Complex dynamics in physical pendulum equation with suspension axis vibrations

The physical pendulum equation with suspension axis vibrations is investigated. By using Melnikov＇s method, we prove the conditions for the existence of chaos under periodic perturbations. By using second-order averaging method and Melinikov＇s method, we give the conditions for the existence of chaos in an averaged system under quasi-periodic perturbations for Ω = nω ＋ εv, n = 1 - 4, where ν is not rational to ω. We are not able to prove the existence of chaos for n = 5 - 15, but show the chaotic behavior for n = 5 by numerical simulation. By numerical simulation we check on our theoretical analysis and further exhibit the complex dynamical behavior, including the bifurcation and reverse bifurcation from period-one to period-two orbits; the onset of chaos, the entire chaotic region without periodic windows, chaotic regions with complex periodic windows or with complex quasi-periodic windows; chaotic behaviors suddenly disappearing, or converting to period-one orbit which means that the system can be stabilized to periodic motion by adjusting bifurcation parameters α, δ, f0 and Ω; and the onset of invariant torus or quasi-periodic behaviors, the entire invariant torus region or quasi-periodic region without periodic window, quasi-periodic behaviors or invariant torus behaviors suddenly disappearing or converting to periodic orbit; and the jumping behaviors which including from period- one orbit to anther period-one orbit, from quasi-periodic set to another quasi-periodic set; and the interleaving occurrence of chaotic behaviors and invariant torus behaviors or quasi-periodic behaviors; and the interior crisis; and the symmetry breaking of period-one orbit; and the different nice chaotic attractors. However, we haven＇t find the cascades of period-doubling bifurcations under the quasi-periodic perturbations and show the differences of dynamical behaviors and technics of research between the periodic perturbations and quasi-periodic perturbations.     

Complex dynamics in Duffing system with two external forcings

Duffing's equation with two external forcing terms have been discussed. The threshold values of chaotic motion under the periodic and quasi-periodic perturbations are obtained by using second-order averaging method and Melnikov's method. Numerical simulations not only show the consistence with the theoretical analysis but also exhibit the interesting bifurcation diagrams and the more new complex dynamical behaviors, including period-n (n=2,3,6,8) orbits, cascades of period-doubling and reverse period doubling bifurcations, quasi-periodic orbit, period windows, bubble from period-one to period-two, onset of chaos, hopping behavior of chaos, transient chaos, chaotic attractors and strange non-chaotic attractor, crisis which depends on the frequencies, amplitudes and damping. In particular, the second frequency plays a very important role for dynamics of the system, and the system can leave chaotic region to periodic motions by adjusting some parameter which can be considered as an control strategy of chaos. The computation of Lyapunov exponents confirm the dynamical behaviors.     

Bifurcations and Chaos in Duffing Equation

The Duffing equation with even-odd asymmetrical nonlinear-restoring force and one external forcingis investigated.The conditions of existence of primary resonance,second-order,third-order subharmonics,m-order subharmonics and chaos are given by using the second-averaging method,the Melnikov method andbifurcation theory.Numerical simulations including bifurcation diagram,bifurcation surfaces and phase portraitsshow the consistence with the theoretical analysis.The numerical results also exhibit new dynamical behaviorsincluding onset of chaos,chaos suddenly disappearing to periodic orbit,cascades of inverse period-doublingbifurcations,period-doubling bifurcation,symmetry period-doubling bifurcations of period-3 orbit,symmetry-breaking of periodic orbits,interleaving occurrence of chaotic behaviors and period-one orbit,a great abundanceof periodic windows in transient chaotic regions with interior crises and boundary crisis and varied chaoticattractors.Our results show that many dynamical behaviors are strictly departure from the behaviors of theDuffing equation with odd-nonlinear restoring force.     

