Acute Inhalation (µg/m 3) Date c Value Reviewed [Added] 8-Hour Inhalation (µg/m 3) Date c Value R... more Acute Inhalation (µg/m 3) Date c Value Reviewed [Added] 8-Hour Inhalation (µg/m 3) Date c Value Reviewed [Added] Chronic Inhalation (µg/m 3) Date c Value Reviewed [Added] Chronic Oral (mg/kg-d) Date c Value Reviewed [Added] Inhalation d Unit Risk (µg/m 3)-1 Inhalation d Cancer Potency Factor (mg/kg-d)-1 Date c Value Reviewed [Added] Oral Slope Factor (mg/kg-d)-1
Acute Inhalation (µg/m 3) Date c Value Reviewed [Added] 8-Hour Inhalation (µg/m 3) Date c Value R... more Acute Inhalation (µg/m 3) Date c Value Reviewed [Added] 8-Hour Inhalation (µg/m 3) Date c Value Reviewed [Added] Chronic Inhalation (µg/m 3) Date c Value Reviewed [Added] Chronic Oral (mg/kg-d) Date c Value Reviewed [Added] Inhalation d Unit Risk (µg/m 3)-1 Inhalation d Cancer Potency Factor (mg/kg-d)-1 Date c Value Reviewed [Added] Oral Slope Factor (mg/kg-d)-1
Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics, 2000
We study the influence of internal fluctuations on phase synchronization in oscillatory reaction-... more We study the influence of internal fluctuations on phase synchronization in oscillatory reaction-diffusion systems through a master equation approach. In the limit of large system size, the probability density is analyzed by means of the eikonal approximation. This approximation yields a Hamilton-Jacobi equation for the stochastic potential, which may be reduced to coupled nonlinear diffusion equations for the phase of oscillation and (conjugate) "momentum." We give explicit expressions for the coefficients of these equations in terms of averages over the deterministic periodic orbit. For one-dimensional systems, we obtain an explicit solution for the stationary stochastic potential: the width in phase, which is defined as the root mean square fluctuation in phase, characterizes the roughness of phase locking, and diverges with the system size L according to a power law w approximately Lalpha, with alpha=1/2. To study higher-dimensional systems, we show that the eikonal ap...
Benzo(ghi)perylene (B(ghi))Per, (191-24-2)) and coronene (Cor, (191-07-1)) are major constituents... more Benzo(ghi)perylene (B(ghi))Per, (191-24-2)) and coronene (Cor, (191-07-1)) are major constituents of the polycyclic aromatic hydrocarbons (PAH) found in automobile exhaust and polluted air [eg, Grimmer et al, 1981]. Nitration of these PAH by NO2 and traces of HNO3, which are also formed in automobile exhaust, seems highly probable. To identify the presence of these nitroarenes in environmental samples and to examine their mutagenic potencies we synthesized and characterized nitro derivatives of both PAH. 5-NO2B(ghi)Per (81316-87-2) and 1-NO2Cor (81316-84-9) produced 405 and 340 revertants/nmole respectively in TA98 in the presence of 0.6 mg of microsomal enzymes (S-9) per plate in the Ames test. 5,8-diNO2B(ghi)Per (83292-25-5) and 5,10-diNO2B(ghi)Per (83292-26-6) produced 21,500 and 4,000 revertants/nmole in TA98 without microsomal activation. Mutagenicity for the dinitrobenzo(ghi)perylenes was also high in TA98NR and TA97 but was reduced by 97% in TA98-1,8DNP. There is close simila...
