Numerical characterization of nonlinear dynamical systems using parallel computing: The role of GPUs approach

Fazanaro, Filipe Ieda; Soriano, Diogo C.; Suyama, Ricardo; Madrid, Marconi Kolm; Oliveira, J. R. da C.; Muñoz, Ignácio Bravo; Attux, Romis

doi:10.1016/j.cnsns.2015.12.021

Cited by 16 publications

(4 citation statements)

References 75 publications

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“…. Corresponding numerical simulation diagrams were obtained by MATCONT package, XPPAUTO software, and parallel computing [63][64][65], including stability distribution diagrams, two-parameter bifurcation diagrams, and the maximum Lyapunov exponent diagrams, etc. The fourth-order Runge-Kutta algorithm with step size 0.01 and initial value fixed at (0.1, 0.1, 0.1, 0.1, 0.1) is applied in numerical calculation and process.…”

Section: Methodsmentioning

confidence: 99%

Dynamic Analysis and Synchronization with Unidirectional Coupling Involving Energy of a HR Neuron Under Electric Field and Memristive Autapse

Qiao

Gao

et al. 2021

Preprint

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Reliable neuron models play an important role in identifying the electrical activities, global bifurcation patterns, and dynamic mechanisms of neurons in complex electromagnetic environments. Considering the memristive autapse involving magnetic coupling has voltage-controlled, nonlinear, and memory, a 5-D HR neuron model containing magnetic field and electric field variables is established. Detailedly, the existence and stability conditions of the equilibrium point are determined by theoretical analysis, and the complex time-varying stability, saddle-node bifurcation, and Hopf bifurcation behaviors of the model are verified by numerical calculation. Interestingly, the system has a bistable structure consisting of quiescent state and period-1 and period-2 bursting modes near the subcritical Hopf bifurcation. It is noteworthy that the memristive autapse has a complex regulation mechanism for the bistable region so that three kinds of bistable coexisting structures and counterintuitive dynamic phenomena can be induced by appropriately adjusting the memristive autapse. Accordingly, the mechanism of positive feedback memristive autapse decreases its firing frequency, while negative feedback memristive autapse promotes its excitability was revealed by the fast-slow dynamic analysis. Extensive numerical results display that the system generally possesses period-adding bifurcation modes and comb-shaped chaotic structures. Furthermore, it is found that the firing modes and multistability regions of the system can be accurately predicted by analyzing the global dynamic behaviors of Hamilton energy. Importantly, it is verified that the unidirectional coupling controller involving energy is far more efficient and consumes less energy than electrical synaptic coupling in achieving complete synchronization with mismatched parameters.

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Section: Methodsmentioning

confidence: 99%

Dynamic Analysis and Synchronization with Unidirectional Coupling Involving Energy of a HR Neuron Under Electric Field and Memristive Autapse

Qiao

Gao

et al. 2021

Preprint

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“…This phenomenon is called thread divergence. This is why explicit algorithms produces much higher speed-up over CPUs for nonstiff problems (even by a factor of 100 [39,71]) compared to the implicit ones for stiff problems (usually less than a factor 10 [40]). This is a consequence of the much more simple control logic of explicit solvers causing much less hazard for thread divergence.…”

Section: The Integration Algorithmsmentioning

confidence: 99%

“…In this way, only the required special properties are transferred back to the host (main memory of the CPU) from the device (global memory of the GPU) after the end of a simulation instead of the whole trajectories. Although many other implementations (on GPUs) have been already published during the last years [65][66][67][68][69][70][71], according to the best knowledge of the author, such a general technique to handle large number of independent ODEs is still not available in the literature.…”

Section: Introductionmentioning

confidence: 99%

Modular, general purpose ODE integration package to solve large number of independent ODE systems on GPUs

Hegedűs¹

2018

Preprint

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A general purpose, modular program package for the integration of large number of independent ordinary differential equation systems capable of using professional graphics cards is presented. The available numerical schemes are the explicit and adaptive Runge-Kutta-Cash-Karp algorithm and the explicit fourth order Runge-Kutta method with fixed time step. In order to harness the huge processing power of graphics cards, the intermediate points of the computed trajectories are not stored. As a compensate, with pre-declared device functions, the required special features or properties of a solution can be easily extracted and stored each into a dedicated variable. For instance, the maximum and minimum values and/or their time instances. Event handling is also incorporated into the package in order to detect special points which can be stored as well. Moreover, again with pre-declared device function calls at such special points, the efficient handling of nonsmooth dynamics-e.g. impact dynamics-is possible. Several test cases are presented to demonstrate the flexibility of the pre-declared device functions and the strength of the program package. The applied models are the simple Duffing oscillator, the more complex Keller-Miksis equation known in bubble dynamics, and a system describing the behaviour of a pressure relief valve that can exhibit impact dynamics.

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“…Due to the aforementioned reasons, reimplementations of even the simplest Runge-Kutta solvers exist in the literature for many specialised problems: solving chemical kinetics [59][60][61][62], simulation in astrophysics [63,64], epidemiological model fitting [65,66] or non-linear dynamical analysis [67,68], to name a few. Some general-purpose solvers, e.g., ODEINT or DifferentialEquations.jl (written in Julia) offer the possibility to transfer the integration procedure to the GPU.…”

Section: Introductionmentioning

confidence: 99%

Solving large number of non-stiff, low-dimensional ordinary differential equation systems on GPUs and CPUs: performance comparisons of MPGOS, ODEINT and DifferentialEquations.jl

Nagy¹,

Plavecz²,

Hegedűs³

2020

Preprint

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In this paper, the performance characteristics of different solution techniques and program packages to solve a large number of independent ordinary differential equation systems is examined. The employed hardware are an Intel Core i7-4820K CPU with 30.4 GFLOPS peak double-precision performance per cores and an Nvidia GeForce Titan Black GPU that has a total of 1707 GFLOPS peak double-precision performance. The tested systems (Lorenz equation, Keller-Miksis equation and a pressure relief valve model) are non-stiff and have low dimension. Thus, the performance of the codes are not limited by memory bandwidth, and Runge-Kutta type solvers are efficient and suitable choices. The tested program packages are MPGOS written in C++ and specialised only for GPUs; ODEINT implemented in C++, which supports execution on both CPUs and GPUs; finally, DifferentialEquations.jl written in Julia that also supports execution on both CPUs and GPUs. Using GPUs, the program package MPGOS is superior. For CPU computations, the ODEINT program package has the best performance.

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Numerical characterization of nonlinear dynamical systems using parallel computing: The role of GPUs approach

Cited by 16 publications

References 75 publications

Dynamic Analysis and Synchronization with Unidirectional Coupling Involving Energy of a HR Neuron Under Electric Field and Memristive Autapse

Dynamic Analysis and Synchronization with Unidirectional Coupling Involving Energy of a HR Neuron Under Electric Field and Memristive Autapse

Modular, general purpose ODE integration package to solve large number of independent ODE systems on GPUs

Solving large number of non-stiff, low-dimensional ordinary differential equation systems on GPUs and CPUs: performance comparisons of MPGOS, ODEINT and DifferentialEquations.jl

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