Complex dynamics in Duffing–Van der Pol equation

Duffing–Van der Pol equation with fifth nonlinear-restoring force and two external forcing terms is investigated. The threshold values of existence of chaotic motion are obtained under the periodic perturbation. By second-order averaging method and Melnikov method, we prove the criterion of existence of chaos in averaged system under quasi-periodic perturbation for ω₂ = nω₁ + εσ, n = 1, 3, 5, and cannot prove the criterion of existence of chaos in second-order averaged system under quasi-periodic perturbation for ω₂ = nω₁ + εσ, n = 2, 4, 6, 7, 8, 9, 10, where σ is not rational to ω₁, but can show the occurrence of chaos in original system by numerical simulation. Numerical simulations including heteroclinic and homoclinic bifurcation surfaces, bifurcation diagrams, Lyapunov exponent, phase portraits and Poincaré map, not only show the consistence with the theoretical analysis but also exhibit the more new complex dynamical behaviors. We show that cascades of interlocking period-doubling and reverse period-doubling bifurcations from period-2 to -4 and -6 orbits, interleaving occurrence of chaotic behaviors and quasi-periodic orbits, transient chaos with a great abundance of period windows, symmetry-breaking of periodic orbits in chaotic regions, onset of chaos which occurs more than one, chaos suddenly disappearing to period orbits, interior crisis, strange non-chaotic attractor, non-attracting chaotic set and nice chaotic attractors. Our results show many dynamical behaviors and some of them are strictly departure from the behaviors of Duffing–Van der Pol equation with a cubic nonlinear-restoring force and one external forcing.     

Quasi-periodic Solutions of Completely Resonant Wave Equations with Quasi-periodically forced Vibrations

In this paper, we provide the existence of quasi-periodic solutions with two frequencies for a class of completely resonant nonlinear wave equations with quasi-periodically forced vibrations under the spatial periodic boundary conditions. We consider the frequencies vector (ω ₁,ω ₂) near the linear system. The proofs are based on the Variational Lyapunov-Schmidt reduction and Linking Theorem.     

Bifurcations of periodic solutions and chaos in Duffing-van der Pol equation with one external forcing

The Duffing-Van der Pol equation withfifth nonlinear-restoring force and one external forcing term isinvestigated in detail: the existence and bifurcations of harmonicand second-order subharmonic, and third-order subharmonic,third-order superharmonic and $m$-order subharmonic under smallperturbations are obtained by using second-order averaging methodand subharmonic Melnikov function; the threshold values of existenceof chaotic motion are obtained by using Melnikov method. Thenumerical simulation results including the influences of periodicand quasi-periodic and all parameters exhibit more new complexdynamical behaviors. We show that the reverse period-doublingbifurcation to chaos, period-doubling bifurcation to chaos,quasi-periodic orbits route to chaos, onset of chaos, and chaossuddenly disappearing, and chaos suddenly converting to periodorbits, different chaotic regions with a great abundance of periodicwindows (periods:1,2,3,4,5,7,9,10,13,15,17,19,21,25,29,31,37,41, andso on), and more wide period-one window, and varied chaoticattractors including small size and maximum Lyapunov exponentapproximate to zero but positive, and the symmetry-breaking ofperiodic orbits. In particular, the system can leave chaotic regionto periodic motion by adjusting the parameters $p, \beta, \gamma, f$and $\omega$, which can be considered as a control strategy.     

Unfolding the torus: Oscillator geometry from time delays

Summary We present a simple method of plotting the trajectories of systems of weakly coupled oscillators. Our algorithm uses the time delays between the “firings” of the oscillators. For any system ofn weakly coupled oscillators there is an attracting invariantn-dimensional torus, and the attractor is a subset of this invariant torus. The invariant torus intersects a suitable codimension-1 surface of section at an (n−1)-dimensional torus. The dynamics ofn coupled oscillators can thus be reduced,in principle, to the study of Poincaré maps of the (n−1)-dimensional torus. This paper gives apractical algorithm for measuring then−1 angles on the torus. Since visualization of 3 (or higher) dimensional data is difficult we concentrate onn=3 oscillators. For three oscillators, a standard projection of the Poincaré map onto the plane yields a projection of the 2-torus which is 4-to-1 over most of the torus, making it difficult to observe the structure of the attractor. Our algorithm allows a direct measurement of the 2 angles on the torus, so we can plot a 1-to-1 map from the invariant torus to the “unfolded torus” where opposite edges of a square are identified. In the cases where the attractor is a torus knot, the knot type of the attractor is obvious in our projection.     