Twenty-nine derivatives of fluorene were tested for mutagenic potency in four strains of Salmonel... more Twenty-nine derivatives of fluorene were tested for mutagenic potency in four strains of Salmonella typhimurium with and/or without S9 microsomal activation. The effects of a second functional group on the mutagenic activity of an amino-, nitroso-, and nitrofluorene were correlated with its physical and chemical properties. When the functional group is conjugated by resonance, both inductive and resonance effects are determinants of mutagenic potency. Electron-withdrawing groups such as the halogens (F, C1, Br, and I), nitro, nitroso, and cyano at C-7 increased the mutagenic potency of 2-nitrofluorene. Electron-donating substituents such as hydroxy and amino groups at C-7 decreased the mutagenic potency of 2-amino, 2-nitroso-, and 2-nitrofluorene. Acetylation of a hydroxy or an amino group at C-7 increased the mutagenic potency of 2-nitrofluorene, presumably by decreasing the electron-donating properties of these groups. In contrast, acetylation of a nonresonance-conjugated amino gr...
Seventeen structurally homologous nitroaromatics were tested for direct-acting mutagenic potency ... more Seventeen structurally homologous nitroaromatics were tested for direct-acting mutagenic potency in nine strains of Salmonella typhimurium. The following four structural features were determined to have a strong influence on mutagenic activity: physical dimensions of the aromatic rings, isomeric position of the nitro group, conformation of the nitro group with respect to the plane of the aromatic rings, and ability to resonance-stabilize the ultimate electrophile. Progressive addition of five-and six-membered rings to a nitrobenzene nucleus demonstrated that mutagenic activity was a direct function of size. Fluoranthene was of optimal size (four rings) for mutagenicity; an additional benzene ring, giving benzo[k]fluoranthene, reduced mutagenic activity. Nitroaromatics with a nitro group oriented along the long axis of symmetry of the molecule were more potent mutagens than those with the nitro group oriented along the short axis. These results are discussed in light of the insertion-denaturation model for intercalation of certain DNA adducts. Nitroaromatics with nitro groups sterically forced out of the plane of the aromatic rings were weakly mutagenic or nonmutagenic. Nitro groups located between two peri hydrogens or in a bay-region are examples of this conformation. Finally, structural features that contribute to resonance stabilization of the reactive nitrenium ion enhance mutagenic potency. Thus, 6-nitroindene was at least tenfold more mutagenic than hitroindene. These positional isomers are structurally identical with the exception of the position of an olefinic bond in the adjacent five-membered ring which can contribute to resonance stabilization of a carbonium ion formed after bioactivation of 6-nitroindene but not of 5-nitroindene. The predictive value of these structure-activity relationships should permit a first approximation in the assessment of mutagenic potency of nitroaromatics.
... 99, 1989 331 Universal Bifurcation Structures of Forced Oscillators William Vance, George Tsa... more ... 99, 1989 331 Universal Bifurcation Structures of Forced Oscillators William Vance, George Tsarouhas*' and John Ross Department of Chemistry, Stanford University Stanford ... Generally, entrainment regions that emanate from the points (coo/cüe=p/q, £=0) with q=l, 2 and 3 have ...
Proceedings of the National Academy of Sciences, 2002
We present an approach to the determination of causal connectivities and part of the kinetics of ... more We present an approach to the determination of causal connectivities and part of the kinetics of complex reaction systems. Our approach is based on analytical and computational methods for studying the effects of a pulse change of concentration of a chemical species in a reaction network, either at equilibrium or in a nonequilibrium stationary state. Such disturbances generally propagate through a few species, depending on the values of the kinetic coefficients, before being broadened and dissipated. This short range gives a local probe of the kinetics and connectivity of the reaction network. The range of propagation also indicates species to perturb in further experiments. From piecing together these local connectivities, the global structure of the network can be constructed. The experimental design allows deduction of both reaction orders and rate constants in many cases. An example of the usefulness of the approach is illustrated on a model of a part of glycolysis.
