The Motion Constants of Linear Autonomous Dynamical Systems

2013 ◽  
Vol 65 (4) ◽  
Author(s):  
Andrew J. Sinclair ◽  
John E. Hurtado
Keyword(s):  
Dynamical Systems ◽  
Canonical Form ◽  
Frequency Ratio ◽  
Autonomous Systems ◽  
State Transformation ◽  
Close Inspection ◽  
Ergodic Behavior ◽  
Single Exception ◽  
Comprehensive Development ◽  
Autonomous Dynamical Systems

The time-independent integrals, here referred to as motion constants, for general nth-order linear autonomous systems are developed. Although it is commonly believed that this topic has been fully addressed, close inspection of the literature reveals that a comprehensive development is missing. This paper provides a complete tutorial treatment of the calculation of these motion constants. The process involves a state transformation to a canonical form of uncoupled real subsystems. Following this, motion constants that are internal to each subsystem are found, after which motion constants that connect the subsystems to each other are computed. Complete sets of n − 1 real single-valued motion constants can be formed for all linear autonomous systems with a single exception. The exception is systems composed of undamped oscillators whose frequency ratio is irrational. Such systems are known to exhibit ergodic behavior and lack a number of analytic motion constants.

scholarly journals Physical invariant measures and tipping probabilities for chaotic attractors of asymptotically autonomous systems

2021 ◽  
Author(s):  
Peter Ashwin ◽  
Julian Newman
Keyword(s):  
Dynamical Systems ◽  
Invariant Measures ◽  
Autonomous Systems ◽  
Chaotic Attractors ◽  
Physical Measure ◽  
Multiple Attractors ◽  
The Past ◽  
Rate Dependent ◽  
Typical Behaviour ◽  
Autonomous Dynamical Systems

AbstractPhysical measures are invariant measures that characterise “typical” behaviour of trajectories started in the basin of chaotic attractors for autonomous dynamical systems. In this paper, we make some steps towards extending this notion to more general nonautonomous (time-dependent) dynamical systems. There are barriers to doing this in general in a physically meaningful way, but for systems that have autonomous limits, one can define a physical measure in relation to the physical measure in the past limit. We use this to understand cases where rate-dependent tipping between chaotic attractors can be quantified in terms of “tipping probabilities”. We demonstrate this for two examples of perturbed systems with multiple attractors undergoing a parameter shift. The first is a double-scroll system of Chua et al., and the second is a Stommel model forced by Lorenz chaos.

scholarly journals The Planar Inverse Problem for Autonomous Systems

1983 ◽  
Vol 74 ◽  
pp. 353-367 ◽  
Author(s):  
Basilis C. Xanthopoulos ◽  
George Bozis
Keyword(s):  
Inverse Problem ◽  
Dynamical Systems ◽  
Autonomous Systems ◽  
Sufficient Conditions ◽  
Parametric Family ◽  
General Version ◽  
Conic Sections ◽  
Family Of Curves ◽  
Necessary And Sufficient ◽  
Autonomous Dynamical Systems

AbstractWe study the general version of the inverse problem for planar trajectories and for autonomous dynamical systems possessing three integrals, i.e., for a given three-parametric family of curves f(x,y,a,b)=c we find the potential V(x,y) for which these curves are orbits of a unit mass. All possible cases, depending on the preassigned function f, are classified and in each case the necessary and sufficient conditions for the.existence of a solution are established. Among the examples is the case of the Keplerian conic sections which is studied in detail.

On the Roughness of Quasinilpotency Property of One-parameter Semigroups

2017 ◽  
Vol 60 (2) ◽  
pp. 364-371 ◽  
Author(s):  
Ciprian Preda
Keyword(s):  
Dynamical Systems ◽  
Systems Theory ◽  
Infinitesimal Generator ◽  
Autonomous Systems ◽  
Dynamical Systems Theory ◽  
Extinction Property ◽  
Quasinilpotent Operators ◽  
Strong Concept ◽  
Time Extinction

AbstractLet S := {S(t)}t≥0 be a C0-semigroup of quasinilpotent operators (i.e., σ(S(t)) = {0} for eacht> 0). In dynamical systems theory the above quasinilpotency property is equivalent to a very strong concept of stability for the solutions of autonomous systems. This concept is frequently called superstability and weakens the classical ûnite time extinction property (roughly speaking, disappearing solutions). We show that under some assumptions, the quasinilpotency, or equivalently, the superstability property of a C0-semigroup is preserved under the perturbations of its infinitesimal generator.

scholarly journals State space decomposition for non-autonomous dynamical systems

2011 ◽  
Vol 141 (5) ◽  
pp. 957-974 ◽  
Author(s):  
Xiaopeng Chen ◽  
Jinqiao Duan
Keyword(s):  
Dynamical Systems ◽  
Differential Equations ◽  
State Space ◽  
Decomposition Theorem ◽  
Navier Stokes ◽  
Locally Compact ◽  
Infinite Dimensional ◽  
The Lorenz System ◽  
A Chain ◽  
Autonomous Dynamical Systems

The decomposition of state spaces into dynamically different components is helpful for understanding dynamics of complex systems. A Conley-type decomposition theorem is proved for non-autonomous dynamical systems defined on a non-compact but separable state space. Specifically, the state space can be decomposed into a chain-recurrent part and a gradient-like part. This result applies to both non-autonomous ordinary differential equations on a Euclidean space (which is only locally compact), and to non-autonomous partial differential equations on an infinite-dimensional function space (which is not even locally compact). This decomposition result is demonstrated by discussing a few concrete examples, such as the Lorenz system and the Navier–Stokes system, under time-dependent forcing.

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