The evolution of a novel four-dimensional autonomous system: Among 3-torus, limit cycle, 2-torus, chaos and hyperchaos

In this paper, a novel four-dimensional autonomous system in which each equation contains a quadratic cross-product term is constructed. It exhibits extremely rich dynamical behaviors, including 3-tori (triple tori), 2-tori (quasi-periodic), limit cycles (periodic), chaotic and hyperchaotic attractors. In particular, we observe 3-torus phenomena, which have been rarely reported in four-dimensional autonomous systems in previous work. With the parameter r varying in quite a wide range, the evolution process of the system begins from 3-tori, and after going through a series of periodic, quasi-periodic and chaotic attractors in so many different shapes coming into being alternately, it evolves into hyperchaos, finally it degenerates to periodic attractor. Moreover, when the system is hyperchaotic, its two positive Lyapunov exponents are much larger than those of the hyperchaotic systems already reported, especially the largest Lyapunov exponents. We also observe a chaotic attractor of a very special shape. The complex dynamical behaviors of the system are further investigated by means of Lyapunov exponents spectrum, bifurcation diagram and phase portraits.     

Sectional-Anosov flows

We define sectional-Anosov flow as a vector field on a manifold, inwardly transverse to the boundary, whose maximal invariant set is sectional-hyperbolic (Metzger and Morales in Ergodic Theory Dyn Syst 28:1587–1597, 2008). We obtain properties of sectional-Anosov flows without null-homotopic periodic orbits on compact irreducible 3-manifolds including: incompressibility of transverse torus, non-existence of genus 0 transverse surfaces nor hyperbolic attractors nor hyperbolic repellers and sufficient conditions for the existence of singularities non-isolated in the nonwandering set. These generalize some known facts about Anosov flows.     

More Denjoy minimal sets for area preserving diffeomorphisms

For an area preserving, monotone twist diffeomorphism and an irrational number ω, we prove that if there is no invariant circle of angular rotation number ω, then there are uncountably many Denjoy minimal sets of angular rotation number ω. For each pair of positive integersn andR we prove that the space (with the vague topology) of Denjoy minimal sets of angular rotation number ω and intrinsic rotation number (ω+R)/n (mod. 1) contains a disk of dimensionn−1. Partially supported by NSF contract #MCS82-01604.     

Complex dynamics in a discrete-time predator-prey system without Allee effect

In this paper, complex dynamics of the discrete-time predator-prey system without Allee effect are investigated in detail. Conditions of the existence for flip bifurcation and Hopf bifurcation are derived by using center manifold theorem and bifurcation theory and checked up by numerical simulations. Chaos, in the sense of Marotto, is also proved by both analytical and numerical methods. Numerical simulations included bifurcation diagrams, Lyapunov exponents, phase portraits, fractal dimensions display new and richer dynamics behaviors. More specifically, this paper presents the finding of period-one orbit, period-three orbits, and chaos in the sense of Marotto, complete period-doubling bifurcation and invariant circle leading to chaos with a great abundance period-windows, simultaneous occurrance of two different routes (invariant circle and inverse period- doubling bifurcation, and period-doubling bifurcation and inverse period-doubling bifurcation) to chaos for a given bifurcation parameter, period doubling bifurcation with period-three orbits to chaos, suddenly appearing or disappearing chaos, different kind of interior crisis, nice chaotic attractors, coexisting (2,3,4) chaotic sets, non-attracting chaotic set, and so on, in the discrete-time predator-prey system. Combining the existing results in the current literature with the new results reported in this paper, a more complete understanding is given of the discrete-time predator-prey systems with Allee effect and without Allee effect.     