We develop and discuss the generation of new limit cycles in chemical systems by external periodi... more We develop and discuss the generation of new limit cycles in chemical systems by external periodic perturbations. The deterministic analysis is based on prior work by Loud. We choose a model derived from experiments, the conversion of 2,3 epoxy-1-propanol to glycerine. The autonomous system of equations which govern the dynamics of this system has a limit cycle arising from a normal Hopf bifurcation, and a stable focus, and a limit cycle associated with an inverted Hopf bifuraction. For specific choices of periodic perturbations a prescribed number of periodic attractors can be generated from a single periodic attractor (normal Hopf bifurcation); and two periodic attractors, a periodic and biperiodic, or two biperiodic attractors can be generated from a stable focus and periodic attractor (inverted Hopf bifurcation). Noise is then imposed on the attractors of the autonomous system and the multiple attractors of the externally forced system. Noise imposed on the autonomous system with a stable focus and limit cycle causes random transitions, analogous to first order phase transitions. This behavior has some resemblance to intermittency noted in some biological systems. From the results of noise imposed on systems with different numbers of isoperiodic limit cycles we suggest, after a limited number of observations, the trend that as the number of attractors increases, the lines in the power spectrum narrow and the area in the phase plane within contours of equal probability increases.
... Scientific Co.; carbon tetrafluo-ride with a stated minimum purity of 99.7%, from Linde Speci... more ... Scientific Co.; carbon tetrafluo-ride with a stated minimum purity of 99.7%, from Linde Specialty Gases ... 6 A liquid nitrogen trap inside the diffraction cham-ber effectively condensed SF6 and SiF4, It was ineffec-tive in trapping CF4 for which chamber pressures rose to values ...
We study experimentally continuous transitions from quasiperiodic to periodic states for a time-p... more We study experimentally continuous transitions from quasiperiodic to periodic states for a time-periodically forced chemical oscillator. The chemical reaction is the hydration of 2,3-epoxy-1-propanol, and is carried out in a continuous stirred tank reactor (CSTR). Periodic oscillatory states are observed to arise in the autonomous system through supercritical Hopf bifurcations as either the total flow rate or the cooling coil temperature is changed. Under conditions of oscillation for the autonomous system, small-amplitude periodic variation of the total flow rate generates an attracting two-torus from the stable limit cycle. From the experiments we determine the structure of the toroidal flow, stroboscopic phase portraits, and circle maps as a function of the forcing amplitude and period. A continuous transition from the quasiperiodic to a periodic state, in which the two-torus contracts to a closed curve (Neimark–Sacker torus bifurcation), is observed as the forcing amplitude is increased at a constant forcing period, or as the forcing period is changed at a constant moderate forcing amplitude. Qualitative theoretical predictions compare well with the experimental observations. This paper presents the first experimental observation of a Neimark–Sacker torus bifurcation in a forced chemical oscillator system, and relates the bifurcation diagram of the unforced system to that of the forced system.
We investigate stationary and nonstationary probability densities for a weakly forced nonlinear p... more We investigate stationary and nonstationary probability densities for a weakly forced nonlinear physical or chemical system that displays self-oscillations in the absence of forcing. The period and amplitude of forcing are taken as adjustable constraints. We consider a homogeneous reaction system described by a master equation. Our method of solution is based on the Wentzel-Kramers-Brillouin (WKB) expansion of the probability density with the system size as the expansion parameter. The first term in this expansion is the stochastic potential (eikonal). In the absence of forcing, the probability density is logarithmically flat on the limit cycle. With periodic forcing, the phenomenon of phase locking can occur whereby a stable cycle, which is close to the unforced cycle, adopts a constant relative phase to the forcing. A saddle cycle also exists and has a different constant relative phase. For such phase-locked solutions, the distribution over the relative phases is peaked on the stable cycle and exhibits a logarithmically flat region (a plateau) that originates on the saddle cycle. This plateau is due to a nonzero relative phase slippage: large fluctuations from the stable cycle over the saddle cycle are overwhelmingly more probable in a certain relative phase direction, which depends upon the location of the parameters within an entrainment region. This distribution of relative phases is logarithmically equivalent to that of a Brownian particle in a periodic potential with a constant external force in the strong damping and weak noise limits. For parameter values outside of an entrainment region (for which a quasiperiodic solution exists), the distribution in relative phase is logarithmically flat. For this regime, we investigate the evolution of an initially localized density and show that the width grows proportionally with the square root of time. The proportionality factor depends upon both the position (phase) on the cross section of the peak of the density and the distance in parameter space from the boundary of the entrainment region. For parameter values that approach the boundary of an entrainment region, this proportionality factor tends to infinity. We also determine an expression for the first order correction to the stochastic potential for both entrained and quasiperiodic solutions. A thermodynamic interpretation of these results is made possible by the equality of the stochastic potential with an excess work function.
