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Left: Approximate spectrum of monodromy operator for the linearized predator-prey model (98)-(99) with parameters from Theorem 8.1.1, τ = 3 and h = 0.060, with a unit circle for scale and visual instability reference. Right: zoomed in portion of the region in the dashed box showing the unstable Floquet multiplier (black dot).
Source publication
We develop validated numerical methods for the computation of Floquet multipliers of equilibria
and periodic solutions of delay differential equations, as well as impulsive delay differential equations.
Using our methods, one can rigorously count the number of Floquet multipliers outside a closed disc
centered at zero or the number of multipliers c...
Contexts in source publication
Context 1
... results of the computer-assisted proof are tabulated in Table 1 and the approximate spectrum for h = 0.060 and τ = 3 is plotted in Figure 3. ...
Citations
... The impulsive differential equations have been widely studied and have seen significant progress in recent years by many authors [10][11][12][13]. It has found widespread application in a variety of fields, including biological technology, medicine dynamics, physics, economy, population dynamics, and epidemiology [14]. ...
This paper presents an exploitation model with a stage structure to analyze the dynamics of a fish population, where periodic birth pulse and pulse fishing occur at different fixed time. By utilizing the stroboscopic map, we can obtain an accurate cycle of the system and investigate the stability thresholds. Through the application of the center manifold theorem and bifurcation theory, our research has shown that the given model exhibits transcritical and flip bifurcation near its interior equilibrium point. The bifurcation diagrams, maximum Lyapunov exponents and phase portraits are presented to further substantiate the complexity. Finally, we present high-resolution stability diagrams that demonstrate the global structure of mode-locking oscillations. We also describe how these oscillations are interconnected and how their complexity unfolds as control parameters vary. The two parametric planes illustrate that the structure of Arnold’s tongues is based on the Stern–Brocot tree. This implies that the periodic occurrence of birth pulse and pulse fishing contributes to the development of more complex dynamical behaviors within the fish population.
... Bifurcation theory for periodic solutions of impulsive functional differential equations has undergone some new developments in recent years [5,7], but such results are only useful if one has computed a periodic solution to begin with. This is a main motivation for considering here the problem of computation and continuation of periodic solutions for impulsive delay differential equations. ...
... Methods for computer-assisted proofs of stability for linear impulsive delay differential equations has recently been accomplished [7] using Chebyshev spectral collocation techniques, and the approach we take here shares some similarities. The idea is as follows. ...
... Converting problems in nonlinear dynamics into zero-finding problems in sequence spaces is not a new idea. In the context of sequence spaces representing Chebyshev series coefficients, see [2,7,15,20] for a few recent applications. One solitary application of Chebyshev expansions in nonlinear impulsive dynamical systems we could find appears in [24], where it was used to generate a simplified approximation of an optimal control problem involving impulsive integrodifferential equations. ...
We develop a rigorous numerical method for periodic solutions of impulsive delay differential equations, as well as parameterized branches of periodic solutions. We are able to compute approximate periodic solutions to high precision and with computer-assisted proof, verify that these approximate solutions are close to true solutions with explicitly computable error bounds. As an application, we prove the existence of a global branch of periodic solutions in the pulse-harvested Hutchinson equation, connecting the state at carrying capacity in the absence of harvesting to the transcritical bifurcation at the extinction steady state.
... Bifurcation theory for periodic solutions of impulsive functional differential equations has undergone some new developments in recent years [5,7], but such results are only useful if one has computed a periodic solution to begin with. This is a main motivation for considering here the problem of computation and continuation of periodic solutions for impulsive delay differential equations. ...
... Methods for computer-assisted proofs of stability for linear impulsive delay differential equations has recently been accomplished [7] using Chebyshev spectral collocation techniques, and the approach we take here shares some similarities. The idea is as follows. ...
... Converting problems in nonlinear dynamics into zero-finding problems in sequence spaces is not a new idea. In the context of sequence spaces representing Chebyshev series coefficients, see [2,7,15,20] for a few recent applications. One solitary application of Chebyshev expansions in nonlinear impulsive dynamical systems we could find appears in [24], where it was used to generate a simplified approximation of an optimal control problem involving impulsive integrodifferential equations. ...
