The orientation-preserving Hénon map is shown to be equivalent to the period one return map of a ... more The orientation-preserving Hénon map is shown to be equivalent to the period one return map of a periodically kicked harmonic oscillator with a suitable nonlinear coupling to the kicking term. The conservative, dissipative, and extremely dissipative (logistic limit) cases are each considered. All can be derived from a single differential equation describing the kicked oscillator system. In each case explicit coordinate and parameter transformations are given relating the return map of the kicked harmonic oscillator to a standard form of the Hénon map. In the dissipative case two studies are presented with periodic kicking replaced by a smooth periodic function. Each is found to yield a “Hénon-like” chaotic attractor.
Thesis Drexel University 1989 Source Dissertation Abstracts International Volume 50 05 Section B Page 1983, 1989
We investigate the classical dynamics of a one -dimensional Morse oscillator subjected to periodi... more We investigate the classical dynamics of a one -dimensional Morse oscillator subjected to periodic impulsive (delta function) forcing. This system serves as a rough model for the interaction of diatomic molecule (or single bond of a polyatomic molecule) with an intense laser field. The dynamics of the system are shown to be governed by an area -preserving map (period 1 return map) of the flow generated by the underlying Hamiltonian. The properties and behavior of this map are the main focus of study. The map is shown to be reversible and we obtain an explicit factorization of the map into a product of two orientation-reversing involutions. The invariant sets (symmetry lines) of these involutions are then determined analytically. The so called dominant symmetry line is identified as one of the symmetry lines. Many of the structures within phase space are organized by the symmetry lines and their iterates under the map. Among these structures are the (Poincare-Birkhoff) periodic orbits, resonance islands, and (primary) homoclinic orbits. We classify all of the Poincare-Birkhoff orbits according to the symmetry lines which they visit. The resonance islands, which are 'built' around these orbits, are similarly organized. We also give a similar symmetry classification for the primary homoclinic orbits. We extend these observations by proving that the homoclinic orbits in all reversible area-preserving maps fall on symmetry lines. Bifurcation behavior is examined in light of the symmetry line structure of phase space. The generation (or destruction) of period one orbits with winding numbers n/1 (n >=q 1), via tangent bifurcations, is found to be governed by systematic rules. These rules relate to the crossing of two symmetry lines as the control parameters of the map (amplitude and period of the driving term) are varied. We generalize some of the symmetry observations to a larger class of 'kicked' Hamiltonian systems. We provide evidence that the dominant symmetry line for this class of systems is determined solely by the form of the coupling function to the periodic forcing term, and we give an expression for the dominant symmetry line. We conjecture that the dominant symmetry line for all maps within this class is given by this expression. Finally, we examine the dissociation behavior of the Morse oscillator as a function of the control parameters. This study results in a boundary in parameter space which separates parameters which lead to dissociation from those which do not. A series of numerical dissociation experiments carried out over successively smaller regions of parameter space reveals that this boundary has a considerable amount of structure, even at very fine scales, and may possibly be fractal.
For the past 3 years our group at IDA has been involved in validation efforts associated with sev... more For the past 3 years our group at IDA has been involved in validation efforts associated with several Urban Transport and Dispersion (T&D) modeling systems. Models under study include MESO/RUSTIC, QUIC-URB/QUIC-PLUME, CT-Analyst, and four sub-models within HPAC, the Urban Canopy Model, Micro-Swift/Spray, the Urban Dispersion Model, and the Urban Windfield Module. Our main efforts have centered on supplying sponsors, and the T&D community as a whole, credible, protocol-driven comparisons of model predictions and field trial observations. I will review our most recent Urban T&D comparison work, with particular attention paid to comparisons of QUIC-URB/QUIC-PLUME predictions to the 29 continuous SF6 releases carried out during the Joint Urban 2003 (JU2003) field experiment in Oklahoma City.
Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics, 1996
We introduce a general technique to alter the properties of chaotic signals used with coupled cha... more We introduce a general technique to alter the properties of chaotic signals used with coupled chaotic systems. The changes we introduce allow one to vary the synchronization properties of synchronized chaotic circuits, synchronize chaotic systems that do not otherwise synchronize, vary the spectral properties of chaotic signals, and produce a variety of chaotic signals from one chaotic circuit. The transformations
Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics, 1995
Synchronous chaotic behavior is often interrupted by bursts of desynchronized behavior. We invest... more Synchronous chaotic behavior is often interrupted by bursts of desynchronized behavior. We investigate the role of unstable periodic orbits in bursting events and show that they serve as sources of local transverse instability within a synchronous chaotic attractor. Analysis of bursts in both model and experimental studies of two coupled Rössler-like oscillators reveals the importance of unstable periodic orbits in
The orientation-preserving Hénon map is shown to be equivalent to the period one return map of a ... more The orientation-preserving Hénon map is shown to be equivalent to the period one return map of a periodically kicked harmonic oscillator with a suitable nonlinear coupling to the kicking term. The conservative, dissipative, and extremely dissipative (logistic limit) cases are each considered. All can be derived from a single differential equation describing the kicked oscillator system. In each case explicit coordinate and parameter transformations are given relating the return map of the kicked harmonic oscillator to a standard form of the Hénon map. In the dissipative case two studies are presented with periodic kicking replaced by a smooth periodic function. Each is found to yield a “Hénon-like” chaotic attractor.