Chaos: An Interdisciplinary Journal of Nonlinear Science, 1991
A theoretical investigation of bifurcation structures of periodically forced oscillators is prese... more A theoretical investigation of bifurcation structures of periodically forced oscillators is presented. In the plane of forcing frequency and amplitude, subharmonic entrainment occurs in v-shaped (Arnol'd) tongues, or entrainment bands, for small forcing amplitudes. These tongues terminate at higher forcing amplitudes. Between these two limits, individual tongues fit together to form a global bifurcation structure. The regime in which the forcing amplitude is much smaller than the amplitude of the limit cycle is first examined. Using the method of multiple time scales, expressions for solutions on the invariant torus, widths of Arnol'd tongues, and Liapunov exponents of periodic orbits are derived. Next, the regime of moderate to large forcing amplitudes is examined through studying a periodically forced Hopf bifurcation. In this case the forcing amplitude and the amplitude of the limit cycle can be of the same order of magnitude. From a study of the normal forms for this case, it is shown how Arnol'd tongues terminate and how complicated bifurcation structures are associated with strong resonances. Aspects of model and experimental chemical systems that show some of the phenomena predicted from the above theoretical results are mentioned.
The ANNALS of the American Academy of Political and Social Science, 1926
Page 1. The Historical Background of the Legal Aid Movement By WILLIAM R.VANCE Professor of Law, ... more Page 1. The Historical Background of the Legal Aid Movement By WILLIAM R.VANCE Professor of Law, Yale University T HE ... child. Thus if the son refused to obey the father he was branded, bound and sold into slavery. So ...
Acute Inhalation (µg/m 3) Date c Value Reviewed [Added] 8-Hour Inhalation (µg/m 3) Date c Value R... more Acute Inhalation (µg/m 3) Date c Value Reviewed [Added] 8-Hour Inhalation (µg/m 3) Date c Value Reviewed [Added] Chronic Inhalation (µg/m 3) Date c Value Reviewed [Added] Chronic Oral (mg/kg-d) Date c Value Reviewed [Added] Inhalation d Unit Risk (µg/m 3)-1 Inhalation d Cancer Potency Factor (mg/kg-d)-1 Date c Value Reviewed [Added] Oral Slope Factor (mg/kg-d)-1
Acute Inhalation (µg/m 3) Date c Value Reviewed [Added] 8-Hour Inhalation (µg/m 3) Date c Value R... more Acute Inhalation (µg/m 3) Date c Value Reviewed [Added] 8-Hour Inhalation (µg/m 3) Date c Value Reviewed [Added] Chronic Inhalation (µg/m 3) Date c Value Reviewed [Added] Chronic Oral (mg/kg-d) Date c Value Reviewed [Added] Inhalation d Unit Risk (µg/m 3)-1 Inhalation d Cancer Potency Factor (mg/kg-d)-1 Date c Value Reviewed [Added] Oral Slope Factor (mg/kg-d)-1
Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics, 2000
We study the influence of internal fluctuations on phase synchronization in oscillatory reaction-... more We study the influence of internal fluctuations on phase synchronization in oscillatory reaction-diffusion systems through a master equation approach. In the limit of large system size, the probability density is analyzed by means of the eikonal approximation. This approximation yields a Hamilton-Jacobi equation for the stochastic potential, which may be reduced to coupled nonlinear diffusion equations for the phase of oscillation and (conjugate) "momentum." We give explicit expressions for the coefficients of these equations in terms of averages over the deterministic periodic orbit. For one-dimensional systems, we obtain an explicit solution for the stationary stochastic potential: the width in phase, which is defined as the root mean square fluctuation in phase, characterizes the roughness of phase locking, and diverges with the system size L according to a power law w approximately Lalpha, with alpha=1/2. To study higher-dimensional systems, we show that the eikonal ap...
