Sequence Of Period-doubling Bifurcations Research Articles

Period-doubling behavior in frontal polymerization of multifunctional acrylates.

Front dynamics in the frontal polymerization of two multifunctional acrylate monomers, 1,6-hexanediol diacrylate (HDDA) and trimethylolpropane ethoxylate triacrylate (TMPTA), with Lupersol 231 [1,1-di(t-butylperoxy)-3,3,5-trimethylcyclohexane] as the initiator, are studied. In most frontal polymerization systems, the dynamics are associated with a planar front propagating through the sample. However, in some cases, front behavior can be altered: the front becomes nonplanar characterized by complex patterns like spin modes and pulsations. To determine how these periodic and aperiodic modes arise, reactant solutions consisting of HDDA diluted with diethyl phthalate (DEP) and TMPTA diluted with dimethyl sulfoxide (DMSO) were used in the study. In the study we reveal frontal behavior characteristic of period-doubling behavior, a doubling of spin heads that degenerate into an apparently chaotic mode. Also, a pulsating symmetric mode has been observed. These observations have a striking similarity to observations made in studies of self-propagating high-temperature synthesis (SHS) in which the addition of an inert diluent afforded a rich variety of dynamical behavior. The degree of cross-linking has also been found to be a bifurcation parameter. The energy of activation of multifunctional acrylate polymerization is a strong function of the degree of polymerization. By adding a monoacrylate (benzyl acrylate: BzAc), such that the front temperature was invariant, we observed a period-doubling bifurcation sequence through changes in the energy of activation, which has not been previously reported. (c) 1999 American Institute of Physics.

Numerical study of reverse period doubling route from chaos to stability in a two-mode intracavity doubled Nd-YAG laser.

We have numerically studied the behavior of a two-mode Nd-YAG laser with an intracavity KTP crystal. It is found that when the parameter, which is a measure of the relative orientations of the KTP crystal with respect to the Nd-YAG crystal, is varied continuously, the output intensity fluctuations change from chaotic to stable behavior through a sequence of reverse period doubling bifurcations. The graph of the intensity in the X-polarized mode against that in the Y-polarized mode shows a complex pattern in the chaotic regime. The Lyapunov exponent is calculated for the chaotic and periodic regions. (c) 1999 American Institute of Physics.

Chaos in Relativity and Cosmology

Chaos appears in various problems of Relativity and Cosmology. Here we discuss (a) the Mixmaster Universe model, and (b) the motions around two fixed black holes, (a) The Mixmaster equations have a general solution (i.e. a solution depending on 6 arbitrary constants) of Painleve type, but there is a second general solution which is not Painleve. Thus the system does not pass the Painleve test, and cannot be integrable. The Mixmaster model is not ergodic and does not have any periodic orbits. This is due to the fact that the sum of the three variables of the system (α + β + γ) has only one maximum for τ = τ m and decreases continuously for larger and for smaller τ. The various Kasner periods increase exponentially for large τ. Thus the Lyapunov Characteristic Number (LCN) is zero. The “finite time LCN” is positive for finite τ and tends to zero when τ → ∞. Chaos is introduced mainly near the maximum of (α + β + γ). No appreciable chaos is introduced at the successive Kasner periods, or eras. We conclude that in the Belinskii-Khalatnikov time, τ, the Mixmaster model has the basic characteristics of a chaotic scattering problem, (b) In the case of two fixed black holes M 1 and M 2 the orbits of photons are separated into three types: orbits falling into M 1 (type I), or M 2 (type II), or escaping to infinity (type III). Chaos appears because between any two orbits of different types there are orbits of the third type. This is a typical chaotic scattering problem. The various types of orbits are separated by orbits asymptotic to 3 simple unstable orbits. In the case of particles of nonzero rest mass we have intervals where some periodic orbits are stable. Near such orbits we have order. The transition from order to chaos is made through an infinite sequence of period doubling bifurcations. The bifurcation ratio is the same as in classical conservative systems.

