to integrate different control strategies with partial objectives or "behaviors" using Lebesgue automatons overlooking achieve certain operational objectives. These automatons can identify optimal sequences of modes, also called control programs, using system simulations. Multimodal control programs consist of a sequence of modes, each of which comprises a feedback control law□(x) and relevant termination conditions□(x,T). In this paper the use of a reinforcement learning algorithm is described via to find, by simulations, optimal control program aligned to maximize productivity in a buffer system composed of two tanks.

Low-thrust displaced orbits by weak Hamiltonian-Structure-Preserving control

For the requirements from some missions, the spacecraft is placed near the unstable region around the hyperbolic equilibrium. To stabilize the motion, a novel concept of the weak Hamiltonian-Structure-Preserving (HSP) control parameterized by the Coriolis acceleration is introduced. Based on the dynamical model of low-thrust displaced orbits, this paper searches for the mechanism in orbital stability change by the weak HSP control and provides specific strategies on how the control is implemented. Since the weak control has no effect on the topology of the original system, two invariant equilibria, a hyperbolic one and an elliptic one are solved. Dynamical system techniques are employed to investigate the controlled motions near the two equilibria, illustrating the Lyapunov orbit in different Coriolis acceleration cases and presenting two basic periodic modes measuring the bounded trajectories in the interior region. Controlled trajectories in the neck region are analyzed for their characteristics, classification and transition to make preparation for the stability change. One of the important contributions of this paper is to numerically demonstrate that the weak HSP control preserves the homoclinic orbits to the hyperbolic equilibrium as well as to the Lyapunov orbit, and another is to achieve the orbital transfer within, or even beyond KAM tori based on two methods in determining whether the controlled trajectory integrated from an initial point can exhibit transit.

Is uniform persistence a robust property in almost periodic models? A well-behaved family: almost-periodic Nicholson systems

Using techniques of non-autonomous dynamical systems, we completely characterize the persistence properties of an almost periodic Nicholson system in terms of some numerically computable exponents. Although similar results hold for a class of cooperative and sublinear models, in the general non-autonomous setting one has to consider persistence as a collective property of the family of systems over the hull: the reason is that uniform persistence is not a robust property in models given by almost periodic differential equations.

Dark energy with a gradient coupling to the dark matter fluid: cosmological dynamics and structure formation

We consider scalar field models of dark energy interacting with dark matter through a coupling proportional to the contraction of the four-derivative of the scalar field with the four-velocity of the dark matter fluid. The coupling is realized at the Lagrangian level employing the formalism of Scalar-Fluid theories, which use a consistent Lagrangian approach for relativistic fluid to describe dark matter. This framework produces fully covariant field equations, from which we can derive unequivocal cosmological equations at both background and linear perturbations levels. The background evolution is analyzed in detail applying dynamical systems techniques, which allow us to find the complete asymptotic behavior of the universe given any set of model parameters and initial conditions. Furthermore we study linear cosmological perturbations investigating the growth of cosmic structures within the quasi-static approximation. We find that these interacting dark energy models give rise to interesting phenomenological dynamics, including late-time transitions from dark matter to dark energy domination, matter and accelerated scaling solutions and dynamical crossing of the phantom barrier. Moreover we obtain possible deviations from standard ΛCDM behavior at the linear perturbations level, which have an impact on the dynamics of structure formation and might provide characteristic observational signatures.

Trajectory design for a cislunar CubeSat leveraging dynamical systems techniques: The Lunar IceCube mission

Lunar IceCube is a 6U CubeSat that is designed to detect and observe lunar volatiles from a highly inclined orbit. This spacecraft, equipped with a low-thrust engine, is expected to be deployed from the upcoming Exploration Mission-1 vehicle. However, significant uncertainty in the deployment conditions for secondary payloads impacts both the availability and geometry of transfers that deliver the spacecraft to the lunar vicinity. A framework that leverages dynamical systems techniques is applied to a recently updated set of deployment conditions and spacecraft parameter values for the Lunar IceCube mission, demonstrating the capability for rapid trajectory design.

Time-dependent propagators for stochastic models of gene expression: an analytical method

The inherent stochasticity of gene expression in the context of regulatory networks profoundly influences the dynamics of the involved species. Mathematically speaking, the propagators which describe the evolution of such networks in time are typically defined as solutions of the corresponding chemical master equation (CME). However, it is not possible in general to obtain exact solutions to the CME in closed form, which is due largely to its high dimensionality. In the present article, we propose an analytical method for the efficient approximation of these propagators. We illustrate our method on the basis of two categories of stochastic models for gene expression that have been discussed in the literature. The requisite procedure consists of three steps: a probability-generating function is introduced which transforms the CME into (a system of) partial differential equations (PDEs); application of the method of characteristics then yields (a system of) ordinary differential equations (ODEs) which can be solved using dynamical systems techniques, giving closed-form expressions for the generating function; finally, propagator probabilities can be reconstructed numerically from these expressions via the Cauchy integral formula. The resulting ‘library’ of propagators lends itself naturally to implementation in a Bayesian parameter inference scheme, and can be generalised systematically to related categories of stochastic models beyond the ones considered here.

Traveling Fronts of a Real Supercritical Ginzburg–Landau Equation Coupled by a Slow Diffusion

Ginzburg–Landau equation plays a very important role in a wide range of physical contexts. Much attention had been given to explore the dynamics of the waves in Ginzburg–Landau equations as well as its various coupled reaction–diffusion systems. In this article, dynamical system techniques such as the methods of geometric singular perturbation theory and Melnikov function are combined to show the persistence (existence) and bifurcations of traveling fronts in a real supercritical Ginzburg–Landau equation coupled by a slow diffusion mode. It is shown that heteroclinic saddle-node bifurcation occurs under certain parametric values such that 1, 2 and even 3 heteroclinic solutions may exist. The critical manifolds of the parameters corresponding to the heteroclinic saddle-node bifurcations are explicitly determined.

Traveling wave solutions for the dispersive models describing population dynamics of Aedes aegypti

In recent decades the global incidence of dengue has grown dramatically by increased human mobility and urbanization. The study of the mosquitoes population is of great importance for public health in countries where climatic and environmental conditions are favorable for the propagation of this disease. Therefore, this work is based on the study of mathematical models dealing with the life cycle of the mosquito using partial differential equations. We investigate the existence of traveling wave solutions using semi-analytical method combining dynamical systems techniques and numerical integration. Obtained solutions are validated through direct numerical simulations using finite difference schemes. We also present initial study concerning structural stability of traveling wave solution.