Bifurcations and chaos of a discrete mathematical model for respiratory process in bacterial culture

The discrete mathematical model for the respiratory process in bacterial culture obtained by Euler method is investigated. The conditions of existence for flip bifurcation and Hopf bifurcation are derived by using center manifold theorem and bifurcation theory, condition of existence of chaos in the sense of Marotto＇s definition of chaos is proved. The bifurcation diagrams, Lyapunov exponents and phase portraits are given for different parameters of the model, and the fractal dimension of chaotic attractor was also calculated. The numerical simulation results confirm the theoretical analysis and also display the new and complex dynamical behaviors compared with the continuous model. In particular~ we found that the new chaotic attractor, and new types of two or four coexisting chaotic attractors, and two coexisting invariant torus.     

Complex dynamic behaviors of a discrete-time predator-prey system

The dynamics of a discrete-time predator-prey system is investigated in detail in this paper. It is shown that the system undergoes flip bifurcation and Hopf bifurcation by using center manifold theorem and bifurcation theory. Furthermore, Marotto''s chaos is proved when some certain conditions are satisfied. Numerical simulations are presented not only to illustrate our results with the theoretical analysis, but also to exhibit the complex dynamical behaviors, such as the period-6, 7, 8, 10, 14, 18, 24, 36, 50 orbits, attracting invariant cycles, quasi-periodic orbits, nice chaotic behaviors which appear and disappear suddenly, coexisting chaotic attractors, etc. These results reveal far richer dynamics of the discrete-time predator-prey system. Specifically, we have stabilized the chaotic orbits at an unstable fixed point using the feedback control method.     

Composite dynamical behaviors in a simple series–parallel LC circuit

In this paper, we report a variety of dynamical behaviors exhibited in a compact series–parallel LC circuit system comprising of two active elements, one linear negative conductance and one ordinary junction diode with piecewise linear v ? i characteristics. For convenience, we consider the amplitude (E_f) and frequency (f) of the driving force as control parameters amongst various other parameters. We observe the phenomenon of antimonotonicity, torus breakdown to chaos, bubbles to chaos, period doubling to chaos and emergence of multiple attractors which follow a progressive sequence, etc. As an overview to understand many more variety of bifurcations and attractors, the construction of two parameter phase diagram is also shown pictorially. The chaotic dynamics of this circuit is realized by laboratory experiment, numerical and analytical investigations and found that the results are in good agreement with each other.     

CR Runge Sets on Hypersurface Graphs

This work contains an improvement of earlier results of Boggess and Dwilewicz regarding global approximation of CR functions by entire functions in the case of hypersurface graphs. In this work, we show that if ω, an open subset of a real hypersurface in ℂ ⁿ , can be graphed over a convex subset in ℝ²ⁿ⁻¹, then ω is CR-Runge in the sense that continuous CR functions on ω can be approximated by entire functions on ℂ ⁿ in the compact open topology of ω. Examples are presented to show that this approximation result does not hold for graphed CR submanifolds in higher codimension. R. Dwilewicz is partially supported by the Polish Science Foundation (KBN) grant N201 019 32/805.     

Recursive events in random sequences

Let ω be a Kolmogorov–Chaitin random sequence with ω_{1: n} denoting the first n digits of ω. Let P be a recursive predicate defined on all finite binary strings such that the Lebesgue measure of the set {ω|∃nP(ω_{1: n} )} is a computable real α. Roughly, P holds with computable probability for a random infinite sequence. Then there is an algorithm which on input indices for any such P and α finds an n such that P holds within the first n digits of ω or not in ω at all. We apply the result to the halting probability Ω and show that various generalizations of the result fail. Received: 1 December 1998 / Published online: 3 October 2001     

Random attractors of stochastic non-newtonian fluids

The present paper investigates the asymptotic behavior of solutions for stochastic non-Newtonian fluids in a two-dimensional domain. Firstly, we prove the existence of random attractors ΛH(ω) in H; Secondly, we prove the existence of random attractors ΛV (ω) in V . Then we verify regularity of the random attractors by showing that ΛH(ω) = ΛV (ω), which implies the smoothing effect of the fluids in the sense that solution becomes eventually more regular than the initial data.     

