Showing papers by "Steven H. Strogatz published in 1995"

Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry and Engineering

Abstract: Preface 1. Overview 1.0 Chaos, Fractals, and Dynamics 1.1 Capsule History of Dynamics 1.2 The Importance of Being Nonlinear 1.3 A Dynamical View of the World PART I. ONE-DIMENSIONAL FLOWS 2. Flows on the Line 2.0 Introduction 2.1 A Geometric Way of Thinking 2.2 Fixed Points and Stability 2.3 Population Growth 2.4 Linear Stability Analysis 2.5 Existence and Uniqueness 2.6 Impossibility of Oscillations 2.7 Potentials 2.8 Solving Equations on the Computer Exercises 3. Bifurcations 3.0 Introduction 3.1 Saddle-Node Bifurcation 3.2 Transcritical Bifurcation 3.3 Laser Threshold 3.4 Pitchfork Bifurcation 3.5 Overdamped Bead on a Rotating Hoop 3.6 Imperfect Bifurcations and Catastrophes 3.7 Insect Outbreak Exercises 4. Flows on the Circle 4.0 Introduction 4.1 Examples and Definitions 4.2 Uniform Oscillator 4.3 Nonuniform Oscillator 4.4 Overdamped Pendulum 4.5 Fireflies 4.6 Superconducting Josephson Junctions Exercises PART II. TWO-DIMENSIONAL FLOWS 5. Linear Systems 5.0 Introduction 5.1 Definitions and Examples 5.2 Classification of Linear Systems 5.3 Love Affairs Exercises 6. Phase Plane 6.0 Introduction 6.1 Phase Portraits 6.2 Existence, Uniqueness, and Topological Consequences 6.3 Fixed Points and Linearization 6.4 Rabbits versus Sheep 6.5 Conservative Systems 6.6 Reversible Systems 6.7 Pendulum 6.8 Index Theory Exercises 7. Limit Cycles 7.0 Introduction 7.1 Examples 7.2 Ruling Out Closed Orbits 7.3 Poincare-Bendixson Theorem 7.4 Lienard Systems 7.5 Relaxation Oscillators 7.6 Weakly Nonlinear Oscillators Exercises 8. Bifurcations Revisited 8.0 Introduction 8.1 Saddle-Node, Transcritical, and Pitchfork Bifurcations 8.2 Hopf Bifurcations 8.3 Oscillating Chemical Reactions 8.4 Global Bifurcations of Cycles 8.5 Hysteresis in the Driven Pendulum and Josephson Junction 8.6 Coupled Oscillators and Quasiperiodicity 8.7 Poincare Maps Exercises PART III. CHAOS 9. Lorenz Equations 9.0 Introduction 9.1 A Chaotic Waterwheel 9.2 Simple Properties of the Lorenz Equations 9.3 Chaos on a Strange Attractor 9.4 Lorenz Map 9.5 Exploring Parameter Space 9.6 Using Chaos to Send Secret Messages Exercises 10. One-Dimensional Maps 10.0 Introduction 10.1 Fixed Points and Cobwebs 10.2 Logistic Map: Numerics 10.3 Logistic Map: Analysis 10.4 Periodic Windows 10.5 Liapunov Exponent 10.6 Universality and Experiments 10.7 Renormalization Exercises 11. Fractals 11.0 Introduction 11.1 Countable and Uncountable Sets 11.2 Cantor Set 11.3 Dimension of Self-Similar Fractals 11.4 Box Dimension 11.5 Pointwise and Correlation Dimensions Exercises 12. Strange Attractors 12.0 Introductions 12.1 The Simplest Examples 12.2 Henon Map 12.3 Rossler System 12.4 Chemical Chaos and Attractor Reconstruction 12.5 Forced Double-Well Oscillator Exercises Answers to Selected Exercises References Author Index Subject Index

Kink propagation in a highly discrete system: Observation of phase locking to linear waves.

Abstract: We report the first observation of phase locking between a kink propagating in a highly discrete system and the linear waves excited in its wake. The current-voltage ($I\ensuremath{-}V$) characteristics of discrete rings of Josephson junctions have been measured. Resonant steps appear in the $I\ensuremath{-}V$ curve, due to phase locking between a propagating vortex and its induced radiation. Unexpectedly, mode numbers outside the first Brillouin zone are physically relevant, due to the nonlinearity of the system.

Whirling modes and parametric instabilities in the discrete Sine-Gordon equation: Experimental tests in Josephson rings.

