Homoclinic orbits in the dynamics of resonantly driven coupled pendula

We present a plethora of homoclinic and heteroclinic orbits that exist in the phase space of a weakly nonlinear model describing small oscillations of two resonantly driven coupled pendula. These orbits connect equilibria in a resonance band, which is contained in an invariant plane, and is born under perturbation out of a circle of equilibria. Their existence is shown by using a combination of the Melnikov method and singular perturbation techniques.     

Continuation of relative periodic orbits in a class of triatomic Hamiltonian systems

We study relative periodic orbits (i.e. time-periodic orbits in a frame rotating at constant velocity) in a class of triatomic Euclidean-invariant (planar) Hamiltonian systems. The system consists of two identical heavy atoms and a light one, and the atomic mass ratio is treated as a continuation parameter. Under some nondegeneracy conditions, we show that a given family of relative periodic orbits existing at infinite mass ratio (and parametrized by phase, rotational degree of freedom and period) persists for sufficiently large mass ratio and for nearby angular velocities (this result is valid for small angular velocities). The proof is based on a method initially introduced by Sepulchre and MacKay [J.-A. Sepulchre, R.S. MacKay, Localized oscillations in conservative or dissipative networks of weakly coupled autonomous oscillators, Nonlinearity 10 (1997) 679–713] and further developed by Muñoz-Almaraz et al. [F.J. Muñoz-Almaraz, et al., Continuation of periodic orbits in conservative and Hamiltonian systems, Physica D 181 (2003) 1–38] for the continuation of normal periodic orbits in Hamiltonian systems. Our results provide several types of relative periodic orbits, which extend from small amplitude relative normal modes [J.-P. Ortega, Relative normal modes for nonlinear Hamiltonian systems, Proc. Roy. Soc. Edinburgh Sect. A 133 (2003) 665–704] up to large amplitude solutions which are not restrained to a small neighborhood of a stable relative equilibrium. In particular, we show the existence of large amplitude motions of inversion, where the light atom periodically crosses the segment between heavy atoms. This analysis is completed by numerical results on the stability and bifurcations of some inversion orbits as their angular velocity is varied.     

Homotopic Arnold tongues deformation of the MHD α2-dynamo

We consider a mean–field α₂–dynamo with helical turbulence parameter α(r)=α₀+γΔα(r) and a boundary homotopy with parameter β∈[0,1] interpolating between Dirichlet (idealized, β=0) and Robin (physically realistic, β=1) boundary conditions. It is shown that the zones of oscillatory solutions at β=1 end up at the diabolical points for β=0 under the homotopic deformation. The underlying network of the diabolical points for β=0 substantially determines the choreography of eigenvalues and thus the character of the dynamo instability for β=1. Using perturbation theory we derive the first–order approximations to the resonance (Arnold's) tongues in the α₀βγ-space, which turn out to be cones in the vicinity of the diabolical points, selected by the Fourier coefficients of Δα(r). The space orientation of the 3D tongues is determined by the Krein signature of the modes involved in the diabolical crossings at the apexes of the cones. The Krein space induced geometry of the resonance zones explains the subtleties in finding α-profiles leading to oscillatory dynamos, and it explicitly predicts the locations of the spectral exceptional points, which are important ingredients in the recent theories of polarity reversals of the geomagnetic field. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

An Asymptotic Analysis of a 2-D Model of Dynamically Active Compartments Coupled by Bulk Diffusion