A theory for n-dimensional nonlinear dynamics on continuous vector fields

This paper systematically presents a theory for n-dimensional nonlinear dynamics on continuous vector fields. In this paper, a different view to look into the fundamental theory in dynamics is presented. The ideas presented herein are less formal and rigorous in an informal and lively manner. The ideas may give some inspirations in the field of nonlinear dynamics. The concepts of local and global flows are introduced to interpret the complexity of flows in nonlinear dynamic systems. Further, the global tangency and transversality of flows to the separatrix surface in nonlinear dynamical systems are discussed, and the corresponding necessary and sufficient conditions for such global tangency and transversality are presented. The ε-domains of flows in phase space are introduced from the first integral manifold surface. The domain of chaos in nonlinear dynamic systems is also defined, and such a domain is called a chaotic layer or band. The first integral quantity increment is introduced as an important quantity. Based on different reference surfaces, all possible expressions for the first integral quantity increment are given. The stability of equilibriums and periodic flows in nonlinear dynamical systems are discussed through the first integral quantity increment. Compared to the Lyapunov stability conditions, the weak stability conditions for equilibriums and periodic flows are developed. The criteria for resonances in the stochastic and resonant, chaotic layers are developed via the first integral quantity increment. To discuss the complexity of flows in nonlinear dynamical systems, the first integral manifold surface is used as a reference surface to develop the mapping structures of periodic and chaotic flows. The invariant set fragmentation caused by the grazing bifurcation is discussed. The global grazing bifurcation is a key to determine the global transversality to the separatrix. The local grazing bifurcation on the first integral manifold surface in a single domain without separatrix is a mechanism for the transition from one resonant periodic flow to another one. Such a transition may occur through chaos. The global grazing bifurcation on the separatrix surface may imply global chaos. The complexity of the global chaos is measured by invariant sets on the separatrix surface. The invariant set fragmentation of strange attractors on the separatrix surface is central to investigate the complexity of the global chaotic flows in nonlinear dynamical systems. Finally, the theory developed herein is applied to perturbed nonlinear Hamiltonian systems as an example. The global tangency and tranversality of the perturbed Hamiltonian are presented. The first integral quantity increment (or energy increment) for 2n-dimensional perturbed nonlinear Hamiltonian systems is developed. Such an energy increment is used to develop the iterative mapping relation for chaos and periodic motions in nonlinear Hamiltonian systems. Especially, the first integral quantity increment (or energy increment) for two-dimensional perturbed nonlinear Hamiltonian systems is derived, and from the energy increment, the Melnikov function is obtained under a certain perturbation approximation. Because of applying the perturbation approximation, the Melnikov function only can be used for a rough estimate of the energy increment. Such a function cannot be used to determine the global tangency and transversality to the separatrix surface. The global tangency and transversality to the separatrix surface only can be determined by the corresponding necessary and sufficient conditions rather than the first integral quantity increment. Using the first integral quantity increment, limit cycles in two-dimensional nonlinear systems is discussed briefly. The first integral quantity of any n-dimensional nonlinear dynamical system is very crucial to investigate the corresponding nonlinear dynamics. The theory presented in this paper needs to be further developed and to be treated more rigorously in mathematics.     

A scenario for torus T destruction via a global bifurcation

We show a scenario of a two-frequency torus breakdown, in which a global bifurcation occurs due to the collision of a quasi-periodic torus T² with saddle points, creating a heteroclinic saddle connection. We analyze the geometry of this torus-saddle collision by showing the local dynamics and the invariant manifolds (global dynamics) of the saddle points. Moreover, we present detailed evidences of a heteroclinic saddle-focus orbit responsible for the type-II intermittency induced by this global bifurcation. We also characterize this transition to chaos by measuring the Lyapunov exponents and the scaling laws.