Chaos in Relativity and Cosmology

AbstractChaos appears in various problems of Relativity and Cosmology. Here we discuss (a) the Mixmaster Universe model, and (b) the motions around two fixed black holes, (a) The Mixmaster equations have a general solution (i.e. a solution depending on 6 arbitrary constants) of Painlevé type, but there is a second general solution which is not Painlevé. Thus the system does not pass the Painlevé test, and cannot be integrable. The Mixmaster model is not ergodic and does not have any periodic orbits. This is due to the fact that the sum of the three variables of the system (α + β + γ) has only one maximum for τ = τm and decreases continuously for larger and for smaller τ. The various Kasner periods increase exponentially for large τ. Thus the Lyapunov Characteristic Number (LCN) is zero. The “finite time LCN” is positive for finite τ and tends to zero when τ → ∞. Chaos is introduced mainly near the maximum of (α + β + γ). No appreciable chaos is introduced at the successive Kasner periods, or eras. We conclude that in the Belinskii-Khalatnikov time, τ, the Mixmaster model has the basic characteristics of a chaotic scattering problem, (b) In the case of two fixed black holes M1 and M2 the orbits of photons are separated into three types: orbits falling into M1 (type I), or M2 (type II), or escaping to infinity (type III). Chaos appears because between any two orbits of different types there are orbits of the third type. This is a typical chaotic scattering problem. The various types of orbits are separated by orbits asymptotic to 3 simple unstable orbits. In the case of particles of nonzero rest mass we have intervals where some periodic orbits are stable. Near such orbits we have order. The transition from order to chaos is made through an infinite sequence of period doubling bifurcations. The bifurcation ratio is the same as in classical conservative systems.

A simpler derivation of Feigenbaum’s renormalization group equation for the period-doubling bifurcation sequence

One interesting and important property of nonlinear dynamical systems is that they can exhibit universality—behavior that is quantitatively identical for a broad class of systems. The first and most famous example of universality in a dynamical system was identified by Feigenbaum [M. J. Feigenbaum, J. Stat. Phys. 19, 25–52 (1978), 21, 669–706 (1979)] in the period-doubling route to chaos. This note presents a new derivation of Feigenbaum’s renormalization group equation, used to understand this universality. The argument, designed for incorporation into an undergraduate dynamical systems course, is simpler than those in standard textbooks.

Bifurcations and transitions to chaos in an inverted pendulum

We consider a parametrically forced pendulum with a vertically oscillating suspension point. It is well known that, as the amplitude of the vertical oscillation is increased, its inverted state (corresponding to the vertically-up configuration) undergoes a cascade of ``resurrections,'' i.e., it becomes stabilized after its instability, destabilize again, and so forth ad infinitum. We make a detailed numerical investigation of the bifurcations associated with such resurrections of the inverted pendulum by varying the amplitude and frequency of the vertical oscillation. It is found that the inverted state stabilizes via alternating ``reverse'' subcritical pitchfork and period-doubling bifurcations, while it destabilizes via alternating ``normal'' supercritical period-doubling and pitchfork bifrucations. An infinite sequence of period-doubling bifurcations, leading to chaos, follows each destabilization of the inverted state. The critical behaviors in the period-doubling cascades are also discussed.

Universal similarities of transition to chaos of satellite oscillation in elliptic orbit

For the problem on oscillation about mass centre of a satellite moving in the elliptical orbit in the central force field, the scaling functions have been built, which describe the satellite phase trajectory evolution at transition to chaos. Four period-doubling bifurcation sequences are considered, generated both from the stable and unstable steady states. The scaling universal constants for conservative systems are obtained, the multiplicator universal value for the satellite periodical unstable motion around its centroid at an accumulation point on the threshold of chaos is determined as well.

Chaos in ferroresonant wound voltage transformers: effect of core losses and universal circuit behaviour

The damping effects of transformer core losses upon the ferroresonant behaviour of wound instrument transformers, fed from a sinusoidal supply through circuit breaker grading capacitance, are studied using nonlinear dynamical methods. The system is shown to exhibit three types of ferroresonance; fundamental frequency ferroresonance, subharmonic ferroresonance and chaotic ferroresonance. It is also shown that this ferroresonant circuit can be driven into chaotic behaviour through a sequence of period-doubling bifurcations. Values of Feigenbaum universal number δ are calculated and are shown to be in good agreement with those reported from various other disciplines.

Nonlinear cycle dynamics in lean spark ignition combustion

The goal of this study was to experimentally investigate the onset of lean combustion instability in spark ignition engines. New data analysis techniques from nonlinear dynamics and chaos theory were employed to reveal previously unrecognized patterns in cycle-resolved measurements of combustion heat release. The results indicated a transition from stochastic behavior to noisy nonlinear determinism as equivalence ratio was decreased from stoichiometric to very lean conditions. The transition to nonlinear deterministic behavior appeared to occur via a period-doubling bifurcation sequence. Experimentally observed patterns were compared with patterns predicted by a recently proposed engine model. The comparison supported the hypothesis that the combustion instability develops as a noisy period-doubling bifurcation. Experimental results from a single-cylinder research engine and a production eight-cylinder engine showed similar trends in cycle dynamics under lean conditions. The similar behavior of these very different engines suggests that the basic phenomena involved may be common in spark ignition engines. The observation of nonlinear determinism under lean conditions may have important implications for engine diagnostics and control because cyclic dispersion under very lean conditions is not a purely random process.