A class of coupled cell–bulk ODE–PDE models is formulated and analyzed in a two-dimensional domain, which is relevant to studying quorum-sensing behavior on thin substrates. In this model, spatially segregated dynamically active signaling cells of a common small radius \(\epsilon \ll 1\) are coupled through a passive bulk diffusion field. For this coupled system, the method of matched asymptotic expansions is used to construct steady-state solutions and to formulate a spectral problem that characterizes the linear stability properties of the steady-state solutions, with the aim of predicting whether temporal oscillations can be triggered by the cell–bulk coupling. Phase diagrams in parameter space where such collective oscillations can occur, as obtained from our linear stability analysis, are illustrated for two specific choices of the intracellular kinetics. In the limit of very large bulk diffusion, it is shown that solutions to the ODE–PDE cell–bulk system can be approximated by a finite-dimensional dynamical system. This limiting system is studied both analytically, using a linear stability analysis and, globally, using numerical bifurcation software. For one illustrative example of the theory, it is shown that when the number of cells exceeds some critical number, i.e., when a quorum is attained, the passive bulk diffusion field can trigger oscillations through a Hopf bifurcation that would otherwise not occur without the coupling. Moreover, for two specific models for the intracellular dynamics, we show that there are rather wide regions in parameter space where these triggered oscillations are synchronous in nature. Unless the bulk diffusivity is asymptotically large, it is shown that a diffusion-sensing behavior is possible whereby more clustered spatial configurations of cells inside the domain lead to larger regions in parameter space where synchronous collective oscillations between the small cells can occur. Finally, the linear stability analysis for these cell–bulk models is shown to be qualitatively rather similar to the linear stability analysis of localized spot patterns for activator–inhibitor reaction–diffusion systems in the limit of long-range inhibition and short-range activation.     

On the orbital stability of planar periodic motions of a rigid body in the Bobylev-Steklov case

We deal with the problem of orbital stability of pendulum-like periodic motions of a heavy rigid body with a fixed point. We suppose that a mass geometry corresponds to the Bobylev-Steklov case. The stability problem is solved in nonlinear setting. In the case of small amplitude oscillations and rotations with large angular velocities the small parameter can be introduced and the problem can be investigated analytically. In the case of unspecified oscillation amplitude or rotational angular velocity the problem is reduced to analysis of stability of a fixed point of the symplectic map generated by the equations of the perturbed motion. The coefficients of the symplectic map are determined numerically. Rigorous results on the orbital stability or instability of unperturbed motion are obtained by analyzing these coefficients.     

Border-collision bifurcations on a two-dimensional torus

This paper studies resonance phenomena in a piecewise-smooth dynamical system with external periodic action and examines transitions to chaos via border-collision bifurcations of cycles on a two-dimensional torus. As an example we consider a control system with pulse-width modulation described by a three-dimensional set of piecewise-linear non-autonomous equations. It is shown that the domains of synchronization of quasiperiodic oscillations for piecewise-smooth dynamical systems differ in an essential way from the classical Arnol'd tongues. The difference lies in the inner structure and bifurcational transitions. There are two different kinds of synchronization domains, one of which contains regions of bistability. The structure of border-collision bifurcation boundaries of synchronization tongues and transitions to chaos via border-collision bifurcations of cycles on a two-dimensional torus are described in detail.     

Nonlinear Mathieu equation and coupled resonance mechanism

The method of multiple-scales is used to determine a third-order solution for a cubic nonlinear Mathieu equation. The perturbation solutions are imposed on the so-called solvability conditions. Solvability conditions in the non-resonance case yield the standard Landau equation. Several types of a parametric Landau equation are derived in the neighborhood of five different resonance cases. These parametric Landau equations contain a parametric complex conjugate term or a parametric second-order complex conjugate term or a parametric complex conjugate term as well as a parametric second-order term. Necessary and sufficient conditions for stability are performed in each resonance case. Stability criteria correspond to each parametric Landau equation and are derived by linear perturbation. Stability criteria for the non-trivial steady-state response are discussed. The analysis leads to simultaneous resonance. Transition curves are performed in each case. Numerical calculations are made for some transition curves to illustrate the coupled resonance regions, where the induced stability tongues within the instability tongues are observed. The amplitude of the periodic coefficient of Mathieu equation plays a dual role in the stability criteria for nonlinear Mathieu equation.     

Replicate periodic windows in the parameter space of driven oscillators

In the bi-dimensional parameter space of driven oscillators, shrimp-shaped periodic windows are immersed in chaotic regions. For two of these oscillators, namely, Duffing and Josephson junction, we show that a weak harmonic perturbation replicates these periodic windows giving rise to parameter regions correspondent to periodic orbits. The new windows are composed of parameters whose periodic orbits have the same periodicity and pattern of stable and unstable periodic orbits already existent for the unperturbed oscillator. Moreover, these unstable periodic orbits are embedded in chaotic attractors in phase space regions where the new stable orbits are identified. Thus, the observed periodic window replication is an effective oscillator control process, once chaotic orbits are replaced by regular ones.     

