Nonlinear Dynamics and Chaos¶
Strogatz, S. H. (2015). Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering. Westview Press.
Cited by¶
17 citations across 17 artifacts.
Each citation links to the sentence it supports in the citing article.
Primes¶
- Attractor Selection and Basin Control
- The structural mechanism by which a system's long-term dynamics are directed toward one of multiple possible stable states (attractors) through manipulation of initial conditions, boundary conditions, or control inputs that shift the basins of attraction, as developed canonically by Strogatz (2014) in his treatment of nonlinear dynamics.
This sourceCanonical introduction to nonlinear dynamics; develops multiple coexisting stable equilibria (attractors), their basins of attraction, and how varying parameters reshapes the phase portrait — the dynamical-systems vocabulary the prime builds on. SUPPORTS marker 046 (note: prior annotation text was copied from another prime and described caching/parallelization; corrected here).
Supported in partVerified against the publisher's abstract
The catalog summary shows Strogatz covers attractors, phase-plane analysis and bifurcations, but says nothing of directing a system to a chosen attractor via initial conditions or control inputs.
“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.”
- The structural mechanism by which a system's long-term dynamics are directed toward one of multiple possible stable states (attractors) through manipulation of initial conditions, boundary conditions, or control inputs that shift the basins of attraction, as developed canonically by Strogatz (2014) in his treatment of nonlinear dynamics.
- Critical Juncture
- Many systems have both: a firm at a critical juncture (merge or remain independent?) may, having chosen to merge, then approach a tipping point toward integration or fragmentation, a distinction Strogatz (2014) develops formally in his treatment of bifurcations and threshold crossings in nonlinear systems.
This sourceStandard text developing bifurcations and threshold-crossing dynamics in nonlinear systems; supplies the dynamical-systems vocabulary distinguishing a continuous-variable tipping point/phase transition from a discrete-choice critical juncture.
- Many systems have both: a firm at a critical juncture (merge or remain independent?) may, having chosen to merge, then approach a tipping point toward integration or fragmentation, a distinction Strogatz (2014) develops formally in his treatment of bifurcations and threshold crossings in nonlinear systems.
- Cycle
- Systems and control. Every feedback loop is a cycle in the influence graph; a positive cycle amplifies perturbations and a negative cycle counters them, and the number and sign of cycles in a system's Jacobian govern its qualitative stability.
This sourceTreats feedback loops as cycles in the influence graph, with positive cycles amplifying and negative cycles stabilizing, governing qualitative stability.
- Systems and control. Every feedback loop is a cycle in the influence graph; a positive cycle amplifies perturbations and a negative cycle counters them, and the number and sign of cycles in a system's Jacobian govern its qualitative stability.
- Eigenvalue And Eigenvector
- In dynamical systems and control, the eigenvalues of a Jacobian at a fixed point classify stable, unstable, and oscillatory modes, and the stability boundary is a spectral condition.
This sourceClassifies fixed-point stability via the eigenvalues of the Jacobian and treats the stability boundary as a spectral condition.
- In dynamical systems and control, the eigenvalues of a Jacobian at a fixed point classify stable, unstable, and oscillatory modes, and the stability boundary is a spectral condition.
- Multistability
- Path Dependence
- Tools like bifurcation analysis (which branch is selected at a critical juncture?), scenario mapping (which alternative histories are plausible?), and contingency narratives (what chain of events led here?) transfer across domains, drawing on the dynamical-systems vocabulary of basins of attraction and bifurcation that Strogatz (2014) develops in canonical form.
This sourceStandard text on nonlinear coupling and superposition failure; provides the dynamical-systems vocabulary for understanding why combined-resource systems (caching plus parallelization, coupled oscillators) produce joint behavior that diverges from component-wise prediction.
- Tools like bifurcation analysis (which branch is selected at a critical juncture?), scenario mapping (which alternative histories are plausible?), and contingency narratives (what chain of events led here?) transfer across domains, drawing on the dynamical-systems vocabulary of basins of attraction and bifurcation that Strogatz (2014) develops in canonical form.
