The question, however, was a pressing one, whether the Evangelical elements thus taken up could consist with the Arminian principle. Calvinists earnestly urged that the union was an unnatural one, and could not be stable: that either the Evangelical elements ought to rule to the exclusion of the unharmonizable Arminian principle, in which case we should have consistent Calvinism; or else the Arminian principle would inevitably rule to the exclusion of the Evangelical doctrines forced into artificial conjunction with it, and we should have consistent Arminianism.
[B.B. Warfield, Selected Shorter Writings of Benjamin B. Warfield II, ed. John E. Meeter (Phillipsburg, N.J., USA: Presbyterian and Reformed, 1973), p. 315]
They [the Methodists] call themselves Arminians; but it is perfectly obvious that their theology differs widely from that of Limborch, and Whitby, and Warburton, and all the recognized Arminian divines of Holland and England ... They differ widely and radically in principles and in results; whereas when we hear the gospel preached by a Methodist, we feel that it is the very same to which we love to listen, and are accustomed to hear as Presbyterians. ... Man's ruin by the fall, his native depravity and alienation from God, his absolute need of a Saviour, and utter inability to save himself, the necessity of regeneration by the Holy Spirit, justification, not by works, but by faith alone in the blood and righteousness of Jesus, the free offer of the gospel to every human being without money and without price, the necessity of holiness, not to merit heaven, but to become meet for it — these articles constituted the very burden of their preaching.
[Review of Annals of the American Pulpit (Methodist), in the British and Foreign Evangelical Review, vol. xi (London, UK: Nisbet, 1862), pp. 301-2. As cited in Iain H. Murray, The Old Evangelicalism: Old Truths for a New Awakening (Carlisle, PA, USA: Banner of Truth Trust, 2005), p. 156]
Evangelical Arminianism, Olsen's so-called "Arminianism of the heart", is the "Arminianism" founded mainly by Methodist founder John Wesley. A revivalist following up on the First Great Awakening (the so-called Methodist Awakening), John Wesley was an Anglican who focused much on preaching the Gospel to all. While his forerunner George Whitefield was a Calvinist, Wesley due to certain burdens was unable to accept Calvinism even though many of his compatriots did (ie Richard Hutchins and James Hervey, former members of his Holy Club, a few of his own itinerants like John Bennet, John Whitford and John Edwards, and of course the famous John Newton, author of the hymn "Amazing Grace") .
While Wesley was convinced of Arminianism, and his opposition to it stiffened rather than weakened in light of Calvinistic influence , several points can be made to show the differences between Wesley and his Evangelical Arminianism, and the Classical Armininism of former times.
The first point can be found in the fact that Wesley was not a proper theologian. His bifurcation on the topic of full sanctification (embracing then modifying it) shows us that "his beliefs in their totality made up a loose synthesis, an amalgam, rather than a coherent system. Taken as a whole there was an absence of consistency" . The Modern Arminian Roger Olsen similar states that "Wesley was not a systematic thinker" , citing his horrible contradictions in his sermons preaching on the topic of justification as an example that Wesley's "theology often was ad-hoc — determined by perceived errors and imbalances that needed correcting" .
Pursuant to this point is Wesley's historical context, whereby Wesley stubbornly refuse to separate the doctrine of Calvinism from the dying Dissenter congregations they were found. Over time, a form of Hyper-Calvinism and Antinomianism had crept into the Dissenting (formerly Puritan) congregations, and Wesley stubbornly refuse to delink them, thinking for instance that Antinomianism "was a direct consequence of Calvinistic belief" . Such strawman beliefs held by Wesley held him unable to see much good in Calvinism, which together with his inconsistency, caused him to embrace the national "religion" on this subject - Synergism.
The second point we must note was that Wesley was a revivalist, and revivalists focus on the Gospel. As noted by the British and Foreign Evangelical Review, the Gospel proclaimed by Wesley and the Methodist movement was essentially orthodox and no different from that found in for example Presbyterianism, unlike the "gospel" of the Classical Arminians. Keeping the Gospel at the center yielded a benefit for the Evangelical Arminian movement which distinguished it sharply from Classical Arminianism: the presence of the Gospel, with its doctrine of the sinfulness of Man as opposed to the partial sinfulness of Man in Classical Arminianism.
Putting both points together, we can see that Evangelical Arminianism with its focus on the Gospel is thus orthodox. Wesley's inconsistency would ironically save Evangelical Arminianism from heresy. Right from the start however, the forced fusion of Evangelicalism and Arminianism in Wesleyan thought created logical, philosophical and theological contradictions of all kinds.
As B.B. Warfied has said, the Evangelical and Arminian elements in Evangelical Arminianism war against each other. To the extent that Evangelical Arminians are irrational and Gospel-centered, to the same extent they would remain evangelical. To the extent however that they are logically consistent, they ironically lose their faith, or be Calvinists.
The logical irrationality inherent in Evangelical Arminianism also extends to their doctrine of sin. Being Evangelical, a form of Total Depravity was required to be embraced. At least in their Gospel presentation, men must be presented as being totally depraved and unable to save themselves. The Classical Arminian teaching of Partial Depravity with their denial of Original Guilt can have no place in the Evangelical Arminian Gospel. Similarly, the orthodox doctrine of Justification by Faith alone with the teaching of double imputation is believed and taught in Evangelical Arminianism, while such is impossible in the context of Classical Arminianism. Thus, while Classical Arminians reject all five points of the Calvinist TULIP, Evangelical Arminianism embrace the first point however inconsistently— as long as they remain Evangelical that is.
Being an Evangelical Arminian and living with its logical inconsistencies is indeed unstable, as what Warfield himself has said with regards to the situation. Wesley himself was led into two errors — viz weakening the finality of justification and embracing some form of Christian perfection . The Modern Arminian Roger Olsen evidently has little problems with the heresy of Open Theism , a sober reminder to us of the tenacious nature of the "Evangelical" in Evangelical Arminianism indeed.
In conclusion therefore, Evangelical Arminianism is indeed orthodox, though at a very steep cost in terms of logical and theological inconsistencies. Due to its instability, Evangelical Arminianism tend to settle into either of two trajectories: towards Calvinism or some form of Liberalism (or it could remain irrationally evangelical). While we would gladly call Evangelical Arminians our brothers and sisters in Christ, we know that they are constantly on a precipice near error, and we should strive to bring them closer to the truth of Scripture, instead of toying with teachings that come, as the Canons of Dordt put it, from the "pits of hell".
 Iain H. Murray, Wesley and Men Who Followed (Carlisle, PA, ISA: Banner of Truth Trust, 2003), p. 69
 Ibid., p. 68
 Ibid., p. 76
 Roger E. Olsen, Arminian Thelogy: Myths and Realities (Downers Grove, IL, USA: IVP Academic, 2006), p. 213.
 Murray, p. 65
 Ibid., p. 66
 Olsen, pp. 198-199