Thesis Drexel University 1989 Source Dissertation Abstracts International Volume 50 05 Section B Page 1983, 1989
We investigate the classical dynamics of a one -dimensional Morse oscillator subjected to periodi... more We investigate the classical dynamics of a one -dimensional Morse oscillator subjected to periodic impulsive (delta function) forcing. This system serves as a rough model for the interaction of diatomic molecule (or single bond of a polyatomic molecule) with an intense laser field. The dynamics of the system are shown to be governed by an area -preserving map (period 1 return map) of the flow generated by the underlying Hamiltonian. The properties and behavior of this map are the main focus of study. The map is shown to be reversible and we obtain an explicit factorization of the map into a product of two orientation-reversing involutions. The invariant sets (symmetry lines) of these involutions are then determined analytically. The so called dominant symmetry line is identified as one of the symmetry lines. Many of the structures within phase space are organized by the symmetry lines and their iterates under the map. Among these structures are the (Poincare-Birkhoff) periodic orbits, resonance islands, and (primary) homoclinic orbits. We classify all of the Poincare-Birkhoff orbits according to the symmetry lines which they visit. The resonance islands, which are 'built' around these orbits, are similarly organized. We also give a similar symmetry classification for the primary homoclinic orbits. We extend these observations by proving that the homoclinic orbits in all reversible area-preserving maps fall on symmetry lines. Bifurcation behavior is examined in light of the symmetry line structure of phase space. The generation (or destruction) of period one orbits with winding numbers n/1 (n >=q 1), via tangent bifurcations, is found to be governed by systematic rules. These rules relate to the crossing of two symmetry lines as the control parameters of the map (amplitude and period of the driving term) are varied. We generalize some of the symmetry observations to a larger class of 'kicked' Hamiltonian systems. We provide evidence that the dominant symmetry line for this class of systems is determined solely by the form of the coupling function to the periodic forcing term, and we give an expression for the dominant symmetry line. We conjecture that the dominant symmetry line for all maps within this class is given by this expression. Finally, we examine the dissociation behavior of the Morse oscillator as a function of the control parameters. This study results in a boundary in parameter space which separates parameters which lead to dissociation from those which do not. A series of numerical dissociation experiments carried out over successively smaller regions of parameter space reveals that this boundary has a considerable amount of structure, even at very fine scales, and may possibly be fractal.
For the past 3 years our group at IDA has been involved in validation efforts associated with sev... more For the past 3 years our group at IDA has been involved in validation efforts associated with several Urban Transport and Dispersion (T&D) modeling systems. Models under study include MESO/RUSTIC, QUIC-URB/QUIC-PLUME, CT-Analyst, and four sub-models within HPAC, the Urban Canopy Model, Micro-Swift/Spray, the Urban Dispersion Model, and the Urban Windfield Module. Our main efforts have centered on supplying sponsors, and the T&D community as a whole, credible, protocol-driven comparisons of model predictions and field trial observations. I will review our most recent Urban T&D comparison work, with particular attention paid to comparisons of QUIC-URB/QUIC-PLUME predictions to the 29 continuous SF6 releases carried out during the Joint Urban 2003 (JU2003) field experiment in Oklahoma City.
Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics, 1996
We introduce a general technique to alter the properties of chaotic signals used with coupled cha... more We introduce a general technique to alter the properties of chaotic signals used with coupled chaotic systems. The changes we introduce allow one to vary the synchronization properties of synchronized chaotic circuits, synchronize chaotic systems that do not otherwise synchronize, vary the spectral properties of chaotic signals, and produce a variety of chaotic signals from one chaotic circuit. The transformations
Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics, 1995
Synchronous chaotic behavior is often interrupted by bursts of desynchronized behavior. We invest... more Synchronous chaotic behavior is often interrupted by bursts of desynchronized behavior. We investigate the role of unstable periodic orbits in bursting events and show that they serve as sources of local transverse instability within a synchronous chaotic attractor. Analysis of bursts in both model and experimental studies of two coupled Rössler-like oscillators reveals the importance of unstable periodic orbits in
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