- Potentiation
- In neuroscience education and research, the potentiation construct directly supports hypothesis formation and mechanistic investigation, paralleling how Strogatz (2014) describes positive-feedback amplification as a structurally diagnostic regime in nonlinear dynamics.
This sourceStandard text on nonlinear coupling and superposition failure; provides the dynamical-systems vocabulary for understanding why combined-resource systems (caching plus parallelization, coupled oscillators) produce joint behavior that diverges from component-wise prediction.
- In neuroscience education and research, the potentiation construct directly supports hypothesis formation and mechanistic investigation, paralleling how Strogatz (2014) describes positive-feedback amplification as a structurally diagnostic regime in nonlinear dynamics.
- Recurrence
- Recurrence is the structural property by which a pattern, event, condition, or value reappears across time, iterations, or instances, often with predictable spacing or in response to identifiable triggers, a structural notion Strogatz (2014) develops as foundational to nonlinear dynamics.
This sourceStandard text on nonlinear coupling and superposition failure; provides the dynamical-systems vocabulary for understanding why combined-resource systems (caching plus parallelization, coupled oscillators) produce joint behavior that diverges from component-wise prediction.
- Recurrence is the structural property by which a pattern, event, condition, or value reappears across time, iterations, or instances, often with predictable spacing or in response to identifiable triggers, a structural notion Strogatz (2014) develops as foundational to nonlinear dynamics.
- Regime Change
- Second, threshold or tipping point: the transition occurs abruptly once some control parameter crosses a critical value, often with hysteresis (the forward and backward thresholds differ). Third, attractor switching: the basin of attraction changes; trajectories that once converged toward the old regime now converge toward the new one, a dynamical-systems framing developed by Strogatz (2015) in his canonical introduction to nonlinear dynamics.
This sourceStandard treatment of the structural prerequisites for nonlinear, multi-scale chaotic-coherent dynamics—nonlinearity, sufficient degrees of freedom, persistent driving away from equilibrium—and the boundary conditions under which such dynamics do not arise (purely linear, fully equilibrated, or low-dimensional systems).
- Second, threshold or tipping point: the transition occurs abruptly once some control parameter crosses a critical value, often with hysteresis (the forward and backward thresholds differ). Third, attractor switching: the basin of attraction changes; trajectories that once converged toward the old regime now converge toward the new one, a dynamical-systems framing developed by Strogatz (2015) in his canonical introduction to nonlinear dynamics.
- Reversibility and Irreversibility
- A system can be stable (resistant to perturbations) yet reversible (actions are undoable) or unstable (sensitive to perturbations) yet irreversible (actions are binding)—a separation Strogatz (2015) develops formally in his treatment of fixed points, stability, and bifurcation in nonlinear dynamics.
This sourceStandard treatment of the structural prerequisites for nonlinear, multi-scale chaotic-coherent dynamics—nonlinearity, sufficient degrees of freedom, persistent driving away from equilibrium—and the boundary conditions under which such dynamics do not arise (purely linear, fully equilibrated, or low-dimensional systems).
- A system can be stable (resistant to perturbations) yet reversible (actions are undoable) or unstable (sensitive to perturbations) yet irreversible (actions are binding)—a separation Strogatz (2015) develops formally in his treatment of fixed points, stability, and bifurcation in nonlinear dynamics.
- Saddle Point
- The linear system \(\dot{x} = x\), \(\dot{y} = -y\) is the saddle point stripped to its skeleton,
This sourceStandard text presenting the saddle equilibrium, its mixed-sign eigenvalues, and stable/unstable manifolds (incl. the canonical x' = x, y' = -y example).
- The linear system \(\dot{x} = x\), \(\dot{y} = -y\) is the saddle point stripped to its skeleton,
- Self-Organized Criticality
- It applies to systems with the structural prerequisites Strogatz (2015) identifies for nonlinear, multi-scale dynamical organization—nonlinearity, sufficient degrees of freedom, and persistent driving—rather than to monoscale or purely linear/stochastic systems:
This sourceStandard treatment of the structural prerequisites for nonlinear, multi-scale chaotic-coherent dynamics—nonlinearity, sufficient degrees of freedom, persistent driving away from equilibrium—and the boundary conditions under which such dynamics do not arise (purely linear, fully equilibrated, or low-dimensional systems).