Abstract: We analyze the damped driven discrete sine-Gordon equation. For underdamped, highly discrete systems, we show that whirling periodic solutions undergo parametric instabilities at certain drive strengths. The theory predicts novel resonant steps in the current-voltage characteristics of discrete Josephson rings, occurring in the return path of the subgap region. We have observed these steps experimentally in a ring of 8 underdamped junctions. An unusual prediction, verified experimentally, is that such steps occur even if there are no vortices in the ring. Numerical simulations indicate that complex spatiotemporal behavior occurs past the onset of instability.

The Birth of Period Three

Abstract: Xn+l = rXnO -Xn),(1) where 0 < x_ 1 and 0 < r < 4. In other words, given some starting number 0 < xl < 1, we generate a new number x2 by the rule x2 = rx1(1 x 1), and then repeat the process to generate x3 from x2, and so on. This equation has many virtues: 1) It is accessible. High school students can explore its patterns, as long as they have access to a hand calculator or a small computer. 2) It is exemplary. This single example illustrates many of the fundamental notions of nonlinear dynamics, such as equilibrium, stability, periodicity, chaos, bifurcations, and fractals. May [6] was the first to stress the pedagogical value of (1). 3) It is living mathematics. Most of the important discoveries about the logistic map are less than 20 years old. Certain aspects of (1) are still not understood rigorously, and are being pursued by a few of the finest living mathematicians. 4) It is relevant to science. Predictions derived from the logistic map have been verified in experiments on weakly turbulent fluids, oscillating chemical reactions, nonlinear electronic circuits, and a variety of other systems [8].

Scaling laws for dynamical hysteresis in a multidimensional laser system.

Abstract: We examine scaling laws for dynamical hysteresis in an optically bistable semiconductor laser. An analytic derivation of these laws from multidimensional laser equations is outlined and they are expected to be universal for systems that exhibit a cusp catastrophe. The scaling laws for the hysteresis loop area or width are numerically verified and experimentally measured for operation of the bistable laser above and below threshold. Excellent agreement with theory is obtained in the limit of low switching frequencies.

Stability of Synchronization in Networks of Digital Phase-Locked Loops

Abstract: We analyze the linear stability of the synchronized state in networks of N identical digital phase-locked loops. These are pulse-coupled oscillator arrays in which the frequency (rather than the phase) of each oscillator is updated discontinuously whenever that oscillator reaches a specific phase in its cycle. Three different coupling configurations are studied: one-way rings, two-way rings, and globally coupled arrays. In each case we obtain explicit formulas for the transient time to lock, the critical gain at which the synchronized state loses stability, and the period of the bifurcating solution at the onset of instability. Our results explain the numerical observations of de Sousa Vieira, Lichtenberg, and Lieberman.

Vortex propagation in discrete josephson rings

Abstract: Superconducting systems built of underdamped Josephson junctions are of interest for ballistic and quantum vortex experiments. The dissipation in a Josephson tunnel junction can be extremely small because its subgap resistance can easily be made of the order of 1 MΩ which for a typical junction translates into a McCumber parameter as high as 107. In a Josephson system with such little dissipation, the vortex velocity can be high enough so that the energy stored in the electric field generated by the moving vortex has to be taken into account. This electric field contribution to the energy acts like a kinetic-energy term and the vortex mass is found to be proportional to the junction capacitance [1, 2, 3]. Measurements have indicated the existence of a mass term in the equation of motion for a vortex [4, 5]. The concept of a massive vortex has also been tested in quantum experiments: there is evidence for quantum tunneling of vortices [4, 5], for a collective, quantum phase transition triggered by a Bose-condensation of vortices [6], and for vortex interference (Aharonov-Casher effect) [7].

Resonant steps in parallel Josephson junction arrays: parametric instabilities of whirling modes

Abstract: Circular arrays of underdamped Josephson junctions exhibit a series of resonant steps in the return path of the subgap region in the current-voltage characteristics. We show that the voltage locations of the steps can be predicted by studying the parametric instabilities of whirling periodic solutions, and experimentally verify the prediction in a ring of 8 underdamped junctions. The whirling modes become unstable in certain voltage intervals, and a branch (a resonant step) of more complicated solutions emerges from the endpoint of each interval. We extend the analysis to open-ended arrays and find that for f=0, the onset of a zero-field step has the same underlying mechanism. For f>0, combinations of lattice eigen-frequencies are excited. >