Delay-induced chaos in catalytic surface reactions

Deterministic chaos related to a sequence of period-doubling bifurcations (the Feigenbaum transition) has been observed in the NO+CO and NO+H2 reactions on Pt(100). On a microscopic scale, these reactions are accompanied by the formation of 1×1 adsorbate islands due to the properties of the 1×1↔hex phase transition of Pt(100). A simple skeleton model is constructed that describes the behavior of a population of reacting islands which are globally coupled together via the gas phase. Investigations of this model show that the experimentally observed chaotic behavior can result from delays in the response of the reacting islands to partial pressure variations in the gas phase.

Some special features of the transition to chaos in the self-modulation of surface spin waves

The results of experimental investigations of the transition from regular to stochastic self-modulation of intense surface spin waves are presented. It is found that the transition to chaos follows the scenario of a sequence of period-doubling bifurcations. The fractal dimensions and the Kolmogorov entropy are determined for different regimes. The experiments are performed on an apparatus consisting of a microwave oscillator with a spin-wave delay line in the feedback circuit.

Numerical modeling of self-propagating polymerization fronts: The role of kinetics on front stability.

Frontal propagation of a highly exothermic polymerization reaction in a liquid is studied with the goal of developing a mathematical model of the process. As a model case we consider monomers such as methacrylic acid and n-butyl acrylate with peroxide initiators, although the model is not limited to these reactants and can be applied to any system with the similar basic polymerization mechanism. A three-step reaction mechanism, including initiation, propagation and termination steps, as well as a more simple one-step mechanism, were considered. For the one-step mechanism the loss of stability of propagating front was observed as a sequence of period doubling bifurcations of the front velocity. It was shown that the one-step model cannot account for less than 100% conversion and product inhomogeneities as a result of front instability, therefore the three-step mechanism was exploited. The phenomenon of superadiabatic combustion temperature was observed beyond the Hopf bifurcation point for both kinetic schemes and supported by the experimental measurements. One- and two-dimensional numerical simulations were performed to observe various planar and nonplanar periodic modes, and the results for different kinetic schemes were compared. It was found that stability of the frontal mode for a one-step reaction mechanism does not differ for 1-D and 2-D cases. For the three-step reaction mechanism 2-D solutions turned out to be more stable with respect to the appearance of nonplanar periodic modes than corresponding 1-D solutions. Higher Zeldovich numbers (i.e., higher effective activation energies or lower initial temperatures) are necessary for the existence of planar and nonplanar periodic modes in the 2-D reactor with walls than in the 1-D case. (c) 1997 American Institute of Physics.

Bifurcations of Numerically Simulated Marangoni Flows in Floating Zones

Marangoni or thermocapillary convection in a two-dimensional cylindrical half zone configuration under microgravity is studied numerically. The time-dependent simulations take into account convection and conduction in the melt, heat transfer between the melt and the ambient, and deformations of the free melt/gas surface of the half zone. A modified Marker and Cell (MAC) method is used to compute the flow and the temperature fields. The algorithm is applied especially to silicon melts. Above a critical temperature difference in the melt, the steady state becomes unstable and oscillatory thermocapillary convection occurs. The relevant control parameter for the onset of oscillations is the Marangoni number. As the Marangoni number increases, the phenomenon of period doubling is observed in the simulations. After a sequence of period doubling bifurcations, the flow becomes turbulent.

Homoclinic chaos in a model of natural selection

We modify a simple mathematical model for natural selection originally formulated by Robert M. May in 1983 by permitting one homozygote to have a larger selective advantage when rare than the other, and show that the new model exhibits dynamical chaos. We determine an open region of parameter space associated with homoclinic points, and prove that there are infinite sequences of period-doubling bifurcations along selected paths through parameter space. We also discuss the possibility of chaos arising from imbalance in the homozygote fitnesses in more realistic biological situations, beyond the constraints of the model.

A quantitative bifurcation analysis of H�non-like 2D maps

We numerically study the bifurcations of two nonlinear maps, with the same linear part, which depend on a parameter namely the Henon quadratic map and the so called ‘beam-beam’ map. Many families of periodic orbits which bifurcate from the central family, are studied. Each family undergoes a sequence of period doubling bifurcations in the quadratic map, But the behavior of the ‘beam-beam’ map is completely different. Inverse bifurcations occur in both maps. But some families of the same type which bifurcate inversely in the quadratic map do not bifurcate inversely in the ‘beam-beam’ map, even though both maps have common linear part.