We develop a rigorous numerical method for periodic solutions of impulsive delay differential equations, as well as parameterized branches of periodic solutions. We are able to compute approximate periodic solutions to high precision and with computer-assisted proof, verify that these approximate solutions are close to true solutions with explicitly computable error bounds. As an application, we prove the existence of a global branch of periodic solutions in the pulse-harvested Hutchinson equation, connecting the state at carrying capacity in the absence of harvesting to the transcritical bifurcation at the extinction steady state.
... To accompany this publication we have implemented the numerical discretization of the monodromy operator in MATLAB. It can be found at the author's GitHub [9]. The implementation can handle arbitrary period p and delay q, provided these are integers. ...
... The code is general-purpose, taking as input the matrix-valued functions A(t) and B(t) appropriately pre-processed, impulse matrices C1 and C2 and various other user-specified data. The code can be found at [9]. INTLAB is required for rigorous proof, but is not required for monodromy operator discretization, so the latter is suitable for eigenvalue (Floquet multiplier) estimation. ...
... The number of points in this mesh has a significant impact on computation time, and this number is generally different for each proof. See the associated MATLAB code [9] for documentation. ...
We develop validated numerical methods for the computation of Floquet multipliers of equilibria and periodic solutions of delay differential equations, as well as impulsive delay differential equations. Using our methods, one can rigorously count the number of Floquet multipliers outside a closed disc centered at zero or the number of multipliers contained in a compact set bounded away from zero. We consider systems with a single delay where the period is at most equal to the delay, and the latter two are commensurate. We first represent the monodromy operator (period map) as an operator acting on a product of sequence spaces that represent the Chebyshev coefficients of the state-space vectors. Truncation of the number of modes yields the numerical method, and by carefully bounding the truncation error in addition to some other technical operator norms, this leads to the method being suitable to computer-assisted proofs of Floquet multiplier location. We demonstrate the computer-assisted proofs on two example problems. We also test our discretization scheme in floating point arithmetic on a gamut of randomly-generated high-dimensional examples with both periodic and constant coefficients to inspect the precision of the spectral radius estimation of the monodromy operator (i.e. stability/instability check for periodic systems) for increasing numbers of Chebyshev modes.
This article investigates the exponential stability of nonlinear discrete‐time systems with time‐varying state delay and delayed impulses, where the delays in impulses are not fixed. Specifically, the study is separated into two cases: (1) stability of delayed discrete‐time systems with destabilizing delayed impulses, where the time delays in impulses can be flexible and even larger than the length of the impulsive interval; (2) stability of delayed discrete‐time systems with mixed delayed impulses (means stabilizing and destabilizing delayed impulses exist simultaneously), where the time delays in impulses are unfixed between two adjacent impulsive instants. First, the concept of average impulsive delay (AID) is extended to discrete‐time systems with delayed impulses. Then, some Lyapunov‐based exponential stability criteria are provided for delayed discrete‐time systems with delayed impulses, where the impulses satisfy the average impulsive interval (AII) condition and the delays in impulses satisfy the AID condition. For the delayed discrete‐time systems with mixed delayed impulses, the exponential stability criterion is provided by limiting the geometric average of the impulses' strengths. The obtained results can be used to study the delayed discrete‐time systems with some large impulses which may even be unbounded. It's also shown that the delays in impulses can have a stabilizing effect on the stability of delayed discrete‐time systems. Some numerical examples are also provided to illustrate the effectiveness and advantage of the theoretical results.
We consider nonlinear impulsive systems on Banach spaces subjected to disturbances and look for dwell-time conditions guaranteeing the ISS property. In contrary to many existing results our conditions cover the case where both continuous and discrete dynamics can be unstable simultaneously. Lyapunov type methods are used for this purpose. The effectiveness of our approach is illustrated on a rather nontrivial example, which is feedback connection of an ODE and a PDE system.
In this chapter, we develop theoretical and computational aspects of Floquet theory for periodic linear systems.