Stability condition for vertical oscillation of 3-dim heavy spring elastic pendulum

Equations of motion for 3-dim heavy spring elastic pendulum are derived and rescaled to contain a single parameter. Condition for the stability of vertical large amplitude oscillations is derived analytically relating the parameter of the system and the amplitude of the vertical oscillation. Numerical continuation is used to find the border of the stability region in parameter space with high precision. The stability condition is approximated by a simple formula valid for a large range of the parameter and of the amplitude of oscillation. The bifurcation responsible for the loss of stability is identified.      

Bifurcation continuation,chaos and chaos control in nonlinear Bloch system

A detailed analysis is undertaken to explore the stability and bifurcation pattern of the nonlinear Bloch equation known to govern the dynamics of an ensemble of spins, controlling the basic process of nuclear magnetic resonance. After the initial analysis of the parameter space and stability region identification, we utilize the MATCONT package to analyze the detailed bifurcation scenario as the two important physical parameters γ (the normalized gain) and c (the phase of the feedback field) are varied. A variety of patterns are revealed not studied ever before. Next we explore the structure of the chaotic attractor and how the identification of unstable periodic orbit (UPO) can be utilized to control the onset of chaos.     

On a measure of the closeness of neutral systems to internal resonance

For certain classes of parametrically perturbed resonance systems that are neutral in a linear approximation, a quantitative characteristic is introduced for the closeness of the system of resonance: the magnitude of the critical detuning value for resonance δ* at which the change in stability occurs as the system withdraws from resonance. The problem of finding this critical value is made complicated by the non-linear nature of the change in stability in neutral systems. It is solved below for third-order resonances in a situation that guarantees the passage of instability into asymptotic stability as the system withdraws from resonance.

Knowledge of the quantity δ* enables the strong instability domain /1, 2/ in parameter space to be estimated, enables the danger of resonance to be characterized, and enables the structural parameter in the system, the shift of the resonance phases, to be clarified, whose variation would enable the danger of resonance to be increased or reduced.     

Passive aerodynamics control of plasma instabilities

The possibility of using a smart-damping scheme to modify the dynamic responses of plasma oscillations governed by a two-fluid model is considered. The passive aerodynamics control strategy is used to address this issue. The control efficiency is found by analyzing the conditions satisfied by the control gain parameters for which, the amplitude of oscillations is reduced both in the harmonic and chaotic states. In the regular state, the analytical stability analysis uses for linear oscillations the Routh-Hurwitz criterion while the Whittaker method and Floquet theory are utilized for nonlinear harmonic oscillations. The stability boundaries in the control gain parameter space is derived. The agreement between the analytical and numerical results is good. In the chaotic states, numerical simulations are used to perform quenching of chaotic oscillations for an appropriate set of control parameters.     

Approximation of vectors fields by thin plate splines with tension

We study a vectorial approximation problem based on thin plate splines with tension involving two positive parameters: one for the control of the oscillations and the other for the control of the divergence and rotational components of the field. The existence and uniqueness of the solution are proved and the solution is explicitly given. As special cases, we study the limit problems as the parameter controlling the divergence and the rotation converges to zero or infinity. The divergence-free and the rotation-free approximation problems are also considered. The convergence in Sobolev space is studied.     

Post-Double Hopf Bifurcation Dynamics and Adaptive Synchronization of a Hyperchaotic System

In this paper a four-dimensional hyperchaotic system with only one equilibrium is considered and its double Hopf bifurcations are investigated. The general post-bifurcation and stability analysis are carried out using the normal form of the system obtained via the method of multiple scales. The dynamics of the orbits predicted through the normal form comprises possible regimes of periodic solutions, two-period tori, and three-period tori in parameter space. Moreover, we show how the hyperchaotic synchronization of this system can be realized via an adaptive control scheme. Numerical simulations are included to show the effectiveness of the designed control.     