Chaotic Dynamics of a Harmonically Excited Spring-Pendulum System with Internal Resonance

An investigation into chaotic responses of a weakly nonlinear multi-degree-of-freedom system is made. The specific system examined is a harmonically excited spring pendulum system, which is known to be a good model for a variety of engineering systems, including ship motions with nonlinear coupling between pitching and rolling motions. By the method of multiple scales the original nonautonomous system is reduced to an approximate autonomous system of amplitude and phase variables. The approximate system is shown to have Hopf bifurcation and a sequence of period-doubling bifurcations leading to chaotic motions. In order to examine what happens in the original system when the approximate system exhibits chaos, we compare the largest Lyapunov exponents for both systems.

Global scaling behaviors and chaotic measure characterized by the convergent rates of period-p-tupling bifurcations.

The Derrida-Gervois-Pomeau composition of the sequences Q* is a topological conjugate compression operation. On the basis of the numerical calculation for the quadratic map f(lambda)(x) = 1 - lambda x(2), it is found that Q* has the uniform compression ratio delta(-1)(Q), i.e., the reciprocal of the universal convergent rate. A series of global scaling behaviors relating to the periodic windows, the window bands, and the steps of equal topological entropy class (ETEC) are revealed to be independent of the sequences. In particular, the geometric interpretation for the Yorke-Grebogi-Ott-Tedeschini-Lalli normalized crisis value of ''windows'' mu(c) = 9/4 is given clearly through the scaling for the ETEC steps. The universal positions of superstable points in all the periodic windows are determined by the window scaling ratios gamma(PDB)=1 for the period-doubling bifurcation (PDB) sequences and gamma(N)=1/3 for the non-PDB sequences. The approximate analytic formula of the chaotic measure is obtained by employing the convergent rates delta of periodic sequences. The singularity spectrum and the generalized fractal dimension of the chaotic set are also calculated. These results imply that the metric universality of the periodic sequences may be a very good complete description for the topological space of two symbols.

Bifurcation and Chaos in Externally Excited Circular Cylindrical Shells

The nonlinear response of an infinitely long cylindrical shell to a primary excitation of one of its two orthogonal flexural modes is investigated. The method of multiple scales is used to derive four ordinary differential equations describing the amplitudes and phases of the two orthogonal modes by (a) attacking a two-mode discretization of the governing partial differential equations and (b) directly attacking the partial differential equations. The two-mode discretization results in erroneous solutions because it does not account for the effects of the quadratic nonlinearities. The resulting two sets of modulation equations are used to study the equilibrium and dynamic solutions and their stability and hence show the different bifurcations. The response could be a single-mode solution or a two-mode solution. The equilibrium solutions of the two orthogonal third flexural modes undergo a Hopf bifurcation. A combination of a shooting technique and Floquet theory is used to calculate limit cycles and their stability. The numerical results indicate the existence of a sequence of period-doubling bifurcations that culminates in chaos, multiple attractors, explosive bifurcations, and crises.

Delay-induced chaos in catalytic surface reactions: NO reduction on Pt(100).

Deterministic chaos has been observed in the $\mathrm{NO}+\mathrm{CO}$ and $\mathrm{NO}+{\mathrm{H}}_{2}$ reactions on Pt(100). A mathematical model is proposed that explains the origin as being due to delays in the response of a population of reacting adsorbate islands globally coupled via the gas phase. The dynamical equations of this model yield a sequence of period-doubling bifurcations resulting in chaos.

THE RESPONSE OF A NONLINEAR SYSTEM WITH A NONSEMISIMPLE ONE-TO-ONE RESONANCE TO A COMBINATION PARAMETRIC RESONANCE

The Galerkin procedure is used to discretize the nonlinear partial differential equation and boundary conditions governing the flutter of a simply supported panel in a supersonic stream. These equations have repeated natural frequencies at the onset of flutter. The method of multiple scales is used to derive five first-order nonlinear ordinary-differential equations governing the modulation of the amplitudes and phases of the excited modes. Then, the modulation equations are used to calculate the equilibrium solutions and their stability, and hence to identify the excitation parameters that suppress flutter and those that lead to complex motions. A combination of a shooting technique and Floquet theory is used to calculate limit cycles and their stability. The numerical results indicate the existence of a sequence of period-doubling bifurcations that culminates in chaos. The complex motions are characterized by using phase planes, power spectra, Lyapunov exponents, and dimensions.