Nonlinear Dynamics and Chaos¶
Strogatz, S. H. (1994). Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering. Addison-Wesley.
Cited by¶
13 citations across 13 artifacts.
Each citation links to the sentence it supports in the citing article.
Primes¶
- Damping
This sourceStandard introduction to nonlinear dynamics, phase-plane analysis, bifurcations, and perturbation methods; widely used for damped/nonlinear oscillators.
- Equilibrium
- Collapses long-run behavior to a small set of equilibrium states, summarizing trajectory details into attractor identity. Provides a reference frame: deviations from equilibrium become the signal of interest, not the state itself. Separates fast transients from slow structure; transients relax, the equilibrium structure persists. Enables comparative statics — changing a parameter moves the equilibrium, and the new equilibrium can be described without re-simulating the whole trajectory. Bifurcation theory and dynamical systems
This sourceStandard modern text on dynamical systems: fixed points and their stability, phase-plane analysis, bifurcations (how equilibria are born, collide, and change stability as parameters vary), limit cycles, and chaos.
Supported in partVerified against the publisher's abstract
Strogatz's publisher description centres on bifurcations, limit cycles, chaos and strange attractors, backing the bifurcation/dynamical-systems clause but not the equilibrium-abstraction claims.
“The theory is developed systematically, starting with first-order differential equations and their bifurcations, followed by phase plane analysis, limit cycles and their bifurcations, and culminating with the Lorenz equations, chaos, iterated maps, period doubling, renormalization, fractals, and strange attractors.”
- Collapses long-run behavior to a small set of equilibrium states, summarizing trajectory details into attractor identity. Provides a reference frame: deviations from equilibrium become the signal of interest, not the state itself. Separates fast transients from slow structure; transients relax, the equilibrium structure persists. Enables comparative statics — changing a parameter moves the equilibrium, and the new equilibrium can be described without re-simulating the whole trajectory. Bifurcation theory and dynamical systems
- Fixed Point
- The attracting case is what an observer sees as a "stable state"; the repelling case explains why some equilibria are formally present yet never observed; the saddle case explains why a system can sit near an equilibrium for a long time before suddenly departing.
This sourceStandard text on fixed points and their stability classification (attracting, repelling, saddle) via local linearization and the multiplier/eigenvalues, and on basins of attraction.
- The attracting case is what an observer sees as a "stable state"; the repelling case explains why some equilibria are formally present yet never observed; the saddle case explains why a system can sit near an equilibrium for a long time before suddenly departing.
- Instability
- Modern bifurcation theory treatments by Strogatz in 1994
This sourceModern comprehensive treatment of perturbation analysis in nonlinear dynamical systems; covers regular and singular perturbation theory, phase-plane analysis, bifurcations, and chaos; widely used text unifying perturbation methods across disciplines.
- Modern bifurcation theory treatments by Strogatz in 1994
- Logistic Growth
- Setting \(\frac{dN}{dt} = 0\) gives the fixed-point pair: \(N = 0\) (unstable — any perturbation grows) and \(N = K\) (stable — the system settles there asymptotically).
This sourceStandard text providing the phase-line / linear-stability analysis of the logistic equation: the unstable fixed point at zero and stable fixed point at K, and the peak growth rate (inflection) at the half-ceiling N = K/2. ISBN 978-0-201-54344-5.
- Setting \(\frac{dN}{dt} = 0\) gives the fixed-point pair: \(N = 0\) (unstable — any perturbation grows) and \(N = K\) (stable — the system settles there asymptotically).
- Nonlinearity
- … Nonlinear does not mean unsolvable; many nonlinear systems have exact solutions (solitons, integrable systems via inverse-scattering transform, the Lotka-Volterra conservation law) or well-developed approximation schemes (perturbation theory, asymptotic analysis, multiple-scales methods, numerical integration
This sourceModern comprehensive treatment of perturbation analysis in nonlinear dynamical systems; covers regular and singular perturbation theory, phase-plane analysis, bifurcations, and chaos; widely used text unifying perturbation methods across disciplines.
- … Nonlinear does not mean unsolvable; many nonlinear systems have exact solutions (solitons, integrable systems via inverse-scattering transform, the Lotka-Volterra conservation law) or well-developed approximation schemes (perturbation theory, asymptotic analysis, multiple-scales methods, numerical integration
- Oscillation
- Strogatz's 1994 pedagogical treatment
This sourceModern comprehensive treatment of perturbation analysis in nonlinear dynamical systems; covers regular and singular perturbation theory, phase-plane analysis, bifurcations, and chaos; widely used text unifying perturbation methods across disciplines.
- Strogatz's 1994 pedagogical treatment
- Overshoot and Collapse
- The fold-bifurcation geometry is exact: the two saddle-node points where the branches appear and disappear sit at different values of the input, and the gap between them is the hysteresis.
This sourceSaddle-node (fold) bifurcations and the hysteresis arising from bistability in dynamical systems.
- The fold-bifurcation geometry is exact: the two saddle-node points where the branches appear and disappear sit at different values of the input, and the gap between them is the hysteresis.
- Perturbation
- Linearizes difficult problems: many systems are analytically intractable in general but tractable to leading order near a solvable reference — perturbation theory is the systematic exploitation of this.
This sourceModern comprehensive treatment of perturbation analysis in nonlinear dynamical systems; covers regular and singular perturbation theory, phase-plane analysis, bifurcations, and chaos; widely used text unifying perturbation methods across disciplines.
- Linearizes difficult problems: many systems are analytically intractable in general but tractable to leading order near a solvable reference — perturbation theory is the systematic exploitation of this.
- Phase Diagram
This sourceModern comprehensive treatment of perturbation analysis in nonlinear dynamical systems; covers regular and singular perturbation theory, phase-plane analysis, bifurcations, and chaos; widely used text unifying perturbation methods across disciplines.
- Phase Space
This sourceModern comprehensive treatment of perturbation analysis in nonlinear dynamical systems; covers regular and singular perturbation theory, phase-plane analysis, bifurcations, and chaos; widely used text unifying perturbation methods across disciplines.
- Stability
- Consider the damped pendulum, governed by \(\ddot{\theta} + b\dot{\theta} + \frac{g}{L}\sin\theta = 0\).
This sourceStandard text covering the damped pendulum, linearization, basins of attraction, and bifurcation as loss of stability.
- Consider the damped pendulum, governed by \(\ddot{\theta} + b\dot{\theta} + \frac{g}{L}\sin\theta = 0\).
- Tipping Points (or Phase Transitions)
- The concept spans social systems (adoption cascades, protest thresholds, segregation dynamics) and economics (asset bubbles, currency crises, market crashes ) because the underlying bifurcation structure is domain-independent.
This sourceModern comprehensive treatment of perturbation analysis in nonlinear dynamical systems; covers regular and singular perturbation theory, phase-plane analysis, bifurcations, and chaos; widely used text unifying perturbation methods across disciplines.
- The concept spans social systems (adoption cascades, protest thresholds, segregation dynamics) and economics (asset bubbles, currency crises, market crashes ) because the underlying bifurcation structure is domain-independent.
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Links previously used in the corpus¶
Before the registry existed this work was also linked 2 other ways.
- https://openlibrary.org/books/OL1398563M ×1
- https://www.taylorfrancis.com/books/mono/10.1201/9780429492563/nonlinear-dynamics-chaos-steven-strogatz ×1
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