Benzo(ghi)perylene (B(ghi))Per, (191-24-2)) and coronene (Cor, (191-07-1)) are major constituents... more Benzo(ghi)perylene (B(ghi))Per, (191-24-2)) and coronene (Cor, (191-07-1)) are major constituents of the polycyclic aromatic hydrocarbons (PAH) found in automobile exhaust and polluted air [eg, Grimmer et al, 1981]. Nitration of these PAH by NO2 and traces of HNO3, which are also formed in automobile exhaust, seems highly probable. To identify the presence of these nitroarenes in environmental samples and to examine their mutagenic potencies we synthesized and characterized nitro derivatives of both PAH. 5-NO2B(ghi)Per (81316-87-2) and 1-NO2Cor (81316-84-9) produced 405 and 340 revertants/nmole respectively in TA98 in the presence of 0.6 mg of microsomal enzymes (S-9) per plate in the Ames test. 5,8-diNO2B(ghi)Per (83292-25-5) and 5,10-diNO2B(ghi)Per (83292-26-6) produced 21,500 and 4,000 revertants/nmole in TA98 without microsomal activation. Mutagenicity for the dinitrobenzo(ghi)perylenes was also high in TA98NR and TA97 but was reduced by 97% in TA98-1,8DNP. There is close simila...
Twenty-nine derivatives of fluorene were tested for mutagenic potency in four strains of Salmonel... more Twenty-nine derivatives of fluorene were tested for mutagenic potency in four strains of Salmonella typhimurium with and/or without S9 microsomal activation. The effects of a second functional group on the mutagenic activity of an amino-, nitroso-, and nitrofluorene were correlated with its physical and chemical properties. When the functional group is conjugated by resonance, both inductive and resonance effects are determinants of mutagenic potency. Electron-withdrawing groups such as the halogens (F, C1, Br, and I), nitro, nitroso, and cyano at C-7 increased the mutagenic potency of 2-nitrofluorene. Electron-donating substituents such as hydroxy and amino groups at C-7 decreased the mutagenic potency of 2-amino, 2-nitroso-, and 2-nitrofluorene. Acetylation of a hydroxy or an amino group at C-7 increased the mutagenic potency of 2-nitrofluorene, presumably by decreasing the electron-donating properties of these groups. In contrast, acetylation of a nonresonance-conjugated amino gr...
Seventeen structurally homologous nitroaromatics were tested for direct-acting mutagenic potency ... more Seventeen structurally homologous nitroaromatics were tested for direct-acting mutagenic potency in nine strains of Salmonella typhimurium. The following four structural features were determined to have a strong influence on mutagenic activity: physical dimensions of the aromatic rings, isomeric position of the nitro group, conformation of the nitro group with respect to the plane of the aromatic rings, and ability to resonance-stabilize the ultimate electrophile. Progressive addition of five-and six-membered rings to a nitrobenzene nucleus demonstrated that mutagenic activity was a direct function of size. Fluoranthene was of optimal size (four rings) for mutagenicity; an additional benzene ring, giving benzo[k]fluoranthene, reduced mutagenic activity. Nitroaromatics with a nitro group oriented along the long axis of symmetry of the molecule were more potent mutagens than those with the nitro group oriented along the short axis. These results are discussed in light of the insertion-denaturation model for intercalation of certain DNA adducts. Nitroaromatics with nitro groups sterically forced out of the plane of the aromatic rings were weakly mutagenic or nonmutagenic. Nitro groups located between two peri hydrogens or in a bay-region are examples of this conformation. Finally, structural features that contribute to resonance stabilization of the reactive nitrenium ion enhance mutagenic potency. Thus, 6-nitroindene was at least tenfold more mutagenic than hitroindene. These positional isomers are structurally identical with the exception of the position of an olefinic bond in the adjacent five-membered ring which can contribute to resonance stabilization of a carbonium ion formed after bioactivation of 6-nitroindene but not of 5-nitroindene. The predictive value of these structure-activity relationships should permit a first approximation in the assessment of mutagenic potency of nitroaromatics.