A mechanical system containing weakly coupled subsystems

The concept of a mechanical system (model), containing coupled subsystems (MSCCS) is introduced. Examples of such a system are the Sun–planets–satellites system, a system of interacting moving objects, a system of translationally and rotationally moving celestial bodies, chains of coupled oscillators, Sommerfeld pendulums, spring systems, etc. The MSCCS subsystems and the entire system are analysed, and problems related to the investigation of the oscillations, bifurcation, stability, stabilization and resonance are stated. A solution of the oscillations problem is given for a class of MSCCS, described by reversible mechanical systems. It is proved that the autonomous MSCCS retains its family of symmetrical periodic motions (SPMs) under parametric perturbations, while in the periodic MSCCS a family of SPMs bifurcates by producing two families of SPMs. The two-body problem and the N-planet problem are investigated as applications. The generating properties of the two-body problem are established. For the N-planet problem it is proved that an (N + 1)-parametric family of orbits exists, close to elliptic orbits of arbitrary eccentricity, the family being parametrized by energy integral constant, and a syzygy of planets occurs.     

Heteroclinic Transition Motions in Periodic Perturbations of Conservative Systems with an Application to Forced Rigid Body Dynamics

We consider periodic perturbations of conservative systems. The unperturbed systems are assumed to have two nonhyperbolic equilibria connected by a heteroclinic orbit on each level set of conservative quantities. These equilibria construct two normally hyperbolic invariant manifolds in the unperturbed phase space, and by invariant manifold theory there exist two normally hyperbolic, locally invariant manifolds in the perturbed phase space. We extend Melnikov’s method to give a condition under which the stable and unstable manifolds of these locally invariant manifolds intersect transversely. Moreover, when the locally invariant manifolds consist of nonhyperbolic periodic orbits, we show that there can exist heteroclinic orbits connecting periodic orbits near the unperturbed equilibria on distinct level sets. This behavior can occur even when the two unperturbed equilibria on each level set coincide and have a homoclinic orbit. In addition, it yields transition motions between neighborhoods of very distant periodic orbits, which are similar to Arnold diffusion for three or more degree of freedom Hamiltonian systems possessing a sequence of heteroclinic orbits to invariant tori, if there exists a sequence of heteroclinic orbits connecting periodic orbits successively.We illustrate our theory for rotational motions of a periodically forced rigid body. Numerical computations to support the theoretical results are also given.     

The calculation of resonance separatrices for the near-integrable case of the standard mapping

The standard mapping arises in many physical applications, including the analysis of nonlinear resonant acoustic oscillations in a closed tube. A perturbation expansion, in powers of the amplitude parameter, is given for the calculation of the fixed points of various orders and the associated separatrices. It is shown how exact homoclinic orbits can be calculated numerically. Explicit analytic expressions are given for the separatrices associated with the first four resonances when the perturbation parameter is small.     

On the boundaries of the parametric resonance domain

The problem of stability for a system of linear differential equations with coefficients which are periodic in time and depend on the parameters is considered. The singularities of the general position arising at the boundaries of the stability and instability (parametric resonance) domains in the case of two and three parameters are listed. A constructive approach is proposed which enables one, in the first approximation, to determine the stability domain in the neighbourhood of a point of the boundary (regular or singular) from the information at this point. This approach enables one to eliminate a tedious numerical analysis of the stability region in the neighbourhood of the boundary point and can be employed to construct the boundaries of parametric resonance domains. As an example, the problem of the stability of the oscillations of an articulated pipe conveying fluid with a pulsating velocity is considered. In the space of three parameters (the average fluid velocity and the amplitude and frequency of pulsations) a singularity of the boundary of the stability domain of the “dihedral angle” type is obtained and the tangential cone to the stability domain is calculated.     

Parametric resonance in the problem of a heavy rigid body rolling along a straight line on a plane

The problem of the stability of a heavy rigid body, bounded by the surface of an ellipsoid and with a cavity in the form of a coaxial ellipsoid, rolling along a straight line on a horizontal rough plane is investigated. It is shown that in the case of a body that is close to being dynamically symmetrical, parametric resonance always occurs leading to instability of the rolling. Each ellipsoid has its own “individual” resonance angular velocity. In the general case, regions in which the necessary stability conditions are satisfied can be distinguished in parameter space. The problem of calculating the resonance coefficient corresponding to instability for parametric resonance in a reversible third-order system is solved.     

A qualitative analysis of the subharmonic oscillations of a parametrically excited flexible rod

A non-autonomous non-linear dynamical system with a small parameter that describes the parametric oscillations of a flexible rod with three static equilibrium positions is obtained. The generating equation of this model is a dynamical system in a plane with a separatrix loop. The qualitative analysis presented includes an investigation of the stability and bifurcation of subharmonic motions at resonance energy levels.