- It applies to systems with the structural prerequisites Strogatz (2015) identifies for nonlinear, multi-scale dynamical organization—nonlinearity, sufficient degrees of freedom, and persistent driving—rather than to monoscale or purely linear/stochastic systems:
- Stochasticity vs. Determinism
- Stochastic systems yield probability distributions, variance analysis, and expected-value reasoning—a partition Strogatz (2014) develops in his canonical text on nonlinear dynamics and chaos.
This sourceStandard text on nonlinear coupling and superposition failure; provides the dynamical-systems vocabulary for understanding why combined-resource systems (caching plus parallelization, coupled oscillators) produce joint behavior that diverges from component-wise prediction.
- Stochastic systems yield probability distributions, variance analysis, and expected-value reasoning—a partition Strogatz (2014) develops in his canonical text on nonlinear dynamics and chaos.
- Synergy and Antagonism
- Mechanism attribution (complementary data-flow paths vs shared resources) is informed by performance benchmarking and tracing, drawing on the nonlinear-coupling intuitions Strogatz (2014) develops for systems whose joint behavior is not the sum of component dynamics.
This sourceStandard text on nonlinear coupling and superposition failure; provides the dynamical-systems vocabulary for understanding why combined-resource systems (caching plus parallelization, coupled oscillators) produce joint behavior that diverges from component-wise prediction.
- Mechanism attribution (complementary data-flow paths vs shared resources) is informed by performance benchmarking and tracing, drawing on the nonlinear-coupling intuitions Strogatz (2014) develops for systems whose joint behavior is not the sum of component dynamics.
- Temporal Dynamics
- The when and Order of actions or conditions often matter as much as the actions themselves.
This sourceStandard text on nonlinear coupling and superposition failure; provides the dynamical-systems vocabulary for understanding why combined-resource systems (caching plus parallelization, coupled oscillators) produce joint behavior that diverges from component-wise prediction.
- The when and Order of actions or conditions often matter as much as the actions themselves.
- Threshold Bounded Vicious Cycle
- Model a system whose resource stock \(x\) evolves as \(\dot{x} = f(x) + u\), where \(u\) is an external intervention rate and \(f(x)\) has the cubic-like shape \(f(x) = -x(x - a)(x - b)\) with $0 < a < b$.
This sourceStandard reference for bistable systems, saddle-node structure, basins of attraction, and unstable separatrices — the cubic-vector-field model of a two-attractor trap with a threshold.
- Model a system whose resource stock \(x\) evolves as \(\dot{x} = f(x) + u\), where \(u\) is an external intervention rate and \(f(x)\) has the cubic-like shape \(f(x) = -x(x - a)(x - b)\) with $0 < a < b$.
Mechanisms¶
- Modal Stability Analysis
- Linearised stability is silent about large disturbances and global behaviour — a locally stable system can still be knocked clean out of its basin of attraction, and marginal modes are honestly undecidable at linear order.
This sourceDistinguishes local linear stability from global behavior and shows that nonhyperbolic or marginal cases require nonlinear analysis beyond first order.
- Linearised stability is silent about large disturbances and global behaviour — a locally stable system can still be knocked clean out of its basin of attraction, and marginal modes are honestly undecidable at linear order.
Verification¶
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Links previously used in the corpus¶
Before the registry existed this work was also linked 4 other ways.
- https://iucat.iu.edu/catalog/16142878 ×1
- https://www.google.com/books/edition/Nonlinear_Dynamics_and_Chaos/A8TVDAAAQBAJ ×1
- https://www.routledge.com/Nonlinear-Dynamics-and-Chaos-With-Applications-to-Physics-Biology-Chemistry-and-Engineering/Strogatz/p/book/9780813349107 ×1
- https://www.taylorfrancis.com/books/mono/10.1201/9780429492563 ×1
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