... 99, 1989 331 Universal Bifurcation Structures of Forced Oscillators William Vance, George Tsa... more ... 99, 1989 331 Universal Bifurcation Structures of Forced Oscillators William Vance, George Tsarouhas*' and John Ross Department of Chemistry, Stanford University Stanford ... Generally, entrainment regions that emanate from the points (coo/cüe=p/q, £=0) with q=l, 2 and 3 have ...
Proceedings of the National Academy of Sciences, 2002
We present an approach to the determination of causal connectivities and part of the kinetics of ... more We present an approach to the determination of causal connectivities and part of the kinetics of complex reaction systems. Our approach is based on analytical and computational methods for studying the effects of a pulse change of concentration of a chemical species in a reaction network, either at equilibrium or in a nonequilibrium stationary state. Such disturbances generally propagate through a few species, depending on the values of the kinetic coefficients, before being broadened and dissipated. This short range gives a local probe of the kinetics and connectivity of the reaction network. The range of propagation also indicates species to perturb in further experiments. From piecing together these local connectivities, the global structure of the network can be constructed. The experimental design allows deduction of both reaction orders and rate constants in many cases. An example of the usefulness of the approach is illustrated on a model of a part of glycolysis.
We develop and discuss the generation of new limit cycles in chemical systems by external periodi... more We develop and discuss the generation of new limit cycles in chemical systems by external periodic perturbations. The deterministic analysis is based on prior work by Loud. We choose a model derived from experiments, the conversion of 2,3 epoxy-1-propanol to glycerine. The autonomous system of equations which govern the dynamics of this system has a limit cycle arising from a normal Hopf bifurcation, and a stable focus, and a limit cycle associated with an inverted Hopf bifuraction. For specific choices of periodic perturbations a prescribed number of periodic attractors can be generated from a single periodic attractor (normal Hopf bifurcation); and two periodic attractors, a periodic and biperiodic, or two biperiodic attractors can be generated from a stable focus and periodic attractor (inverted Hopf bifurcation). Noise is then imposed on the attractors of the autonomous system and the multiple attractors of the externally forced system. Noise imposed on the autonomous system with a stable focus and limit cycle causes random transitions, analogous to first order phase transitions. This behavior has some resemblance to intermittency noted in some biological systems. From the results of noise imposed on systems with different numbers of isoperiodic limit cycles we suggest, after a limited number of observations, the trend that as the number of attractors increases, the lines in the power spectrum narrow and the area in the phase plane within contours of equal probability increases.
... Scientific Co.; carbon tetrafluo-ride with a stated minimum purity of 99.7%, from Linde Speci... more ... Scientific Co.; carbon tetrafluo-ride with a stated minimum purity of 99.7%, from Linde Specialty Gases ... 6 A liquid nitrogen trap inside the diffraction cham-ber effectively condensed SF6 and SiF4, It was ineffec-tive in trapping CF4 for which chamber pressures rose to values ...
We study experimentally continuous transitions from quasiperiodic to periodic states for a time-p... more We study experimentally continuous transitions from quasiperiodic to periodic states for a time-periodically forced chemical oscillator. The chemical reaction is the hydration of 2,3-epoxy-1-propanol, and is carried out in a continuous stirred tank reactor (CSTR). Periodic oscillatory states are observed to arise in the autonomous system through supercritical Hopf bifurcations as either the total flow rate or the cooling coil temperature is changed. Under conditions of oscillation for the autonomous system, small-amplitude periodic variation of the total flow rate generates an attracting two-torus from the stable limit cycle. From the experiments we determine the structure of the toroidal flow, stroboscopic phase portraits, and circle maps as a function of the forcing amplitude and period. A continuous transition from the quasiperiodic to a periodic state, in which the two-torus contracts to a closed curve (Neimark–Sacker torus bifurcation), is observed as the forcing amplitude is increased at a constant forcing period, or as the forcing period is changed at a constant moderate forcing amplitude. Qualitative theoretical predictions compare well with the experimental observations. This paper presents the first experimental observation of a Neimark–Sacker torus bifurcation in a forced chemical oscillator system, and relates the bifurcation diagram of the unforced system to that of the forced system.
We investigate stationary and nonstationary probability densities for a weakly forced nonlinear p... more We investigate stationary and nonstationary probability densities for a weakly forced nonlinear physical or chemical system that displays self-oscillations in the absence of forcing. The period and amplitude of forcing are taken as adjustable constraints. We consider a homogeneous reaction system described by a master equation. Our method of solution is based on the Wentzel-Kramers-Brillouin (WKB) expansion of the probability density with the system size as the expansion parameter. The first term in this expansion is the stochastic potential (eikonal). In the absence of forcing, the probability density is logarithmically flat on the limit cycle. With periodic forcing, the phenomenon of phase locking can occur whereby a stable cycle, which is close to the unforced cycle, adopts a constant relative phase to the forcing. A saddle cycle also exists and has a different constant relative phase. For such phase-locked solutions, the distribution over the relative phases is peaked on the stable cycle and exhibits a logarithmically flat region (a plateau) that originates on the saddle cycle. This plateau is due to a nonzero relative phase slippage: large fluctuations from the stable cycle over the saddle cycle are overwhelmingly more probable in a certain relative phase direction, which depends upon the location of the parameters within an entrainment region. This distribution of relative phases is logarithmically equivalent to that of a Brownian particle in a periodic potential with a constant external force in the strong damping and weak noise limits. For parameter values outside of an entrainment region (for which a quasiperiodic solution exists), the distribution in relative phase is logarithmically flat. For this regime, we investigate the evolution of an initially localized density and show that the width grows proportionally with the square root of time. The proportionality factor depends upon both the position (phase) on the cross section of the peak of the density and the distance in parameter space from the boundary of the entrainment region. For parameter values that approach the boundary of an entrainment region, this proportionality factor tends to infinity. We also determine an expression for the first order correction to the stochastic potential for both entrained and quasiperiodic solutions. A thermodynamic interpretation of these results is made possible by the equality of the stochastic potential with an excess work function.
Chaos: An Interdisciplinary Journal of Nonlinear Science, 1991
A theoretical investigation of bifurcation structures of periodically forced oscillators is prese... more A theoretical investigation of bifurcation structures of periodically forced oscillators is presented. In the plane of forcing frequency and amplitude, subharmonic entrainment occurs in v-shaped (Arnol'd) tongues, or entrainment bands, for small forcing amplitudes. These tongues terminate at higher forcing amplitudes. Between these two limits, individual tongues fit together to form a global bifurcation structure. The regime in which the forcing amplitude is much smaller than the amplitude of the limit cycle is first examined. Using the method of multiple time scales, expressions for solutions on the invariant torus, widths of Arnol'd tongues, and Liapunov exponents of periodic orbits are derived. Next, the regime of moderate to large forcing amplitudes is examined through studying a periodically forced Hopf bifurcation. In this case the forcing amplitude and the amplitude of the limit cycle can be of the same order of magnitude. From a study of the normal forms for this case, it is shown how Arnol'd tongues terminate and how complicated bifurcation structures are associated with strong resonances. Aspects of model and experimental chemical systems that show some of the phenomena predicted from the above theoretical results are mentioned.
The ANNALS of the American Academy of Political and Social Science, 1926
Page 1. The Historical Background of the Legal Aid Movement By WILLIAM R.VANCE Professor of Law, ... more Page 1. The Historical Background of the Legal Aid Movement By WILLIAM R.VANCE Professor of Law, Yale University T HE ... child. Thus if the son refused to obey the father he was branded, bound and sold into slavery. So ...
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